Cohomology of completions

NORTH-HOLLAND

MATHEMATICS STUDIES Notas de Matem6tica editor: Leopoldo Nachbin

Cohomology of Completions

SAUL LUBKIN

NORTH-HOLLAND

42

COHOMOLOGY OF COMPLETIONS

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NORTH-HOLLAND MATHEMATICS STUDIES Notas de Matematica (71) Editor: Leopoldo Nachbin Universidade Federal do Rio de Janeiro and University of Rochester

Cohomology of Completions

SAUL LUBKIN Department of Mathematics University of Rochester Rochester, N. Y. 14620, U.S.A.

1980

NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM. NEW YORK. OXFORD

42

Il

North-Holland Publishing Company. 1980

All rights reserved. No part a/this publication may be reproduced. stored in a retrieval system, or transmilled. in any form or by any means. electronic. mechanical. photocopying. recording or otherwise. without the prior permission a/the copyright owner.

ISBN: 0444860428

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Library or Congress Cataloging in Publieallon Data

Lublrln. Saul., 1939Cohomology of completions. (Notas de matem&tica ; 71) (Noith-Holland mathematics studies ; 42) Bibliography: p. 1. Complexes. Cochain. 2. Modul.es (Algebra) 3. Spectral sequences (Mathematics) I. Title. II. Series. QAl.N86 no. 71 cQAl69J 5l0s C5l2'.55J 80-19364 ISBN 0-444-86042-8

PRINTED IN THE NETHERLANDS

To Laure, my wife

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PREFACE

In this book, we study, primarily, the cohomology of the t-adic completion of cochain complexes of left A-modules, where A

is a ring and

tEA.

To this end, in the Introduction, we develop the necessary basic homological techniques that are used throughout this book. The text is also filled with many examples and applications to, e.g., algebraic geometry and algebraic topology. For example, using the deep, basic finiteness theorem, Chapter 5, Theorem 1, we prove the finite generation of lifted p-adic cohomology "using the Noetherian

O-algebra

A,

/\"

of a scheme proper over a

see [P.A.C.].

This connects up with

research done by the author over the past seventeen years on the Weil zeta function stemming from the original Weil Conjectures, see e.g.,

[W.C.], [C.A.W.], [P.P.W.C.], [B.W.V.], [P.C.T.],

[F.G.P.R.] and [P.A.C.]. Also, the general theory of Poincare duality is studied somewhat in Chapter 7, and this is used, in algebraic geometry, to prove Poincare duality for the lifted p-adic cohomology of complete non-singular algebraic varieties, see [P.A.C.]; as well as applications to the more traditional case, in algebraic topology, of the usual singular cohomology of compact, oriented topological manifolds, see the Examples of Chapter 7. And, in Chapter 8, we easily retrieve, as another application to the field of algebraic geometry, most of the well-known theorems on the finite generation of the cohomology of formal schemes proper over an affine, for finitely generated ideals. The book is written in such a way that a graduate student, vii

viii

Preface

with very little other than basic background, and having only the most rudimentary knowledge of homology, should be able to profit from various parts of it.

And, in part towards this

end, there is an extensive Introduction. The first chapter of the Introduction covers the general theory of abelian categories and may be skipped by those readers who are not interested in that subject.

The second chapter of

the Introduction contains what is perhaps the most thorough study ever made of spectral sequences.

Yet, it is so written

that is should be accessible to a beginner in the subject. However, many researchers, even those who are very advanced in the subject, who have to study or construct new spectral sequences, at any level, may be able to profit from the general study made. The greater part of this book is written at the level of left modules over a ring, rather

than for the more general

abelian categories, so as to make the material more accessible to the beginner.

However, for the readers interested in the

greater generality, the first chapter of the Introduction supplies a thorough initiation to the general theory of abelian categories, including how to work in abelian categories. The chapter on spectral sequences, Chapter 2 of the Introduction, is written at the abstract level of abelian categorieshowever, consistent with the general philosophy, it has been written in such a fashion that the beginner can mentally substitute the words "abelian group" for "object", and "the category

A of abelian groups" for "an abelian category

A", throughout

that chapter, if his only interest is that concrete case.

Preface

ix

(Although, Introduction, Chapter 1, contains all the necessary background material on abelian categories, and more, if desired.)

Description of some selected results; applications. In the Introduction, Chapter 1, the

no~ion

of abelian

category is thoroughly developed, and it is shown how to prove theorems in an abstract abelian category.

In section 2, the

familiar notions of subobject and quotient-object are studied, and in section 3 abelian categories are introduced formally. Then, the previously published paper, "Imbedding of Abelian Categories", chapter.

([I.A.C.)) is reproduced, as section 4 of the

The reader learns, in this section, how to work, in

many ways, as freely in a general abstract abelian category, as in the category of left modules over a ring.

Also, in sec-

tion 5 of this chapter, the notion of a sUbquotient of an object in an abelian category (or of an abelian group) is defined formally and developed (for the first time to the best of the author's knowledge).

It is believed that this will facilitate

the understanding of the definition of the "Eoo-term" of a spectral sequence,

as in the next chapter.

(Another applica-

tion to algebraic topology, of this material (not discussed in the text) is a better understanding of the higher order cohomology operations, as in the

Steenrod algebra.)

In Chapter 2 of the Introduction, what is probably the most extensive study to appear so far, is made of the theory of spectral sequences.

However, this material is suitable to be

read by a graduate student who has not had too extensive a background.

To develop the theory properly, the general notion

Preface

x

of graded abelian category, is introduced in section 3.

Fil-

tered objects, and their completions and co-completions, are (E.g., in section 6 it is shown that,

studied in section 6.

whenever the completion of the co-completion, and the co-completion of the completion, of a filtered object both make sense, then they are canonically isomorphic.)

And the abutment, the

partial abutments, see section 7, the Eoo-term, etc., are all studied for the spectral sequence of an exact couple (section 7), the spectral sequence of a filtered cochain complex (section B)and the spectral sequence of a double complex (section 10) •

In sections 2 and 5, the spectral sequence of an exact couple is studied.

Among other things, one obtains a short

exact sequence, the middle term of which is

and the first

Eoo'

and third terms of which depend entirely on the "top object"

v

of the exact couple and the endomorphism

0<-

Ker (t) n (t] [ divisible part of V)

<-

<-I

E 00

V/(t-torsion) t.(V/t-torsion)

t

I

of

<-

V:

0 •

In section 7, the two partial abutments of the spectral sequence of a conventional, bigraded exact couple are defined and studied.

These are the direct limit abutments

anc the inverse limit abutments

"Hn, n E 7..

'Hn, nE 7. ,

These are two sets

of "would-be" abutments, and are filtered objects.

The associ-

ated gradeds of each of these sets is a different subquotient of

The delicate question of whether or not these par-

tial abutments are perfect is studied (e.g., in section 7, and in section 9, Theorem 4 and Corollary 4.1).

It is equivalent

to ask whether or not the left and right defects are zero

Preface

xi

(section 7). A conventional bigraded exact couple induces three in-

creasing subobjects, The right defect is the subobject

of the and the left defect

(In the category of abelian groups, the right defect-but not in general the left defect- is always zero.) The associated graded of the direct limit abutment is canonically isomorphic to

and the associated graded of

the inverse limit abutment is canonically isomorphic to C~,q/B~,q.

For the spectral sequence of a filtered cochain complex, we also construct the "best possible hope" for an abutment for a filtered cochain complex, the integrated partial abutments Hn,

nE 'l'.

These can be thought of as "tying together" the

direct limit and inverse limit abutmenGof the exact couple of the filtered cochain complex- in the precise sense that, the associated graded of the integrated partial abutments are canonically isomorphic to

see section 8.

And the

integrated partial abutments can also be thought of as "tying together" the dual partial abutments of the filtered cochain complex (section 8, Theorem 10).

In fact, there are four, in

general non-isomorphic, exact couples induced by a filtered cochain complex, all having the same spectral sequence (but in general different sets of partial abutments), see section 8. In section 9, Corollary 4.1, it is also shown that if the spectral sequence is such that the cycles stabilize, then the left defect=O, so that if we are in the cat. of abo gps., then

Preface

xii

the integ. part. abts. are a true abutment.

If also the

spectral sequence is semi-stable (see section 9, Remark 3 after Proposition 5.1), then the integrated partial abutments are also complete (section 9, Proposition 5).

(And then similar

observations also hold for the set of partial abutments, and also

for the dual set of partial abutments, of the completion

of the co-completion of the filtered cochain complex, see near the end of section 9).

The spectral sequence of a double

complex is studied in section 10.

And also, in section 10,

Example 3, the Adams spectral sequence is studied from a spectral sequence-theoretic point of view, as an interesting example from algebraic topology of an important spectral sequence coming from an exact couple in which the inverse limit abutments act as an abutment (in the sense of section 7, Definition 1), rather than the more common direct limit abutments. It is shown by Examples, that most of the pathology not excluded by the theories of Intro. Chap.2, actually do occur. Spectral Sequences were first introduced by J. Leray. After the Introduction, we begin the main theme of the book.

One can begin the book at this point, if one wishes,

looking back when necessary to the Introduction. Let

A

be a ring (not necessarily commutative) with

identi ty, and let Let

C*

t

~

A.

be any cochain complex of left A-modules, in-

dexed by all the integers.

all integers

n,

Then we have the cohomology groups,

which are left A-modules.

We also have the

Preface

xiii

groups

and also

all integers

n, i,

with

i > O.

We study the connections be-

tween these groups. In Chapter 1, we study the Bockstein spectral sequence. In particular, as a special case of Introduction, Chapter 2, section 5, we obtain a short exact sequence, the middle term and the first and last terms of which depend

of which is entirely on

Hn(C*)

and on

ter 1, Theorem 2).

Hn+l(C*)

respectively.

No hypotheses are required.

(Chap-

This essen-

tially determines the Ew-term of the Bockstein spectral sequence in the fullest generality. (It was, essentially, in the making of a study of these results, that I was led to the Introduction, Chapter 2.) In Chapter 2, we study an exact sequence of six terms, interconnecting n,i,

with

Hn(C*)

i > O.

for all integers

I introduced this sequence over thirteen

years ago in [P.P.W.C.], where it appeared as (1.8) (and the actual research had been done earlier, at the University of Oxford, in 1963-64).

We strengthen and improve slightly on

these previous results. In Chapter 3, a well known short exact sequence is noted, in the case that

n.

C

n

is t-adically complete for all integers

Preface

xiv

n C

In Chapter 4, we assume that tegers

n,

is complete for all in-

and use the results established in Chapters 1,2

3 to obtain some new results. Chapter 4.

and

Most striking is Theorem 6 of

See also Proposition 5, and Corollary 6.1 of Chap-

ter 4. In Chapter 5, we combine the results of Chapter 1 (especially Proposition 6) with those of Chapter 4 (especially Proposition 2 - a corollary of Theorem 1, which is a special case of Theorem 6) to obtain the theorem (Theorem 1) that gives the chapter its heading.

We state it here.

Chapter 5, Theorem 1. let A

t~

A

Let

A

be an element in the center of

is t-adically complete.

Let

C*

complex of left A-modules such that for all integers by

t":Ci+c

ring

A

be a ring with identity and

i

i,

A.

~-indexed)

be a c

i

Suppose that cochain

is t-adically complete

and that the endomorphism "multiplication

is injective, all integers

i.

Suppose that the

is left Noetherian (i.e., that all left ideals are

finitely generated.

This assumption can be deleted if we com-

plicate the statement of the Theorem somewhat- see the Remarks following Theorem 1 of Chapter 5). teger.

Let

n

be any fixed in-

Then if

Hn(C*/tC*)

is finitely generated as left (A/tA)-module,

for the fixed integer

n,

then Hn(C*)

is finitely generated as left A-module.

This Theorem is, perhaps, a somewhat surprising result.

Preface Can it be generalized? "multiplication by be eliminated?

xv

For example, can the hypothesis that

t:C

i

-+C

i

is injective, for all integers

. "

1,

Chapter 5, Theorem 4 answers this in the affir-

mative, if one assumes in addition that for the fixed integer

n, is finitely generated as left (A/tA)-module, where

0*

is the subcochain complex of

C*

such that

Also in Chapter 5, we introduce a new concept, "the percohomology groups of a cochain complex of left A-modules with coefficients in a right A-module,

M,"

which coincides with

is bdd. above and is flat

Hi (M

@

C*) if

C*

Hi (M,C*),

C* i E 'J'

A

over

A,

for all integers

invariant.

i,

but which in general is a new

Then, in the above theorem,

5), if we assume that the element

and if we replace

"Hn(C*/tC*)

tE A with

(Theorem 1 of Chapter is a non-zero divisor,

"H~(A/tA,C*)'" i

hypothesis that "mulitplication by t:c -+ c all integers

. "

1,

can be deleted.

i

then the

is injective, for

This is a somewhat finer

result than Theorem 1.

This is Chapter 5, Proposition 2.

( If the element

is not a non-zero divisor, then that

tE A

assumption can still be deleted if we replace with

"Hn{C*/tC*)"

"H;[T] {C*, .?'[T] / (T • .?' [T]) ) ") . As an application of the finiteness theorems of Chapter 5

(particularly Chapter 5, Theorem I), we prove, in the notations of [P.A.C.]'

xvi

Preface

Theorem AA.

Let

0

be an unramified(*), complete discrete

valuation ring of mixed characteristic, let and let

A = A ®l< (0).

-

0

Suppose that

A

A

be an O-algebra

is Noetherian.

Let

X

be a scheme simple and proper over

A

and embeddable over

(for this latter, it suffices that

X

be either quasiprojec-

tive over

A

Hh (X,~"),

h ~ 0,

"I\n,

or liftable over

see

[P.~C.l),

and let

be the lifted p-adic cohomology using the

as defined in

generated as

~,

A

[P.A.CII!].

Then

Hh (X,!2.")

~"-module, all integers

is finitely

h.

This is proved in Examples 1 and 2 at the end of Chapter 5.

In fact, it was in the course of proving Theorem A"

above

that we were led to prove the very general Chapter 5, Theorem 1,

(which will probably also have applications to other branches

of mathematics) . I also state Chapter 4, Theorem 6: Chapter 4, Theorem 6. let

tE A

Let

be any element.

A Let

be a ring with identity and C*

complex of left A-modules such that for all integers (1) (2)

n.

be any (~indexed) cochain n C is t-adically complete

Then for every integer

n,

Hn(C*)!(t-divisible elements) is t-adically complete. Hn(C*)

has no non-zero infinitely t-divisib1e e1e-

ments. (3)

[H n (C*)/(t-divisib1e e1ements)1~ Hn(C*)"t.

(3') For every integer

n,

we have an epimorphism from

(*)The hypothesis "unramified" can be replaced by the weaker condition "ord O (p) ~p - 1, where p is the characteristic of the residue class field of 0".

Preface

xvii

the group of equation (3) onto

lim [Hn(C*/tic*)].

i>O (4)

Also, for every integer

n,

there are induced

natural isomorphisms of abelian groups (or of left A-modules if

t

is in the center of

(t-divisible part of

A):

Hn(C*»~

liml(precise ti-torsion in

Hn(C*».

i>O (5)

Also, for every integer Hn(C*),

n,

the subgroup of

the t-divisible part of

Hn(C*),

is con-

tained in the subgroup liml Hn-l(C*/tic*)

i>O of

Hn(C*).

And Note 2 of Theorem 6 states, Note 2. If for any fixed integer

n,

we have that

Im(dn:C n ~ c n + l )

has no non-zero t-torsion elements, then for

that integer

the epimorphism of equation (3') is an iso-

n

morphism, and then also the two subgroups of

Hn(C*) discussed

in conclusion (5) coincide. A special case of Chapter 4, Theorem 6 is that in which (multiplication by gers

n.

t):

Cn~Cn

is injective, for all inte-

Then Note 2 above applies for all integers

n.

That

special case is Chapter 4, Theorem 1 and Theorem 1'. Under the more general hypotheses of Theorem 6, one might wonder whether one can obtain the stronger conclusions of Note 2 above, whether or not (multiplication by is injective.

n n t): C ~ C

The answer is "NO", as is shown in Remark 2

xviii

Preface

following Corollary 6.1 ("Example 2"). Theorem 6'

However in Chapter 4,

(in Remark 4 following Corollary 6.1), it is shown

that, if the hypotheses are as in Chapter 4, Theorem 6 and if in addition the element center of the ring

A,

tE A

is a non-zero divisor in the

and if we work with the appropriate

percohomology groups as in Chapter 5, then we obtain all the conclusions of Note 2, where in conclusion (3'), respectively: (5), we must replace

"Hn(C*/tic*)"

spectively:

"Hn-l(C*/tiC*)"

replace

ttH~(A/tiA,C*)'"

by

re-

"H~-l(A/tiA,C*)'"

by

(In Chapter 4, Remark 7 following Corollary 6.1, it is shown further that the hypotheses that and that

Itt

Itt

is in the center of

is a non-zero divisor"

the ring

A"

can be elimi-

nated if one uses the percohomology groups and

instead,

in conclusions (3') and (5) respectively.) Perhaps also relevant are Chapter 5, Corollaries 1.1 and 1.2.

In Chpater 5, Corollary 1.1, it is shown, e.g., that,

under the hypotheses of Chapter 5, Theorem I, we have that (l)

and

(2)

Does this result hold if we have the more general hypotheses of Chapter 5, Proposition 2?

(I.e., when we delete the hypo-

thesis that "multiplication by all integers

n"?)

t:C

n

-+

n C

is injective, for

The answer is "no", as is shown by an

example (Chapter 5, Remark 4 following Theorem 4).

(However,

if one uses the appropriate percohomology groups instead,

Preface

xix

then one obtains such generalizations, see Chapter 5, Corollaries 1.1' and 1.1".)

E~(ti)

Under the hypotheses of Chapter 5, Theorem 1, if

denotes the abutment of the Bockstein spectral sequence of the cochain complex plication by

C*

ti",

with respect to the endomorphism, "multias defined in Chapter 1, then in Chapter

5, Corollary 1.2, we show that for the fixed integer

n

of

Chapter 5, Theorem I, we have that (1)

[lj,m E~(ti) 1 ",Hn(C*)/(t-torsion), i>O

and

(2)

This Corollary generalizes, Corollary 1.2'

(respectively:

lary 1.2"), to the situation of Chapter 5, Theorem 6' tively:

Corol-

(respec-

Theorem 6), if the Bockstein spectral sequences are

defined suitably (in essentially the only way that is possible, see Chapter 1). Under the more general hypotheses of Chapter 4, Theorem 1, for all integers

and

(1)

[lj,m i>O

(2)

[lj,m i>O

n

we have that

E~(ti) 1'" Hn(C*)/(topological t-torsion), l

E~(ti)l'" ~iml (the kernel of the endomorphism 1>0

induced by "multiplication by {t-divisible elements in This is Chapter 4, Corollary 1.1.

ti"

on

Hn+l(C*)}). The

conclusio~of

Chapter

4, Corollary 1.1 hold equally well in the more general situation of the hypotheses of Chapter 4, Theorem 6, see Chapter 4,

xx

Preface

Remark 7 following Corollary 6.1.

(See also Chapter 4, Corol-

lary 1.1' in Remark 5 following Corollary 6.1.)

(Perhaps also

of interest is Chapter 4, Proposition 3, that under the hypotheses of Chapter 2, Corollary 1.2, if

n

is a fixed integer

such that

then

(Chapter 4, Remark 6 following Corollary 6.1, is also of some relevance to Chapter 4, Theorem 6.) A somewhat amusing set of side results in Chapter 4 are several lemmas and theorems that give information about tdivisible elements, etc., in left A-modules with identity

A,

where

t

is an element of

M

over a ring A.

These in-

clude Lemma 1.1.1 (which asserts that if "left mUltiplication by

t":

M-+M

is injective, then the same is true of

the t-adic completion of

M),

MAt,

Theorem 4, Proposition 5, Corol-

lary 5.1 and Corollary 5.2 of Chapter 4.

E.g., in Chapter 4,

Proposition 5, it is shown that if every t-divisible element of M

is infinitely t-divisible (e.g., this is the case if, either

M has no non-zero t-divisible elements (as for example if is t-adically complete), or if (left multiplication by M-+ M

t):

is inj ecti ve), then liml (kernel of the endomorphism, "multiplication by

i>O till,

of

M)

M

Preface

""

i~~

where

(kernel of the endomorphism, "multiplication by

"MAt/Mil

from

xxi t

i

,"

denotes the cokernel of the natural mapping

M into its t-adic completion

MAt

(whether or not that

mapping is injective). Chapters 1-5 constitute Part I of the book, and is the main emphasis of the book.

Part II consists of Chapters6 and 7.

In Chapters 6 and 7, we are concerned with the highest nonvanishing cohomology group, and with Poincare duality, respectively.

Some of the results are of particular use in applica-

tions to algebraic geometry, and in particular to p-adic cohomology (e.g., see [P.A.C.]), and also to more traditional resuI ts in algebraic topology, as we show in some examples. E.g., in Chapter 6, suppose that we have an integer

n,

such that Hi (C* /tC*)

=

0

for

i> n,

Then what can we say about i ::.. n)?

and such that

Hn(C*)

(and about

Hi(C*)

for

The most general such question is answered in Lemma 1.

The case in which

Hn(C*/tC*)

is finitely generated is covered

in Proposition 2.

The later theorems and propositions of Chap-

ter 6 deal with the case in which (A/tA)-module of rank one.

Hn(C*/tC*)

is a free

Chpater 6, Theorem 4, is particu-

larly useful in [P.A.C.]. In Chapter 7 we study Poincare duality.

The most notable

theorem is Theorem 3 and Corollary 3.1, which study the

xxii

Preface

problem: ring

A

is a differential graded algebra over the cmv.

C*

If

and i f

A

ci

and

are t-adically complete, for all

i, and i f the graded (A/tA)-algebra

integers

Hi (C*) ,

obeys Poincare duality over the quotient ring

i E 7/,

(A/tA), then

when can one conclude that the graded A-algebra, Hi(C*)/(topological t-torsion), obeys Poincare duality over the ring that:

iE7/, A?

Basically, one needs

Hn(C*)/(topological t-torsion) has annihilator ideal

zero; that

Hi(C*)

has no non-zero t-divisible, t-torsion ele-

ments, for all integers

i;

and that the ring

tive as left (A/tA)-module.

A/tA

The first two of these conditions

are reasonable, but the third is very restrictive. this latter condition holds in the case that valuation ring and lary 3.1).

t"F 0

is injec-

A

However,

is a discrete

(the resulting statement is Corol-

This result, and Chapter 6, Theorem 4, are used

to prove that: If

X

is a complete, non- singular (not necessarily lift-

able) embeddable ([P.A.C.]) algebraic variety over a field of characteristic

p,

and if

0

is an unramified, complete

discrete valuation ring of mixed characteristic having residue class field, then if

k

Hi(X,OA),

iE7/,

k

for

is the author's

lifted p-adic cohomology using the "A", as defined in [P.A.C.], then, for

Hi(X,OA)/(t-torsion),

duality over the ring

O.

iE7,

See [P.A.C.].

we have Poincare This is the principle

application of Chapter 7 to our general study of the Weil zeta function [W.C.].

{Chapter 6, Theorem 4 also has application

to algebraic families; i.e., in the notation of [P.A.C.], to

Preface X

over

A

simple, proper,

embeddable over

~).

xxiii

(not necessarily liftable), and

And of course there are more traditional

applications to algebraic topology. In Chapters 1-7, we have dealt with completions with respect to an ideal that is generated by a single element

t.

In

Part III, we make a study of the analogous situation, for completions with respect to finitely generated ideals that are not necessarily simply generated.

Part III consists entirely of

Chapter 8. In Chapter 8, we return to more fundamental considerations as in Chapters 4 and 5, but generalized as follows. a ring (not necessarily commutative) and let finitely generated two-sided ideal in I-adically complete.

Let

C* i C

of left A-modules such that integers

i.

be a

be

be a left

such that

~-indexed)

A

A

is

cochain complex

is I-adically complete for all

Then we state a generalization of Chapter 5, Theo-

(Chapter 8, Theorem 1.

rem 1.

A

I

Let

generated by an "r-sequ~nce for

The ideal i C ",

I

is required to be

all integers

i, see

Chapter 8, Definition 2.) If the ideal

I

admits a set of generators contained in

the center of the ring

A,

then one can state a generalization

of Chapter 5, Theorem 4, namely Chapter 8, Corollary 1.3. with identity and let A

I

Let

A

be a left Noetherian ring

be an ideal in the ring

A

such that

is I-adically complete; and such that we have an integer

r> 0 ideal

and a finite sequence I

tl, ... ,t

r

of generators of the

contained in the center of the ring

¢:7[Tl, .•. ,Trl

~A

A.

Let

be the homomorphism of rings with identity

xxiv

Preface

that sends

Ti

into

t

i

,

1 < i < r.

Let

C*

be a ':1'-indexed) c i is I-adically

cochain complex of left A-modules such that complete, all integers

i.

Let

[ ,···,T lilT 1' ... ' T) , Cj ) , l ,···,T r I(1'T D.lj = Tor.7.I'[T l r r l i, j E 1', (1)

i > O. H

n +i

for (2)

Let

(D~)

Hn(C*)

~

be any fixed integer.

Then if

is finitely generated as left (A/I)-module,

l

0

n

i

~

r,

then

is finitely generated as left A-module.

More general theorems are demonstrated, if merely

I

ad-

mits a finite set of generators that mutually commute, and a less restrictive set of hypotheses apply, assuming that one uses the percohomology~: "Hn(C*/IC*) ", sition 2.

"H~(A/I,C*)"

in lieu of

see Corollary 1.1 and Remark 2 following Propo-

Also, see Chapter 8, Corollaries 1.2 and 1.3'.

(As

a fairly straightforward application of the general theory developed in Chapter 8 to algebraic geometry, in Examples 1-4 of Chapter 8 we deduce by a new method almost all of the wellknown finiteness theorems about cohomology of coherent formal sheaves over proper formal schemes.) Theorem 6 and its Corollaries in Chapter 8 generalize Corollary 1.1 of Chapter 5. and the

liml

(They concern the inverse limit

of the "percohomology groups

mod In".)

results in Chapters 4 and 5 (particularly Chapter 5) generalized to the situation of Chapter 8.

Other are

E.g., Chapter 5,

Proposition 3 is so generalized, see Chapter 8, Proposition 4. Concerning the level of generality of the various chapters:

Preface

xxv

Most of Chapter 1 generalizes to virtually any abelian category, and this is proved.

Chapter 2 requires an abelian category such

that denumerable direct products exist, and such that denumerable direct product is an exact functor, see Introduction, Chapter I, section 7, and this generalization is noted in Remarks. Also, in Chapter I, only the cohomology sequence and not necessarily the cochains, is needed; and in Chapter 2, any cohomology theory,

(not necessarily the cohomology of cochain complexes)

will do.

Chapter 3 again requires an abelian category such

that denumerable direct products exist and such that the functor, "denumerable direct product" is exact,

(but now the coho-

mology theory must be the cohomology of cochain complexes).

The

same is true for all of Chapter 4, as is noted in the text But Chapter 5 requires that we be in the category of (7-indexed) cochain complexes of left A-modules, where that

tEA

tl\ CAt).

be in the center of

A

A

is a ring,

and

(or at least, be such that

Part of Chapter 6 is at the same level of generality

as Chapter 4; but the rest of Chapter 6, and Chapters 7 and 8, again require a ring

A

and the cohomology of cochain complexes.

In keeping with the general philosophy to make the material accessible to beginners, the main text of the book, Chapters 1-8, has been written for the most part at the level of generality of left modules over a ring, even when the more general situation of abelian categories was possible; and the generalizations to abelian categories have been kept in Remarks, that may be ignored by the reader who is not interested in, or cognizant with, these generalizations. Some of the results of this book are somewhat surprising.

xxvi

Preface

These include Chapter 5, Theorem 1; Chapter 4, Theoremsl and 6 and Corollary 6.1.

Perhaps also interesting are Chapter 4,

Proposition 5; Chapter 6, Theorem 4; and Chapter 7, Corollary 3.1; and the theorems of Chapter 8.

We give examples to show

that most of the pathology that would not be excluded by these theorems actually occurs - a very partial list is Chapter 4, Examples 1 and 2, and Example 2 in Remark 2 following Corollary 6.1: and Chapter 7, the Example following Theorem 3 (e.g., the latter example shows the difficulties in attempting to improve Chapter 7, Theorem 3). My special thanks to Mrs. Marion Lind and Mrs. Sandi Agostinelli for their excellent work in typing up this manuscript; and especially to Mrs. Sandi Agostinelli, for putting in so much overtime work, and more than usual care and patience.

Terminology. if

A

Ingeneral, we use conventional terminology.

is a ring and

tEA

and if

M

E.g.,

is a left A-module, then

°,

is t-divisible iff for every integer i > -i v. EM such that t • v. = u. The element u is

uEM

an element there exists

1

1

infinitely t-divisible iff there exists a sequence of elements of and such that v

M such that

°= u.

t · v + =v ' i l i

vi'

i> 0,

all integers

i.::O,

It is easy to give examples of t-di visible

elements that are not infinitely t-divisible. Similarly, if and if

t

M is a left A-module, where

is an element of

terminology, an element exists an integer element

u EM

i >

°

A,

uE M

A

is a ring,

then, following the usual is a t-torsion element iff there

such that

t

i

·u=O.

a precise t-torsion element iff

We will call an t · u = 0.

Thus,

Preface if

i> 1,

then an element

ment if and only if

uE M

t i . u = O.

xxvii

is a precise ti-torsion ele-

Thus,

"u

a precise t-torsion

element" implies "u a precise ti-torsion element" implies "u . a preclse

. t i+l -torslon e 1 ement " .lmp l'les

element", for all integers ti-torsion elements in

M

i > 1.

is the precise ti-torsion part of

(Thus, for each integer

of

M is the kernel of the endomorphism of the abelian group

is at-torsion

The set of all precise

M.

tin

"u

i::.l,

the precisE'! ti-torsion part "multiplication by

M).

However, we do deviate slightly from the most commonly accepted notational convention in our manner of indexing a bigraded exact couple.

E.g., what would be written as

in the original reference [E.C.], is denoted in this text as

That is, we have as usual emphasized the filtration degree

p,

but we have also chosen to emphasize the complimentary degree q

rather than the more traditionally emphasized total degree

n = p + q.

We believe that our indexing notation helps make

comprehension overall a bit easier, perhaps because of the even more rigidly entrenched indexing notational convention, "E~,q". The research for the main text of this book was done mostly at the Pennsylvania State University in 1973-74, except for virtually all of Chapter 2 which was done at the University of Oxford, England in 1963-64, and was partially supported by NSF Research Contracts, respectively an NSF Postdoctoral

xxviii Fellowship.

Preface An early version of this manuscript was pre-

pared in 1975, and is substantially the same as Chapters 1-7 of this ms., and contains the most important parts (Theorem 1 and its Corollaries) of Chapter 8.

Some extensive "touching up" of

Chapter 8 was done at the university of Rochester in the academic year 1977-78. The research for Introduction, Chapter 1, section 4, was done at Columbia College, in the academic year 1958-59, and was supported by a Pulitzer Scholarship and a New York State Regents Scholarship for Education in Engineering, Chemistry, Physics and Mathematics.

(The Exact Imbedding Theorem was

also proved, roughly simultaneously, by a somewhat different method, in [P.F.]).

Introduction, Chapter 1, section 5 was

done at the University of Rochester in Spring, 1979.

Intro-

duction, Chapter 2, sections 1-7, were mostly worked out at the University of California at Berkeley in the 1960's, and owe a debt to an early version of [O.A.L.l, and were partly supported by NSF grants. of Rochester in spring NSF grant.

They were written up at the University to fall, 1979, partly supported by an

Introduction, Chapter 2, sections 8-10 were com-

pleted at the University of Rochester in summer and fall, 1979, and were partially supported by an NSF grant.

TABLE OF CONTENTS

v

DEDICATION

vii

PREFACE INTRODUCTION Chapter 1

Manipulating in Abelian Categories

1

Categories

1

Section 1

Chapter 2

Section 2

Subobjects and Quotient-Objects

Section 3

Abelian Categories

4 11

Section 4

Imbedding of Abelian Categories

44

Section 5

Subquotients

56

Section 6

Left Coherent Rings

66

Section 7

Denumerable Direct Product and Denumerable Inverse Limit

80

Theory of Spectral Sequences Section 1 Section 2

Spectral Sequences in the Ungraded Case

97 97

The Spectral Sequence of an Exact Couple, Ungraded Case

111

Section 3

Graded Categories

137

Section 4

Spectral Sequences in a Graded Abelian Category

185

Section 5

The Spectral Sequence of an Exact Couple, Graded Case

208

Section 6

Filtered Objects

235

Section 7

The Partial Abutments of the Spectral Sequence of an Exact Couple

278

The Spectral Sequence of a Filtered Cochain Complex

339

Convergence

400

Section 8 Section 9

Section 10: Some Examples

441

PART I Chapter 1

The Generalized Bockstein Spectral Sequence

465

Chapter 2

The Short Exact Sequence 1.8

493

xxix

xxx Chapter 3 Chapter 4 Chapter 5

Table of Contents Cohomology of an Inverse Limit of Cochain Complexes

531

Cohomology of Cochain Complexes of t-Adically Complete Left A-Modules

539

Finite Generation of the Cohomology of Cochain Complexes of t-Adically Complete Left A-Modules

607

PART II Chapter 6

The Highest Non-Vanishing Cohomology Group

659

Chapter 7

Poincare Duality

687 PART III

Chapter 8

Finite Generation of the Cohomology of Cochain Complexes of I-Adically Complete Left A-Modules for a Finitely Generated Ideal I

BIBLIOGRAPHY

737 801

CHAPTER 1 MANIPULATING IN ABELIAN CATEGORIES Section 1 Categories

Most of the material of Parts I, II, and III of this book, is about cochain complexes of left modules over a ring.

It is

noted, in the various chapters, that some material generalizes to suitable abelian categories.

Therefore, we write this brief

introductory chapter to acquaint the reader unfamiliar with abelian categories with this important concept.

(The reader

who does not wish to learn about abelian categories can skip this chapter, and can read Introduction, Chapter 2 with "abelian groups" replacing "objects" throughout.) We will assume in this chapter that the reader is familiar with the concept of "category", and of the dual of a category, as defined in the original, still excellent, reference [C.A.]. We recall a few definitions from [C.A.], those of monomorphism and epimorphism. Defini tion. then and

=f

and 0

h,

A map f

C

is a category and

f: A -+ B

f is a monomorphism iff whenever g

fog

If

h

are any maps from

then f:A-+B

g

0

0 into

C,

is any object in A,

C,

such that

= h. in the category

C

is an epimorphism iff

is a monomorphism considered as a map from

the dual category

is a map in

B

into

A

in

Co.

Equivalently a map,

f:A -+ B 1

is an epimorphism iff given

Section 1

2

any object then

D

and maps

g,h:B

->-

D

such that

go f =h

0

f,

g = h. A map

f:A

B

->-

there exist a map

in the category g:B

-+

A

C

such that

is an isomorphism iff

gO f = idA

and

fog = id . B

A map that is an isomorphism is always both an epimorphism and a monomorphism, but the converse is in general false. Examples.

1.

If

C

is the category of sets and functions

(or

the category of groups and homomorphisms; or the category of abelian groups and homomorphisms; or the category of rings and homomorphisms; or the category of commutative rings and homomorphisms), then a map is

a monomorphism iff it is one-to-one,

and is an epimorphism iff it is onto.

In these cases, a map is

an isomorphism iff it is both an epimorphism and a monomorphism. 2.

If

C

is the category of all topological spaces

and continuous functions, then a map is a monomorphism iff it is one-to-one; an epimorphism iff it is onto; and an isomorphism iff it is a homeomorphism.

In this case, not every map that

is both an epimorphism and a monomorphism (i.e., that is oneto-one and onto) is an isomorphism (i.e., is a homeomorphism). 3.

If

C

is the category of all Hausdorff topo-

logical spaces and continuous functions, then let a map.

Then

f

is a monomorphism iff

is an epimorphism iff the is dense in IIf

Y

is onto").

f

f: X ->- Y

be

is one-to-one.

set-theoretic image

f(X)

of

f f

(this is in general weaker than the condition, f

is an isomorphism iff

f

is a homeomorphism.

Therefore in this category also, not every map that is both a monomorphism and an epimorphism is an isomorphism. We will also assume, in this chapter, that the reader is

Categories

3

familiar with the notions of the direct product and direct sum of an indexed family of objects in a category, as presented in [C.A.], and also with the definitions and elementary properties of the inverse limit (or direct limit) of an inverse (or direct) system in a category indexed by a directed set, as covered in [C.A.].

(This generalizes without change to the case in which

the indexing directed set is replaced by a directed class (i.e., a non-empty class (see [K.G.]), perhaps even a proper class, with a preorder such that for every x,y there exists a z such that

x:::. z,

y:::. z)

.)

Section 2 Subobjects and Quotient-Objects

Let

A

be a category and let

Then consider the class UK.G.]) B is an object in

A

and

MA

f:B -+ A

be an object in

A

(B',f')

and

(B",f")::. (B' ,f') category

A

is a monomorphism.

are elements of

f'

f' 0 g = f".

Let

Then we

M , A

that

g:B" -+B'

If such a map

in the g

exists,

is a monomorphism) it is unique; and (since

A

(B', f'),

where

by defining, when-

is a monomorphism) i t is a monomorphism.

Lemma 1. let

M , A

iff there exists a map

such that

then (since fll

(B",f")

(B, f)

of all pairs

have a natural pre-order on the class ever

A.

be a category, let

(B" , fll) E M . A

A

Therefore,

be an object in

A

and

Then the following two conditions

are equivalent.

1)

Both

2)

There exists an isomorphism

the category

(B' ,f')::. (B",f")

A,

such that

and

(B",f") 8

~

from

(B' ,f') B"

in

onto

M . A

B',

in

f' 0 8 = f ".

When these equivalent conditions hold, then the isomorphism of condition 2) is uniquely determined. Proof:

2)

=-

that

=-

f' 08 = fll,

(B",f") < (B',f'),

and therefore,

1)

We have

1)

2):

f'o8=f"

Let and

8 :B" -+ B' f"oT=f'. 4

respectively

respectively

and Then

T:B' -+ B" f'o8

o

f' = f" 08- 1 ,

(B',f') < (B",f"),

be maps such T=f"oT=f'=f' oid

B

Subobjects and Quotient-Objects whence since

f'

is a monomorphism,

Therefore follows since

f'

Definition 1.

Let

in

A.

8

M , A

= id ,

8T

Similarly

B

is an isomorphism.

Uniqueness of

is a monomorphism.

A

8

Q.E.D.

A be a category and let

Then a subobject of

pairs in

5

A

be an object

is an equivalence class of

where two pairs

and

(B' ,f')

(B",fll)

are

equivalent iff the two equivalent conditions of Lemma 1 hold. Given two sUbobjects there exist

S'

and

S"

of

(equivalently: for all)

(B",f")ES",

we have

A,

we say

(B' , f ') E S'

Given a category

mayor may not be a set) and an object

A

A,

in

strongest form of the Axiom of Choice (see [K.G.]) Mi

of

MA

~

sentative pair A,

A

(which

then by the (*)

,

there

We will call

a complete class of representatives for the

such a class

object

and

that contains exactly one repre-

sentative element from each equivalence class.

subobjects

iff

(B',f') 2. (B",f")

We next make a convention.

exists a subclass

S' < S"

It is a class containing exactly one reprefor each subobject

(B',f')

{(B' ,f')}

of the

and consisting only of such representative pairs.

Having chosen such a fixed complete class of representatives for the subobjects of A

A,

by a class (or set) of subobjects of

we will mean a subclass (or subset) of the complete set of

(*)In [K.G.], Godel proves that this strong form of the Axiom of Choice is consistent with the other, standard, axioms of his set theory, assuming that those other axioms are themselves consistent. (The statement of this strongest form of the Axiom of Choice is, that "There exists a well-ordering on the class of all sets".)

5ection 2

6

representatives M~

sentatives

M~.

Also, since the complete class of repre-

is a subclass of

M , A M'

pre-ordered class, we have that

(B', f')

(B", f")

and

5'

iff

and

5"

of

A,

if M'

(B',f') < (B",f")

5' < 5"

A

as subobjects of

in the complete class of representa-

M~.

tives

A and an object

It might be, given a category

that a complete classof representative;; M'

A

of

is a

are their unique representatives in

we have by Definition 1 above that A

MA

is a pre-ordered class.

A

In addition, given two subobjects

and since

A

is a set.

in

A,

for the subobjects

If this is so, then it is independent of the

choice of the class of representati ves the subobjects of cardinality of

A

M~

A

form a set.

M~,

and we say that

Also, in this case, the

is independent of the choice of the complete

M'

class of representatives

and we then call this cardinal

A

number the cardinality of the set of all subobjects of

A.

(The proof of these observations is that given any two complete classes of representatives A,

M'

and

A

for the subobjects of

then there is a unique bijection

from

M'

A

onto

M"A'

such that

S(B,f)

Remarks 1.

The reason for our introducing the notion of "a

is equivalent to

(B,f),

fora11

(B,F) in

complete class of representatives for the subobjects" is as follows.

In Godel's set theory,

[K.Gl, one cannot speak of a

class some of the elements of which are proper classes.

A

by Definition 1 above, if and

A

is an object in

A,

But,

is a category that is not a set, then it is possible for some, or

even all, of the subobjects of

A

to be proper classes.

(E.g. ,

Subobjects and Quotient-Objects

A is the

this is the case if if

A

and if

is any object in A

A;

ca~egory

7

of abelian groups and

A is the category of sets

or if

is any non-empty set).

Therefore, in Godel's set

theory, in such circumstances, one cannot speak literally of "a class, or set, of subobjects of

A"

in the literal sense of

Definition 1; so instead we speak of "a class, or set, of the representatives for the subobjects of

A", as in the above

convention. 2.

In the most common categories

A,

there is usually

a "natural choice" of a complete class of representatives for every object

A,

M'

A

making it unnecessary to invoke the

Axiom of Choice in that case. v·lhere not inconvenient, we will state results in

3.

such a way so as to not necessitate using the above convention (i.e., so as not to speak of "classes or sets of subobjects"). This is usually not di fficul t to do--al though some theorems become sloppy in their statement unless one uses the above convention, and then we will use the above convention. Examples. object

1.

A

A,

in

subobjects of a subset of

A is the category of sets, then for every

If

A A

a complete class of representatives for the is the set of all pairs

and

l:B ... A

A

where

B

is

is the inclusion.

Notice in this case that if every subobject of

(B,l)

A

is a non-empty set then

in the sense of Definition 1, except

for the empty subobject, is a proper class. this case, that the subobjects of

A

Notice also, in

form a set, and are in

natural one-to-one correspondence with the set of all subsets of

A.

8

Section 2

2.

A is the category of all groups (or abelian

If

groups, or rings, or commutative rings, or fields), then for every object

A

in

A,

representatives

M'

for the subobjects of

A

where

(B,l)

all pairs

as in Example 1, a complete set of

B

is

A

the set of

is a subgroup (or, respectively,

subgroup, or subring, or subring, or subfield) and the inclusion.

Again, the subobjects of

A

l:B

-+

A

is

form a set, in

natural one-to-one correspondence with the set of all subgroups (or,

resp~

subgroups, subrings, subrings, or subfieldsl of

3. and

A

If

A is the category of all topological spaces, A,

is an object in

tives for the subobjects of

then a complete set of representaA

is given by

{(B,l)}

where

is a topological space such that the underlying set of a subset of the underlying set of of B

A

AD'

Every pre-ordered class

such that the objects in

AD

B

B

is

and such that the topology

is finer than the induced topology from

4.

A.

0

A; and

1

is incl.

defines a category

are the elements of

0,

and such that {(a,en

Hom A (a,S) =

o

Let

0

!

a.:.B,

if

otherwise.

r,1

be the pre-ordered class of all ordinal numbers (as

defined in [K.G.]l with the reverse of the usual pre-ordering. Then for every object

A

in

AD

each Eubobject of

sists of a single element and is therefore a set. all subobjects of

A

e

con-

The class of

is a proper class, and is in natural one-

to-one correspondence with a <

A

{ordinal numbers

for the usual ordering}.

B

such that

Subobjects and Quotient-Objects Let

A be a category and let

A

9

A.

be an object in

Suppose that we have chosen a complete set of representatives for the subobjects of the object subobjects of

A,

A.

Then given a class

then by a supremum,or infimum, of

mean a supremum, or infimum, in the complete class

A.

tives

M

L

We write

B,

0

n B,

respectively

B{:S

S

S

of

we

f representa-

for this

B ES

supremum, respectively infimum, if it exists. A

For example, if remum of a set

H

is the category of sets, then the sup-

of subobjects of a set

A

is the usual

union, and the infimum is the usual intersection.

If

A is

the category of abelian groups, then the supremum of a set of subobjects of an abelian group

A

is their sum,

B= (B,

U

{finite sums of elements of

B};

H

l

)EH

and the infimum is

(B, l) EH

the set-theoretic intersection.

In the category of groups, the

supremum of a collection of subgroups of a group is the subgroup generated by the set-theoretic union; the infimum is the set-theoretic intersection. Definition 10. in

A.

Let

A be a category and let

Then a quotient-object of the object

A is a subobject of

A

(B',p'),

where

is an epimorphism; where

B'

p"=8

o

p'.

A

(B' ,p')

of representatives

AO •

A

E'

A

Thus, ex-

•

and

p':A->-B

is equivalent to

(B",p")

8:B' ->- B"

A

such that E'

A complete class of representatives

quotient-objects of

category

in the category O

A is an equivalence class of

is an object in

iff there exists an isomorphism

be an object

A

in the dual category

plici tly, a quotient-object of pairs

A

in the category

A

for the

--

A is a complete class

for the subobjects of

A

in the dual

This is in a natural way a pre-ordered class;

Section 2

10

we can speak of a class (or set) of guotient-objects of

A,

and of the supremum or infimum of such a class when the supremum or infimum exists. Examp 1es. 1 0.

In Example 1 above, a complete set of representa-

tives for the quotient-objects of a given set

A,

gory of sets

is the set of all pairs

the quotient-set of p:A -+ B

A

(B,p)

in the catewhere

B

is

by some equivalence relation, and where

is the natural mapping. 2

0

•

In Example 2 above, a complete set of reps. for

the quot.-objs. of a given obj. A in (B,p)

A

where

B

A is the set of all pairs

is a quotient-group (resp.:

group; a quotient ring; a quotient ring; natural map into the quotient (resp.:

30

•

A)

a quotient and

p

ibid, ibid, ibid,

In Example 3 above, a complete set of representa-

tives for the quotient-objects of a topological space the pairs

is the

(Y,p)

where the underlying set of

quotient-set of the underlying set of relation, and where the topology of quotient topology; and where

p: X -+ Y

X Y

Y

X

are

is the

for some equivalence

is coarser than the

is the natural mapping.

Section 3 Abelian Categories

Definition 1.

A pointed category is a category

e,

together

with the additional data, the giving, for every pair objects of (Zl)

e,

A,B

of

of

A map,

0A,B

from

A

into

B

in the category

e,

such that (Z2)

For every object

in

e

e

e

g:B+D,

respectively

g oOA,B =OA,D e,

the maps

for all objects

and every map f:D+A,

we have that

are called the points.

0A, A E Home (A, A) , If

in

respectively

Given a pointed category A,B

D

0A,B

0A,B of=OD,B'

for all objects

We often write

0A

for

A.

is a category that admits the structure of pointed

category, then the points are uniquely determined by the category structure of Proof: in

e

In fact,

e. if we had another set of points

obeying axioms (Zl)

and

(Z2)

(0'A,B ) A,B objs.

above, then using

(Z2) ,

°A,B--0 A,B for all objects

00'

-0'

A,A-

A,B

in

A,B'

A.

Q.E.D. 11

Section 3

12 Examples 1. set and

A pointed set is a pair (S,so)

So E S.

S

is a

Then we have the category of pointed sets,

such that for all pointed sets (S,so) -.. (T,t ) o

where

(S,so)

are all functions

f

(T,t )' the maps: o

and S

from

into

T

such

The category of pointed sets is a pointed category. 2.

The category of groups and homomorphisms is a

pointed category. 3.

The category of abelian groups and homomorphisms

is a pointed category. 4.

The category of sets and functions is not a pointed

category (since, e.g., 5.

Hom{sets} (S,<jl) = <jl).

The category of non-empty sets and functions is

not a pointed category. 6.

(Proof left as an exercise) .

The category of rings (or of commutative rings)

with identity and homomorphisms of rings (that preserve the identity element) is not a pointed category. Hom( {oJ ,if) =
(Since e.g.,

Similarly for the category of all non-zero

rings (or of all non-zero commutative rings) and such homomorphisms. Definition 2.

An additive category is a pair

(C , (+ A,B ) A,B objects in

) C'

where

where, for every pair of objects nary composition on the set (ADl) +A,B

HomC(A,B)

C

A,B

is a category, and of

C,

+A,B

HomC(A,B), such that

together with the binary composition

is an additive abelian group, and (AD2)

Whenever

is a bi-

A,B,D

are objects in

C

and

Abelian Categories

13

have that k O(f + g) = k

g

in

Home (A, D)

(f+g) oh=foh+goh

in

Home(D,B).

0

f +k

0

,e,

Given a category

a doubly indexed system that obeys axioms (ADI) and (AD2) ,

e

(+A,B) A.B cbjects in

and

will be called an additive structure on the category

e.

As we shall see in the next Remark, every additive category is pointed; but the converse is in general false (vide infra) . Remarks l .

Let

objects of

e.

be an additive category, and let

e Let

(respectively:

k EHome(B,D)

be

let

Then the first part (respectively:

hE Home(D,A».

A,B,D

second

part) of axiom (AD2) is equivalent to the assertion that the function:

f

+

k .Of

(respectively:

of abelian groups from tively:

into

Home(A,B)

Home(D,B».

f

f oh)

+

into

Home (A,D)

Therefore, if

0A,B

zero element of the additive abelian group objects (1)

whenever (2) whenever

A,B

in

e,

k E HOme (B, D) ,

0

respectively: resp.:

(f-g) =k of-k og,

f, g E Home (A, B),

Equations (1)

(respec-

denotes the

Home(A,B)

for all

then we have that

k oOA,B = 0A,D'

k

is a homomorphism

0A,B Oh=OD,B'

hEHome(D,A), and

and

(f-g) oh=f oh-g oh,

k E Home (B, D)

and

hE Home (D, A) •

imply that every additive category has a

natural structure as pointed category, by taking the additive identity element of objects

A,B

in

A.

HomA(A,B)

for the point

0A,B'

for all

14

Section 3

Remark 2.

suppose, in the definition of "additive category"

above, we weakened axiom (ADl) , by saying instead that (ADl')

HomC(A,B) +A,B

together with the binary composition

is an additive group (that is not necessarily

abelian) . Then we claim that axiom (ADI'), in the presence of axiom (AD2) , is equivalent to axiom (ADl). Proof:

Suppose we have axioms (ADl') and (AD2).

any objects

A,B

f,g€HomC(A,B)'

in the category

C

Then given

and any two maps

by axiom (AD2) and the associative law, we

have that (Hg)

o

idA + (f+g)

0

idA =f+g+£+g.

On the other hand,

Therefore f+g+f+g Since

f+f+g+g

Home (A,B)

in

P.omC(A,B).

is a group, by the left and right cancella-

tion laws we have f+g=g+f. ~xample

call it

7. P,

Let

e

be the category consisting of one object,

and such that

monoid of the ring 7. Home(P,p) =7.

Q.E.D.

Home(P,p)

of integers).

= (the

multiplicative

(This means that

as set, and the composition is given by multiplica-

Abelian Categories tion of integers).

15

Then there exists an additive structure on

C as in Definition 2 above, such that

the category

comes an additive category. "+" of integers.

C be-

Namely, take the usual addition

However, there are actually

additive structures on the category

C

lot

2

different

0

such that

C becomes

an additive category. Proof:

In fact,

if

is any permutation of the set of

IT

rational primes, then by unique factorization there exists a unique automorphism such that

a

IT

(p) =

two integers

IT

n,m,

a IT

of the multiplicative monoid

(p) ,

for all positive primes

"t"

usual multiplication, makes

P into a ring.

"+" IT

'

makes

additive structures

Given

define

Then the binary composition

datum,

p.

( lI, .)

C

lI,

into an additive

"+"

on the category

IT

different permutations

on

IT

together with the Therefore, the

ca~egory;

and the

C are distinct for

of the set of all rational primes. Q.E.D.

Example 7 above shows that, unlike a pointed category, an additive category cannot in general be thought of as being a category that obeys a certain set of axioms; but the additive structure must, in general, be thought of as a

new set of data.

However, we will see in Corollary 1.1 below, that unlike the situation in Example 7 above, for a wide class of categories, there is at most one additive structure.

First, we prove a

Proposition, that is quite important in its own right. Proposition 1.

Let

A be an additive category, let

n

be a

Section 3

16

non-negative integer and let

AI' .•• ,A

n

A.

be objects in

Then (A) maps

If there exists an object

71 i

:S'" Ai

and

(1)

71.

S

1.

OA' ,1\. J 1.

_

01.-

together with

such that

:A. + S,

1. 1.

A,

in

if

i'/-j,

if

i=j,

J

l

{ idA. 1.

1 ~i,j ,::,n,

and (2)

then the data

(S,7I , .•. ,7I ) l n

in the category [C.A.]),

1.:A. +S, l

(S,ll' •.• 'In'

Conversely, if n

l

(cl Al, ••• ,A

7Ii:S+Ai,

A , ... ,A l n

A,

in the category l,::,i,::,n,

is a direct sum of

(S,7I , ••• ,7I ) n l

is a direct product of

then there exist unique maps

such that equation:; (1) and (21 above hold.

Similarly, if n

of

(in the category-theoretic sense, see

and the data

(B)

Al, ... ,A

A

is a direct product

(S,ll, •.. ,ln)

A,

in the category l~is.n,

is a direct sum of

then there exist unique mappings

such that equations (II and (2) above

hold. proof:

Let

(A)=>(B) :

be any object and let

B

n

+

1

f

n n

)

L 7I.1.f.=

j =1

n

L 6 .. f j

j=l

Jl

=f., l

l,::,i,::,n,

l

J J

f. :B l

+

A. l

Abelian Categories

l

OA. ,A.

0 .. =

where

J1

J

~

~

i ,j

n;

i;ij,

if

i=j,

1

idA.

1

if

17

1

tha t i s

f = 11 fl + .•. + 1 f n n

Therefore

a direct product.

satisfies the existence part for

e

Conversely, if

'IIi 06=f

is any map such that

,

i

then 6

= ids

0

6

=

(1 I'll 1 + ... + 1 n 'II n)

11fl + •.• + 1 n f n

0

e = 1 1 ('II 1 e)

+ ... + 1 n ('II n e )

= f,

e=f.

(4)

Equation (4) is the condition of universality. (B)

=!>

(A):

For each integer

(S,'TT , ..• ,'TT ) 1 n

1 i :A ... S i

o J

1

is a direct product of

a unique mapping

A. ,A. 1 J

=

'11.01

i,

,

~

i

~

n,

since

A , ... ,A , l n

there exists

such that

j ;i i,

1

{ idA.

j = i,

1

But then, 1

'11

1 1

'TT

j

(1

'11

1 1

+ ••• + 1n'TTn) = 'TT j ,

+ •.. + 1n'TTn = identity of The proofs that

or can be deduced from

(A)~

1

~

j

~

n,

and therefore

S. (C)

and

~

(B)"

"(A)

(C)~

and

spectively by passing to the dual category.

(A)

"(B)

are similar-=!>

(A)"

re-

Q.E.D.

Section 3

18

Notice that the above Proposition implies that, in an

A,

additive category

if the direct product of finitely many

.objects exists, then so does the direct sum, and they are canonically isomorphic.

Also, conditions (1) and (2) of part

(A) give a way of defining the direct product (equivalently, the direct sum) of finitely many objects in an additive category, in a way not mentioning universal mapping properties. Both of these observations are special to additive categories. Corollary 1.1. every object of

A

Let A

in

with itself.

A be an additive A,

there exists a direct product

category

A

x

A

Then there exists at most one additive

A.

structure on that category Proof:

category such that, for

Suppose that there exists an additive structure on the

A.

If

A

A,

is an object in

be a direct product of

A

then let

(A x A.1T ,1T ) l 2 Then by Proposition 1,

with itself.

: A + A x A such that condi2 tions (1) and (2) of part (A) of Proposition 1 hold. But, since

part (B), we have unique maps

1

1

,

1

is a direct product of

A

with itself, by the

universal mapping property, condition (1) alone determines the mappings

1

1 ,1 2 ,

Since the point

0A=OA,A

depends only on the

category structure, it follows that the mappings depend only on the category structure. tion, part (A), we have that A

with itself.

(A x A,

Let

be the unique mapping such that

1

1

and

Then, by the Proposi,1

2

)

is a direct sum of

Abelian Categories

Then

depends only on the category structure of

direct sum of

If now A,

19

B

A.

But

A with itself, it follows th.at

is any (possibly different) object irt the category

and if

f,g:B->-A

are maps, then let

(f,g):B ->-A x A be the unique map such that

7Tl ° (f,g) =f,7T

0 (f,g) =g. Then 2 depends only on the category structure of A. And by

(f,g)

equation (3) we have that,

(4)

f+g=PA

o

(f,g).

Since the right side of equation (4) depends only on the category structure, it follows from equation (4) that the additive structure of

A

is uniquely determined by the category Q.E.D.

structure. Remark:

Of course, applying Corollary 1.1 to the dual cate-

gory, it follows,

likewise, that:

that, for every object A

A

in

A,

If

A

is any category such

the direct sum:

Ae A

of

with itself exists, then there is at most one additive

structure on the category

A;

i.e., that there is at most one

Section 3

20

A into an additive category.

way of making Let

C

be an arbitrary pointed category.

set, and suppose that category

(Ai)iEI

Let

I

be a

is a family of objects in the

A indexed by the set

I.

Suppose that the direct

sum, 6l A.

iEI ]. and the direct product, IT A. iU ]. of the objects

8:

Ai'

6l A. -+

iO].

all

i E L

where

i E I,

exist.

Then we define the canonical

IT A. iO].

if

i-lj,

if

i=j,

71.: IT A. -+A. , 1. j EI J 1.

the canonical projections, resp.:

resp.:

1.:A. -+

Ell A., jEI J injections, all i E I. ].

are

1.

Then Corollary 1.2.

If

finite set and if

A

is an additive category, if

Ai'

direct sum

i E I,

6l A. or the direct product iEI 1. they both exist, and the natural mapping

8: 6l A. -+

iEI

1.

IT A. i EI 1.

A

are objects in IT A. i EI

l.

I

is a

such that the exists, then

Abelian Categories

21

is an isomorphism. Proof:

By proposition 1, the indicated direct sum and direct

product both exist.

Let

(S 'TIl' ... ,TIn,ll' ... ,In)

part (A) of Proposition 1. of

e,

we have

e

=

be as in

Then by the defining property (3) which is indeed an isomorphism.

ids'

Q.E.D. Remarks 1. set, and

A

If

is an additive category,

i E: I,

A.

1

are objects in

A,

I

is an infinite

such that the direct

Gl A. and the direct product both exist, then the IT A. 1 iEI iEI 1 natural map e is in general neither a monomorphism nor an sum

epimorphism.

(In the category of abelian

groups it is of

course always a monomorphism.) 2.

Corollary 1.2 essentially characterizes all addi-

tive categories such that, for every object product:

AxA

exists.

Corollary 1.3. every obj ect Then

A (1)

Let A

A

inA,

A,

the direct

More precisely, be a pointed category such that, the di rect product

Ax A

for

exi s ts .

admits an additive structure iff For every object

A

in

A

the direct sum

AGlA

exists, and (2 )

For

every object

A

in

A,

the natural map

e:AGlA-+AxA is an isomorphism. Proof:

(We only sketch the proof, since we will make no use

of this elsewhere.)

Necessity follows from Corollary 1.2.

Conversely, suppose that conditions (1) and (2) hold.

Then

Section 3

22 for any object

A

follows that if

A,

in

since

jl,j2:A+A61A

and if we define

8

is an isomorphism, it

are the canonical injections,

11 =8 ojl,12=8 oj2'

a direct sum of

A

then

(AXA,11,12)

A.

with itself in the category

there exists a unique map in the category

is

Therefore

A,

PA:AXA+A, such that

If now from

B

is any other object in into

B

product of

A,

A

then since

A,

and

f

(A x A, 'TT l' 'TT 2)

and

g

are maps

is a direct

with itself, there exists a unique mapping

(f,g):B+AXA 'TTl ° (f,g) = f,

such that

'TT

2

° (£,g) = g.

Define f+g=PAo(£,g). Then we leave it as an exercise for the reader to verify that, for this definition of

A

that indeed Examples:

A,B

for all objects

in

becomes an additive category.

A, Q.E.D.

Example 1 above, the category of pointed sets, is

not an additive category:

Since if

(s,s) o

and

(T,t) 0

are

pointed sets, then the direct sum is "the disjoint union of and

T

with

(T x {U)I "',

So

glued to

where

to"

(s,O) '" (t,l)

(--Le., is iff

s = So

(SX{O})u and

t = to'

S

Abelian Categories all

s E S,

t E T);

23

and since the direct product (SxT,(s ,t », o 0

Cartesian product

is the usual

and the natural mapping is the "inclusion of the axes

into the product"

(i.e., is the function such that

8(t) = (s ,t), o

all

sES,

tET);

8(s) =

and although this

function is a monomorphism, it is not an epimorphism, either

card(S) or

card(T)

is one.

Example 2 above, the category of groups,

G

additive, since if the free product

G

and

*H

Cartesian product:

G

unless

H

is likewise not

are groups, the direct sum is

and the direct product is the usual

x H.

In this case, the natural mapping

G

is an epimorphism, but is not an isomorphism unless either or

H

8

is the trivial group (i.e., is of cardinality one).

Example 3 above, the category of abelian groups, is an additive category, if for groups, one defines

f,gEHomC(A,B),

f+gEHomC(A,B),

(f + g) (x) = f(x) + g(x),

all

A,B

abelian

by requiring that

x (;A.

Examples 4,5 and 6 above are not additive categories, since they are not even pointed categories. Remark:

If

A

is a pointed (respectively:

additive) cate-

gory, then it is obvious that the dual category is also pointed (respectively: to

B

in

AO

is defined to be

if the category

~A,B

on

Horn

additive); where the point

AO

A (A,B)

0B,A

from

in the category

A

in-

A;

is additive, then the binary composition ('" HomA(B,A»

in

is defined to be

and

Section 3

24

the composition

+B,A

additive category Defini tion 3.

where

K

HomA(B,A),

in the

A.

Let

C.

be a map in

on the abelian group

be a pointed category, and let

C

Then by a kernel of

is an object in

and

C

f

f:A -+ B

we mean a pair is a map in

l:K-+A

(K,l)

C

such that (1)

f

(2)

If

0

1

and such that

= OK, B

is any such pair(*),

(H, j)

then there exists a unique map

j

o

:H-+K

is commutative--that is, such that If

f:A -+ B

cokernel of

f

of

f

CO

is a pair

IT:B-+C

from (C,IT)

is a map in

C,

° ,

(1)

IT of = A,C

(2)

If

a unique map

(H,j) j

o

=1

0

jo'

is a map in the pointed category is a kernel of

dual category

j

such that the diagram:

:C-+H

B

f

iLto

where

considered as a map in the A.

C

then a

C,

Equivalently, a cokernel

is an object in

C

and

such that and such that

is any other such pair, then there exists such that the diagram:

(*)that is, if H is any object in a map such that f 0 j = 0H,B '

C

and if

j:H-+A

is

Abelian Categories

is commutative--that is, Proposi tion 2. be a map in (A.)

Let

C.

Let

such that

jo

0

25

IT

= j.

be a pointed category and let

C

f:A

-+

B

Then be an object and l:K

K

category

C.

(a) (K, l)

is a kernel of

-+

A

be a map in the

Then if and only if the fol-

f,

lowing three conditions all hold: (A.O)

f

= 0

1

0

K,B

is a monomorphism, and

(A.I)

(A.2)

Given any pair in f

C

and

j:H

-+

A

where

H

is a map in

is an object such that

C

th(re exists a map

j = 0H, B '

0

(H,j),

j

o

:H

-+

K

such that (B.)

Let

C

be an object and let

category

C.

(S)

is a cokernel of

(C,IT)

IT:B

-+

C

be a map in the

Then f,

if and only if the fol-

lowing three conditions hold:

= 0A,C '

(B.O)

IT

(B.I)

IT

(B.2)

Given any pair

f

0

is an epimorphism, and

in

C

and

j:B

Proof:

j0

0

IT

-+

H

where

H

is a map in

there exists a map

j of=OA,H ' tha t

(H,j)

is an object C j

such that o

:C'" H

such

=j .

Assume condition (a).

Then conditions (A.O) and (A.2)

Section 3

26

follow by Definition 3. ject and

Then

OK, B 0 h = 00, B.

j:D->A

0

be an ob-

1 0 h=l ok.

be maps such that

h,k:D~K

j=1 0 h=1 0 k.

On the other hand, let

is a map, and

Let

foj=f 01 oh=

By the universal mapping property, condition

(2) of Definition 3, we therefore have that there exists a unique map

j

:0

o

~

K

such that

obey this condition.

But both

Therefore

h = k.

This proves

Conversely, suppose that conditions (A. 0), all hold. H

Then by (A.O), we have that

is any object and

j:H

~

A

Suppose that Then

j~

j~,

1 0 jo = 1

this implies

is another such map, i.e.

I

and since by (A.l)

j 0 = j ~.

And, if

f oj = °H, B , 1 ojo=j·

that

1

oj~=j.

is a monomorphism,

Therefore there is a unique map

in condition (2) of Definition 3.

(A.l) and (A.2),

by passing to the dual categoLY.

be a map. (A.)

C

as

0

(B.l), and (B.2), fol-

lows from the equivalence of (a) with (A.O),

Let

j

This proves (4).

The equivalence of (S) with (B.O),

Corollary 2.1.

k

(A.!) and (A.2)

such that

jo

and

(A.2).

f 01 = 0K,B.

any map such that

then by (A.2) there exists a map

h

Q.E.D.

be a pOinted category and let

f:A

~

B

Then If a kernel

(K,l)

all kernels of

f

of

f

exists, then the class of

is a subobject of

A

in the sense

of section 2, Definition 1. (B.)

If a cokernel

(C,n)

of all cokernels of

of f

f

exists, then the class

is a quotient-object of

B

in the sense of section 2, Definition 1°. Proof: and

Suppose that a kernel of

(K',l')

are both kernels of

f

exists. f,

Then if

(K,l)

then by Proposition 2,

Abelian Categories (A.l) , we have that 2,

(K,

(A.O), applied to

And since

(K,l)

and

1 )

(K',

(K',l'),

1 ') E;

M . A

f,

l' o

(K',

1 ,)

.s.

(K,

such that

ment using that

(K',l ')

larly that

.s.

(K, l)

equivalent to

M . A

in

in

MA .

M , A

ThereMA

(see section 2),

Applying the same argu-

is a kernel of

(K' ,l ,)

(K',l ')

in

1 )

f ol' =OK',B

by condition (A.2) we

fore, by definition of the pre-ordering on we have that

By Proposition

we have that

is a kernel of

have that there exists a map

27

f,

we obtain simi-

Therefore

is

(K,l)

i.e., lie in the same sub-

object. Conversely, if and

(K,l)

(K' ,

1 ' ) 'V

(K,l)

is a kernel of

lie in the same subobject of

(K, l),

f, A,

and if i.e.

(K',l ')

if

I

then by section 2, Lemma 1, we have that there

exists a unique isomorphism

e:K' ~ K

therefore clearly by Definition 3 f.

Therefore the set of kernels of

of

A,

such that

(K',l ,) f

lo e =

1 '.

And

is also a kernel of

are an entire subobject

as asserted.

The latter part of Corollary 2.1 follows from the former Q.E.D.

part by passing to the dual category. I f now

in

C,

C

then we define the kernel of

be the class of exists.

all kernels

(K, l)

f, of

And we define the cokernel of

be the class of all cokernels f

exists.

f:A .... B

is a pointed category, and

(C,n)

f, of

Ker f ,

denoted f,

kernel of

f

object of

B.

Ker f,

is a subobject of

exists, then the cokernel,

Cok f,

denoted f,

to

i f a kernel of

f to

i f a cokernel of

Then, by Corollary 2.1. i f a kernel of

then the kernel,

is a map

A;

Cok f,

f

exists,

and if a cois a quotient:-

Section 3

28 Proposition 3. an object in

Let C.

C

be a pointed category and let

Z

be

Then the following eight conditions are equiva-

lent. (1)

OZ. Z '" id Z

in

(2)

Home(Z.Z)

has cardinality one.

( 3)

Home(Z.A)

has cardinality one. for all objects

A

has cardinality one. for all objects

A

in (4)

A.

HomC(A.Z) in

Hom (Z. Z) . C

A.

(5)

Z

is the direct sum of the empty set of objects.

(6)

Z

is the direct product of the empty set of objects.

(7)

There exists (equivalently:

A.

there exists a map

a kernel of (8)

idA:A

In condition (7) idA:A-+A

range)

(3)~(2)~(1).

has cardinality

have that

Horne (Z, A) f

0.

in

(Z. l)

objects

such that

is

A

(Z.71)

in is

(8)) one can replace the

A

5-1.

epimorphism)

A. Suppose (1).

and

Since

Therefore

Then if

f. g E Horne (Z. A),

= f 00 Z.Z '" 0 Z.A = g 0 OZ. Z '" g 0 id z = g.

HomC(Z.A)

(3).

71:A-+Z.

(respectively:

any obj ect in the category Z

For all)

with any monomorphism (respectively:

Obviously

f = f 0 id

that

A

idA:A-+A.

with domain (respectively: Proof:

such

objects

A.

there exists a map

a cokernel of

map

"* A.

There exists (equivalently:

A.

Note:

->

l: Z

For all)

A

is

'then

Therefore

0Z,AEHome(Z,A),

card Horne (Z, A) = 1,

we proving

Passing to the dual category, we see likewise that

(4)~(2)~(I)~{4).

We leave it as an easy exercise to the

reader to prove the equivalence of the remaining conditions and

Abelian Categories

29

(1), (2), (3) and (4).

Z

An object

Q.E.D.

in a pointed category

C

that obeys the

eight equivalent conditions of Proposition 3 will be called a zero object.

It is immediate that,

if a zero object exists,

then it is unique up to a unique isomorphism. symbol

0

Remarks: C,

We shall use the

to stand for the zero object. 1.

If a zero object exists in the pointed category

then every object

A

has a smallest subobject--namely, the

equivalence class of the element is any zero object in object of

A

objects in

We shall denote this smallest sub-

In a pointed category C,

C,

if

A

and

Bare

then when there is no danger of confusion, we 0A,B~

shall denote the point 3.

If

C

HomC(A,B)

O.

simply as

is a pointed category that does not have

a zero object, then we can "adjoin a zero object to wish.

Namely, let

category

C.

Z

be such that

Then define

C'

objects all the objects of object

Z.

Z

O.

by

2.

C.

where

C

Z

C"

if we

is not an object of the

to be the category, having for together with the one additional

Then define

HOmc(A,B), if A and B Hom ' (A, B) = { C (a set of cardinality one),

are objects of C, if

A

or

B = Z.

Then there exists a unique way of determining a composition such that gory.

C'

becomes a category and such that

And

C'

is a sub-cate-

is a pointed category with a zero object.

We do not, however, 4.

C

Let

C

insist on making this construction.

be a category (not necessarily pointed),

30

Section 3

and let

Z

be an object in

C.

Then condition (3) of Proposi-

tion 3 is equivalent to condition (5). equivalent conditions, then the category

C.

Z

If

Z

obeys these

is called a left zero object in

Passing to the dual category, we have like-

wise that conditions (4) and (6) of Proposition 3 are equivalent.

Z

is called a right zero object if it obeys these two

equivalent conditions. 5.

Let

C

be an arbitrary category.

Then the fol-

lowing three conditions are equivalent.

(1)

There exists an object

Z

in

C

that is

both a left zero object and a right zero object. There exists a left zero object

(2)

and a right zero object

ZR

in

C,

and

in

HomC(ZR'ZL)

C

is non-

empty. (3) zero object in that

There exists a right zero object and a left

C,

HomC(A,B)

and for all objects

A,B

in

C,

we have

t 0.

(4)

The category

C

is pointed, and there exists

a zero object. 6. gory

C,

gory

C

Example:

By Remark 5 above, if

then

Z

Z

is an object in a cate-

obeys condition (1) of Remark 5 iff the cate-

is pointed and

Z

is a zero object.

The category of sets has a left zero object (the empty

set) and a right zero object (any set of cardinality one) . Therefore, by Remark 5 above, the category of sets does not have a zero object.

More strongly,

by Proposition 3, it fol-

lows that the category of sets is not pointed (a fact that's obvious anyway).

Similarly, the category of non-empty sets

Abelian Categories

31

has a right zero object (namely, any set of cardinality one), but does not have a left zero object.

Again, by Proposition 3,

it follows that the category of non-empty sets is not pointed (a fact that is also obvious) . Corollary 3.1.

Let

C

a map in the category If

(A.)

f

be a pointed category. C.

is a monomorphism then

phism iff

Ker f

Ker f

exists iff a

C

is additive, and if

exists, then

f

is a monomor-

Ker f = 0.

(A.)

Proof:

be

Ker f = 0.

Conversely, if the category

either a zero object or

f: A -+ B

Then

zero object exists, in which case (B.)

Let

If

f

is a monomorphism and if a zero object

exists then one verifies using Definition 3 that a kernel of

f.

exists then

Z

(Z,OZ,A)

(z,d

On the other hand if a kernel

Z

of

is f

obeys condition (7) of Proposition 3, as modi-

fied in the Note to Proposition 3, and is therefore a zero object. (B.) f

Suppose that the category

be a map in

with

Z

C

such that a kernel

a zero object.

In fact, maps such that

let

D

To show that

be an object in

f ° h = f ok.

Then

C

is additive and let

(z,d

of

f

exists

f

is a monomorphism.

C

and

h,k:D-+A

be

f ° (h - k) = f ° h - f ° k = 0,

so that by the universal mapping property of Definition 3 there exists a map

t:D-+Z

a zero object, both h - k

= lot

Remark.

such that and

t

h-k=l °t.

But since

Z

is

are zero maps, and therefore

is a zero map, and therefore

h

= k.

Q.E.D.

Part (B.) of Corollary 3.1 does hold for some pointed

32

Section 3

categories that are not additive (e.g., the category of groups), but fails to hold for some pointed categories that are not additive. Example. (8,so)

If

is the category of pointed sets, then let 8

be any pointed set with

T={O,l} that

C

and let

f (s ) = t o

0

to={O}.

,

f (x)

of pointed sets, Definition 4. be a map in

= 1,

Let

Let C.

C

f: S

all

Ker f = 0,

of cardinality.:: 3, T

~

be the function such

xE;S-{s}. o

and yet

f

f

Then

f

is a map

is not a monomorphism.

be a pointed category and let (C,n)

Suppose that a cokernel

Then an image of

(I,k)

is a kernel

let

of

of

n.

f:A~B

f

exists.

Then by Corol-

lary 2.1, part (A.), and by section 2, the dual of Lemma I, if (C',n')

is another cokernel of

isomorphism

6

such that

(C.n)

of

f"

f

sufficient that a cokernel

kernel of

n

It follows readily that the

is independent of the choice of

f.

Thus. for an image of

some (equivalently:

then there exists a unique

n' = en.

definition of "an image of a cokernel

f.

all)

to exist,

(C,n)

of

it is necessary and

f

exist. and that for

(C,n)

cokernels

of

f.

that a

exist.

A co-image of

f

is an image of

in the dual category from

B

into

A.

f

considered as a map Using methods similar

to that of the proof of Corollary 2.1, we see that Proposition 4.

If

f:A .... B

if an image (respectively: the class of all images a subobject of

is a map in a pointed category, and a co-image) of

(respectively:

B (respectively:

f

exists. then

co-images) of

quotien~object

of

the sense of section 2. Definition 1 (respectively:

f A)

forms in

section 2,

Abelian Categories Definition 1

0

33

) •

The proof is elementary and is left as an exercise for the reader. If

is a pointed category and

C

C,

category

f:A

~

B

and if an image (respectively:

exists, then by the image (respectively: denoted

1m (f)

ject of

B

(respectively:

(respectively:

is a map in the co-image) of

the co-image) of

Coim(f»,

f,

we mean the subob-

the quotient-object of

ting of the class of all images (respectively: of

f

A)

consis-

all coimages)

in the sense of Definition 4 above.

f

We now make the conventions made in section 2, just after Definitions 1 and 10.

C, fix a com-

I.e., for every obj. A in

plete class of reps. for the subobjects of A, and also fix a complete class of reps. for the quotient-objects of Therefore, in particular, i f f : A ~ B such that

Ker f

(respectively:

Cok f.

Cok f.

(C, n),

Im(f), Coim f).

( I , k),

1m f.

We often call

the kernel (respectively: of

f,

Ker f

Coim f)

for this specific object

The map

l:K

A

K

Ke r f

exists,

Coim f)

(K,l)

(respectively:

(respectively:

c, I,J)

the cokernel, the image, the coimage)

and use the notation

~

in

(J , p) )

C,

is any map in

then we have a distinguished representative element (respectively:

A.

(respectively:

(respectively: K

n:B

Cok f,

(respectively: ~

C, k:I

-+

B, p:A

then called the canonical injection (respectively:

1m f,

C,I,J). -+

J)

is

the canoni-

cal projection; the canonical injection; the canonical projection) . Theorem 5. be a map in

Let C

C

be a pOinted category, and let

such that

Ker f,

Cok f,

1m f

and

f:A

~

B

Coim f

all

34

Section 3

exist.

Let

and let

1:

Ker f ->- A, and

71: B ->- Cok f

tions.

k: 1m f ->- B

be the canonical inj ections be the canonical projec-

p:A ->-Coimf

Then there exists a unique map a:Coim f ->- 1m f

such that the diagram (Ker

f

f)~}

f)

'r~ICOk

(Coim f)_a_:> (1m f)

is commutative. Proof: image,

We have since

e:

map Since

p (1)

Since that

f

01 ~O.

(Coim f, p)

(Coim f) ->- B

Therefore,

is a cokernel of

such that

e

0

p

~

f.

1,

Then

there exists a 71 08 0 P = 71 0 f = O.

is an epimorphism this implies that 7108=0.

(1mf, k) (1m f, k)

is an image of is a kernel of

we have that there exists a map 8 = ka.

by definition of the co-

f, 71.

we have by Definition 4 Therefore by equation (1)

a: Coim f

->

1m f

such that

But then kap=8p=f,

proving existence of from

Coim f

into

kBp=f.

u. 1m f

Now suppose that such that

8

is another map

Abelian Categories

35

Then kap=f=kBp. Since

k

is a monomorphism and

implies that

p

is an epimorphism, this

a = B.

Q.E.D.

If the hypotheses of Theorem 5 hold, a:Coim f

~

1m f

Examples.

deduced is called the factorization map of

In the category of pointed sets,

is a map, Cok f

then

Cok f

(T/~,

is

{to})'

tion on the set

t

T,

t,t' ~f(S); and where T~T/~).

map:

then the map

Coimf =5/f

-1

"~'

where ~

t'

=

if

T/f (S)

iff either

t

=

(T,t ) o

(more precisely,

t'

or both under the natural

the set-theoretic image, and

(i .e. ,

(to)

~

is the equivalence rela-

to is the image of to

Imf =f(S),

f: (S,so)

f.

glued to a point") .

Therefore, in this pointed category, the factorization map a: Coim f

~

1m f

is always an epimorphism-but is not,

in gen-

eral, a monomorphism. In the category of groups, i f f : G ~ H i s a homomorphi sm of groups, then generated by f (G»,

and

Ker f = f

f(Gj~

-1

(1),

Cok f = H/(the normal subgroup

1m f = (the normal subgroup generated by

Coim f = G/Ker f '" f (G) ,

the set-theoretic image.

Therefore, in this pointed category, the factorization map a:Coim f

~

1m f

is always a monomorphism, but is not always

an epimorphism. In the category of abelian groups, morphism of abelian groups, then B/f (A),

Im(f) = f (A)

and

if

Ker f = f-

f:A l

~

(0) ,

Coim f = A/Ker f.

this additive category, the factorization map

B

is a homoCok f =

Therefore in a :Coim f

-+

1m f

Section 3

36

is always an isomorphism. Example 8.

C be the category of all topological abelian

Let

groups and continuous homomorphisms. category, and the description of

Then

C is an additive

Ker, Cok, 1m, and Coim is

similar to case of the category of (abstract) abelian groups, above.

However, if

f:A-+B

is a continuous homomorphism of

topological abelian groups, and if factorization map, then

C!:

is the

Coim f .... 1m f

is always both an epimorphism and

C!

a monomorphism, but is not in general an isomorphism (since has the quotient topology from

Coim f

induced topology from Example 9.

from

C,

Ker f = f

Cok f = B/f (A),

B.

1m f = f (A) ,

to~

A

f:A -+ B

abelian

is a map

with the induced topology

(0)

imag~

B

by the

with the quotient topology

the closure of the set-theoretic image

with the induced topology from quotient-group of

-1

Then if

the quotient-group of

closure of the set-theoretic from

has the

1m f

C be the category of all Hausdorff

Let

we have that A.

while

B).

groups and continuous homomorphisms. in

A

B,

and

Coimf =A/f

-1

(0),

with the quotient topology from

this additive category, the factorization map of a map

the

A.

In

f

is

always a monomorphism,but is not in general an epimorphism. The next definition is very important. Definition 5.

An abelian category is an additive category,

such that (ABl)

Finite direct sums of objects exist.

(AB2)

Kernels and cokernels of maps exist.

(AB3)

If map

f:A-+B C!:

is a map in

(Coim f) -+ (1m f)

A,

then the factorization

is an isomorphism.

Abelian Categories Remarks 1.

and

(AB1)

37

is equivalent to saying,

(ABl.l)

There exists a zero object,

(ABl. 2)

If

A

and

B

objec~

are

exists a direct sum 2.

Since

3.

1m f

and

A,

then there

A 6l B.

Imf =Ker(Cokf) ,

axiom (AB2) implies that

in

Coimf = Cok(Ker f) exist.

Coim f

In view of axiom (ABl) and Corollary 1.1, if a

A admits an additive structure such that it is an

category

abelian category, then that additive structure is unique. Therefore, an abelian category can be thought of, equivalently, as being a category A

structure on

A,

such that there exists an additive

such that axioms (ABl) ,

(AB2) and (AB3) all

hold. 4.

Suppose that we have an additive category

that axioms (ABl) and (AB2) both hold. If

f:A -+ B

is a map in

A

such

Suppose also that

then the factorization map

is both a monomorphism and an epimorphism.

a:(Coimf) -+Im(f) Then is

A,

A

an abelian category?

To the best of my knowledge,

this question has not yet been settled.

(See, however, the

last paragraph of section 4 below.) Example.

Of Examples 1-9 above, the only one that is an abelian

category is Example 3, the category of abelian groups. Example 10.

The category of all left modules over a fixed ring

with identity Example 11.

R

is an abelian category.

The category of all sheaves of abelian groups

on a fixed topological space Example 12.

If

A

X

is an abelian category.

is a category, and if

that isa set, then we let

C A ,

C

is a category

the exponent category, denote

S(X)

38

Section 3

the category having for objects all covariant functors from C

into

tors.

and for maps all natural transformations of func-

A,

(This useful notation is original to Joseph D'Atri of

Rutgers University, Newark.)

A is an abelian (respectively:

Then if

C

pointed) category, and if

C A

then

additive;

is any category that is a set,

~s an abelian (respectively:

additive; pointed) cate-

gory. Example 13. category

A be an abelian category.

Let

AO is an abelian category.

Example 14.

Let

A

pointed) category. complexes

Definition 6.

be

a

Then the category

of all cochain

indexed by all the integers is an abelian nE;?" additive; pointed) category.

Let

A

be an abelian category, and let

sequence of maps in the category (1)

subobjects of

B.

Example.

A

Let

is exact at spot

F~G

B

A. iff

Then we say that Ker g = 1m f

be an abelian category and let

gory that is a set.

AB

Co(A)

additive;

f g A -->B-->C

the sequence

in

be an abelian (respectively:

n (en,d )

(respectively:

(1)

Then the dual

S

as

be a cate-

Then a sequence

__ H

is exact at spot

the sequence F(B) -+G(B) -+H(B)

G

iff for every object

B

of

S,

Abelian Categories is exact at spot

G(B)

39

in the abelian category

A.

A is an abelian category, then a sequence:

Similarly, if A*-+B*-+C*

of

7-indexed cochain complexes in

for every integer A

n

-+B

n

is exact at spot

n,

-+C

A is exact at spot B* iff

the sequence

n

Bn.

In Example 11 above, in the category

Six)

of abelian groups on a fixed topological space

of sheaves of abelian groups iff for every F

x

x E X,

-+G

x

-+H

in

SiX)

of sheaves X, a sequence

is exact at spot

G

we have that the sequence of stalks:

x

is exact in the category of abelian groups at spot

G .

x For the rest of this section, we state several theorems,

for arbitrary abelian categories

A,

all of which are easy to

prove, or well-known, in the case that abelian groups.

A is the category of

By the Exact Imbedding Theorem, see sect.4, we

therefore have these results in all abelian categories. Proposition 7. f (1)

A be an abelian category, and let

Let g

A-->E~C

be a sequence of maps in of

B

iff

A.

Coker f = Coim g

Then

Ker f = 1m g

as subobjects

as quotient-objects of

B.

40

Section 3

Note:

By Definition 6 above, an equivalent statement is:

"Then the sequence (1)

A

category

is exact at spot

B

in the abelian

iff the sequence

g f C_>B-->A in the dual category Proof:

AO

is exact in

The proof is easy if

groups.

A

AO

at spot

B.

is the category of abelian

By the Exact Imbedding Theorem (see 'section 4),

it follows that it is true for all abelian categories.

8.

Proposition

Let

A

be an abelian category, and let

an object in

A.

(K,i)

associates the quotient-object

of

A

A

be

Then the function which, to every subobject is a

Coker(i) ,

one-to-one, order-reversing, correspondence from the class of all subobjects of of

A.

The inverse correspondence is also order-reversing, and

is the one

that, to every quotient-object

ciates the subobject Remark:

A onto the class of all quotient-objects

Ker

~

(C.~)

of

A,

asso-

.

In the statement of Proposition 8 above, we must make

the conventions of section 2, just after Definitions 1 and 1

0

,

since otherwise one cannot speak of the "class of all subobjects" or the "class of all quotient-objects" of an object Proof:

Same as Proposition 7.

Proposi tion Let

A

9.

(Fundamental Theorem of Homological Algebra) .

be an abelian category, and let f*

(1)

A.

0 ....

g*

A*~B*---l>C* -+

0

Abelian Categories

41

be a short exact sequence(*) in the abelian category ~-indexed

of all

category

A

cochain complexes

(Example 14 above).

Co(A)

of objects and maps in the

Then there is induced a

long exact sequence of cohomology: n-l n n n d_ _> Hn (A*) H (f*» Hn (B*) H (g*» Hn (C*) ~>Hn+l (A*)

-l>-

•

Moreover, given a commutative diagram with exact rows,

o-:r rl f*

g*

,:t"

O_ _ 'A*~'B*~> ·C*

0 4-

0

in the abelian category eo(A), we have that the diagrams:

commute in the category Proof:

If

A

A,

for all integers

n.

is the category of abelian groups, then this

theorem is well-known, see [C.E.H.A.l.

By the Exact Imbedding

Theorem (see section 4 below), it follows that it holds for all

abelian categories

A.

Finally, we conclude this section with a few elementary definitions about functors of abelian categories. Definition 7. F:A~>

gory

S

Let

A and S be additive categories and let

be a functor.

Then we say that the functor

F

is

(*)ThiS means that the sequence (1) in the abelian cateCo(A) is exact at spots A*,B* and C*.

••

•

Section 3

42

addi ti ve iff whenever A

f, g:

A

I

~

A

are maps in the category

then we have that F(f + g) = F(f) + F(g)

in Remark.

F:A~>

By Proposition 1, we have that, if

B

is an

additive functor of additive categories, then

F

preserves

finite direct sums when defined.

n

is an integer

~o,

and if

sum sum

AI' ... ,A

Al (Jl ... (Jl An

That is, if

are objects of

n

A

exists in the category

F(A ) al ••• (Jl F(A ) n l

such that the direct A,

t hen the direct

exists in the category

(3,

and the

natural map is an isomorphism:

Conversely, by the proof of Corollary 1.1 , if (3

are additive categories, and if the category

that, for every object then a functor

A

in

A,

the direct sum:

is additive iff

F:A~>(3

A

F

A

and

is such A(JlA

exists,

preserves finite

direct sums when defined iff F preserves A(JlA, for all A in

A.

In particular, a functor of abelian categories is additive iff it preserves finite direct sums of objects. Definition 8. F:A~>

(3

Let

A

and

be a functor.

(respectively:

B

be abelian categories, and let

Then the functor

right exact; exact) iff

F F

is left exact is additive, and in

addition whenever

o~

f I

fll

A'---:>A~A"-"O

is a short exact sequence in the abelian category

A,

then the

Abelian Categories

43

sequence: F(f' ) F(f") 0 .... F(A') ----'"p(A) -----...;*'(A") (respectively:

the sequence

F (f')

F(f")

F(A')--~)F(A)--~)F(A") -+

0;

the sequence F(f' ) 0 .... F(A')

---~*,(A)

F(f") ----'»F(A") .... 0)

is exact in the abelian category Remarks 1.

An additive functor

exact (respectively: (respectively: 2.

B. of abelian categories is left

right exact) iff it preserves kernels

cokernels).

(See [C.E.H.A.l ).

It is not difficult to show that a functor of

abelian categories is left exact (respectively: iff it preserves finite inverse (respectively: limits.

right exact) finite direct)

(This can be interpreted as meaning "direct (respec-

tively: inverse) limits over finite pre-ordered sets, as defined in [c.A.l": or the more general "direct (respectively: inverse) limits over fini te

categorie~1

(meaning categories that

have only finitely many objects and such that all of the Hom's are fini tel ) . 3.

A

functor of abelian

categories is exact iff it

is both left exact and right exact. The

next section is a faithful reproduction of an earlier

paper by the author,

[I.A.C.l.

Appreciation is expressed by

the author to the original publisher for allowing such reproduction.

Section 4 Imbedding of Abelian Categories

1.

Introduction.

In this section we prove the following

EXACT IMBEDDING THEOREM.

Every abelian category (whose objects

form a set) admits an additive imbedding into the category of groups which carries exact sequences into exact se-

a~elian

quences. As a consequence of this theorem, every object has "elements"- namely,

the elements of the image

A A'

A

of of

A

under the imbedding--and all the usual propositions and constructions performed by means of "diagram chasing" may be carried out in an arbitrary abelian category precisely as in the cateqory of abelian groups. In fact,

A

if we identify

with its image

imbedding, then a sequence is exact in an exact sequence of abelian groups. image, and coimage of a map image, and coimage of map

f

f

f

of

A

A'

under the

if and only if it is

The kernel, cokernel,

A are the kernel, cokernel,

in the category of abelian groups; the

is an epimorphism, monomorphism, or isomorphism if and

only if it has the corresponding property considered as a map of abelian groups. of

A

The direct product of finitely many objects

is their direct product as abelian groups.

then every subobject of

A

is a subgroup of

A,

tersection (or sum) of finitely many subobjects of

44

If

A E A,

and the inA

is their

Imbedding of Abelian Categories

45

ordinary intersection (or sum); the direct (or inverse) image of a subobject of

A

by a map of

direct (or inverse) image.

are maps in

A

is the usual set-theoretic

Moreover, if

and the set-theoretic composite

A,

then this composite AO -+ A + l ' 2n is the image of a unique map of A, this map of A being inis a well-defined function

dependent of the exact imbedding

A~>{abelian

groups} chosen.

In particular, many of the proofs and constructions in le.A.)

remain valid in an abstract abelian category-e.g., the

Five Lemma, the construction of connecting homomorphisms, etc. If the abelian category

A

is not a set, then each of its

objects is represented under the imbedding by a group that need not be a set. I am very grateful to Professor S. Eilenberg for the encouragement and patience he has shown during the wri ting of this paper. 2.

Exhaustive systems.

In

§§2-4,

A

denotes a fixed

set-theoretically legitimate abelian category, and

E

the cate-

gory of abelian groups. An exhaustive system of monomorphisms in an abelian cate,) 'ED' of monomorphismf gory A is a non-empty direct system (A"a, ~ ~J ~ in A such that: (E) If iED exists

j,;i

Lemma 1. in

A

such that

If

(1'•. , a .. ) ~

~J

and f:Ai-+B is a monomorphism then therE 0ij=f.

is an exhaustive system of monomorphisms

then (1.1)

there exists

If

f,f':A'-+A iE D

are maps in

such that

A

and

flf'

Hom(f,A ) I Hom(f' ,Ai)' i

then

46

Section 4

(1.2)

f:A .... B

If

is a monomorphism, j > i

a map then there exists

Proof.

Let

(1.1).

iED

A.=A(!)A .. J ~

Hom (f , Ai)

and

(1.2).

A -> B (!) Ai

I

,

D

D

in

(C,y)

3.

so that

h A.--------;>.;C

g

k

k

A-----f

is the composite

Ai .... B (!) Ai .... C,

we have

there exists

kf = Ctijg,

Then

hE Hom(A,A.), J

f

(E).

But

f, -g.

/"

By

be the injection.

i

be the cokernel of the map

A------~

site

i

such that

y

B (!) Ai

h

J

be the injection.

h Ai -----'!>-:.c1.:

Then if

k: B .... A.

g:A->A

such that

disagree on

with coordinates

g

and

A j) .

Let

~

and

g:Ai->A(!)A

h:A->A. J

Hom (f I ,Ai)

Hom (f , A j) "I Hom (f Proof.

Let

and j ~ i

Then by (E) there exists then

in

iED

j > i

B .... B (!) Ai .... C,

and

kf = hg,

k

and

such that

k

the compo-

is a monomorphism.

(). .. = h. 1J

Then

as required.

Q.E.D.

The basic construction.

In this section, we shall

prove Theorem 1.

If

A

is an abelian category then there exists an

exhaustive system of monomorphisms in

A.

We first obtain some preliminary results. Lemma 1. phisms in

If

A,

(Ai,Ctij)i,jEI and

is a directed system of monomorf:A. .... B 10

is a monomorphism,

Imbedding of Abelian Categories

47

then there exists a direct system of monomorphisms extending

Let

I' = I U J*

J*

be a set equipot9nt to

j -> j *

be a bijection from

j, k

i E I,

i

~

For A.

E J,

j

~

~ j

let

J,

j* iff

*

~

and diSjoint from

J

onto

iff

k*

I j ~

J* .

Let

as a directed subj E

for

k;

J

and

i ::. j .

be the cokernel of the map

Aj *

with coordinates the maps

J

-f:A.

and

i,

+ A. Ell B

1.0

J

be the directed set containing

set such that, for *

I'

be the directed subset of r consisting of all

J

i ~ i O' Let r, and let

j

in

j > i

a ij = f.

such that Proof.

and a

a . . :A.->A. 1. 0 J

and

1.0 J

Then we have a commutative diagram of monomorphisms:

+B.

1.0

B

f'

fl A. 1.0 a.1.

oA.

0

,j

J

is functorial,

The assigment

so

that for

we have a monomorphism i E I,

and

i

j*

~

in

I',

then define

a .. * 1., J

<x ••

Ai ~Aj ->A j *.

posite

A.

in

Then

familY of monomorphisms in

in

j ~ k

J,

If to be the comis a direct

A extending

(Ai,aij)i,jEI

and Q.E.D.

Lemma 2. in

A,

If

(Ai'<Xij)i,jEI

then there exists a direct system

monomorphisms extending i E rand j ~ i

is a direct system of monomorphisms

in

f:A.

1.

I'

-+ B

with

(Ai,uij)i,jEI

is a monomorphism in a .. = f. 1.J

(Ai,aij)i,jEI'

such that, whenever

A,

there exists

of

48

Section 4

Proof.

Let

and

is a monomorphism of

f

P

be the set of all pairs

by a relation

P

family

;,.

(A, ,a., ,).

. "I

Having defined put

D(k,f) =IU

family

A . k

k EI

Well-order

define a directed

of monomorphisms as follows:

1J 1,

"

for

(k',f') < (k,f),

(k', f ' )

(k',f'Y«k,f)I(k',f')' Then apply Lemma 1 to the

(A. ,Cl. .. ). 1

(k, f) E p,

(A" Cl. .. ), j"I ].

where

A with domain

For each

1J 1,J" (k,f)

1

(k, f)

let

'CD

1J 1,J" (1<, f)

(A. ,Cl. .. ). 1

'CI

1J 1,J" (k,f)

be the

family obtained. Put

I' =

(k,~)EP I(k,f)" Then the family

(Ai,Cl.ij)i,jEI'

has the desired properties. Proposition 1.

If

££QQf.

(Ai,Cl.ij)i,jED

A,

monomorphisms in

Q.E.D. is any direct system of

then there exists an exhaustive system

For each non-negative integer

let

q,

be defined inductively by: DO = D. (A. ,Cl. .. ). 1

Having defined

'ED

1J 1, J

Lemma 2 to

(Ai' Cl. ij ) i, j ED

'

let

q

be the direct system obtained by applying

q+l

(Ai,Cl.ij)i,jED • q

putting

D' =

~oD

q=

q

is an

we see that

,

Q.E.D.

exhaustive system of monomorphisms. Proof of Theorem 1. single element

i.

Let Let

be the identity map of

D Ai A .• 1

system of monomorphisms of

be a directed set consisting of a be any object of

A,

and

Cl.,

.

1,1

Then

A.

Proposition 1 completes the

proof. Remark.

Note that the cardinal number of the indexing set of

Imbedding of Abelian Categories

49

the exhaustive systems we have constructed is that of the set of monomorphisms of

4.

A.

(This is so even if

The limiting process.

system of

F:A*-+ E

by

(A.

~

D.

is not a set.)

be an exhaustive

,CI. . . ) ~J

in the abelian category

mo~omorphisms

by the directed set

Let

A

A,

indexed

Define a functor F(f) = l,tm Hom(F,A ) . i

F(A) = l,tm Hom(A,Ai)'

~

~

Then it is easy to see that

F,

being the direct limit of

additive, left-exact functors mapping into

E,

is itself left-

exact and additive. Lemma 1.

The functor

F

is an exact imbedding(l) from

A*

E.

into Proof.

If

A-IA'

Hom (A' ,A.) = ¢.

A,

in

i E D,

then for each

Hence, the abelian groups

F (A)

Hom(A, A.)n ~

and

F (A' )

~

are disjoint and therefore distinct. If exists maps:

f

l

,f :A' -+A 2

i ED

are maps in

such that

Hom(A,A ) -+F(A), i

A,

then by (1.1) there

Hom(fl,A ) -I Hom(f ,A ). i 2 i Hom(A',A ) -+F(A'), i

limit of monomorphisms in the category

E,

The canonical

being the direct are monomorphisms.

Hence, considering the commutative diagrams F(f .) F(A)

J

I

>F(A')

1

Hom (f. ,A. ) Hom(A,A. ) _ _ _~J'--_~-7> Hom(A' ,A.) ~

~

(l)Recall that an imbedding from a category A into a category B is an isomorph~sm from A onto a subcategory of

B.

50

Section 4

for

j = 1, 2,

F(f ) i- F(f ). l 2

we see that

Hence,

F

is an im-

bedding. If

f:B-+A

is an epimorphism of abelian groups. then let

g EHom(A,A.)

k:B-+A.

g:A-+A.

1

as in (1.

J

represent

1

morphism and

Hom (A, a .. ) g. lJ by k. Then

Let

A*,

is an epimorphism in

2) •

In fact,

g.

kf = a

k EF(B)

ij

g,

F(f):F(B) -+F(A)

if

g

f:A-+B

Then

is a map; choose Then

then

in

j ~ i

~F(A),

is a monoD

and

Hom ( f , A . ) k =

i.e. ,

J

be the element of

F(B)

represented

chasing elements in the commutative diagram:

kEF (B) _ _F_(.c-f _)_ _----.,) F (A~

1

Hom (A, A. ) 3 g

j

HOm(A, Hom(f, A.) k E Hom (B , A . ) _ _ _ _ _J"'--~) Hom (A, A . )

we see that

J

a:

j)

J

F(f) (k) =g,

as required.

Since we have already observed that

F

is left exact,

the theorem follows.

Q.E.D.

Applying Theorem 1 and Lemma 1 to the dual category

A*,

A admits an (additive) exact imbedding into

we see that

E,

which proves the Exact Imbedding Theorem. 5.

Generalizations.

It should be clear that the proof

of the Exact Imbedding Theorem can be extended to arbitrary categories. If equalizer

In this section, we state such a generalization.

f,g:A-+B E(f,g)

subobject of category,

A

are maps in the category of

f

on which

and

g

f

and

E(Lg) =Ker(f-g),

A,

thenthe

(when defined) is the biggest g

agree.

Ker(£) =E(f,O).

In an additive so that

Imbedding of Abelian Categories

51

"equalizer" is a generalization of "kernel" to arbitrary categories. Theorem.

Let

A be a set-theortically legitimate category

closed under direct products of two objects and equalizers of two maps.

Then the following three conditions are equiva-

lent:

A admits an imbedding into the category of nonempty

1.

sets preserving monomorphisms, epimorphisms, equalizers, and finite direct products.

A

2.

admits an imbedding into the category of nonempty

sets preserving epimorphisms, equalizers, and finite direct products. 3.

with

g

Every diagram:

an epimorphism can be imbedded in a commutative

diagram:

A--+B

with

h

AxB-+A,

an epimorphism.

Moreover, the canonical map,

is an epimorphism.

In addition, if the category then we

A is pointed or additive,

may assume that the imbeddings in 1 and 2 take values

in the category of pointed sets or abelian groups.

52

Section 4 6.

Element techniques in an abstract abelian category.

In this section, we prove the assertions made in

E.

exact imbeddings into

§l

about

We consider only abelian categories.

It is easy to see that, in an abelian category, if with

k

a monomorphism and

image and

i

i

an epimorphism, then

is a coimage of

f.

f = ki

k

is an

This is seen by considering

the commutative diagram: ---~)L

I~----

where

K,L

and

and image of Theorem 2.

f, Let

are, respectively, the kernel, cokernel,

and F

a

Al~A~A3

be an exact imbedding from the abelian

Necessity is clear; let us prove sufficiency.

Hence, if

such that

of

is exact in

.!...LtlF (A ) 3

since

Then a sequence

B.

that fg=O.

B.

A if and only if the image

is exact in

sequence-is exact in Proof.

is uniquely determined.

A into the abelian category

category (1)

I'

k=kernel(f)

g=ki.

But

whence

a monomorphism in and (1) is exact in

i

A,

Then

then there exists

F(k) = kernel (F(f))=image(F(g))

F (g) = F (k) F (i) ,

F(g),

in

B.

Suppose

it follows that

F(i)

is an epimorphism in

A.

Since

9 = ki

A;

Henceforth, fixing an exact imbedding into

S;

is the coimage clearly

it follows that

A.

in

i

k

is

k = image (g) , Q.E.D.

E for each

abelian category, we use freely the set-theoretic properties

53

Imbedding of Abelian Categories of objects and maps in abelian categories. If

A,BEA,

subobject

R

then an (additive) relation

of

AxB.

If

R:A-+B

then their composite

SR

A x B x B x C -+ A x C

(R x S)

diagonal of

of

B x B.

and

R:A-+B

S:B-+C

is a

are relations,

is the image under the projection

n

(P.

x /':, x C),

where

/':,

is the

Since this coincides with the set-theoretic

definition, composition of relations is associative. Similarly, one defines the inverse relation proves that (R-l)-l If A -+ A

x B

f:A-+B

R

-1

and

,

=R.

isamapin

A,

wi th coordinates

then the image of the map is the graph

f

it is the ordinary set-theoretic graph. one-to-one and preserves composites. R · f The relation R

R f

of

f;

Hence,

We identify the map

f

with its graph

the composite F:A

->-

B

is a

R -+ A x B ->- A

(graph of some) map if and only if is an isomorphism.

Hence, if

R

is an exact imbedding of abelian categories, then

is a map if and only if

F(R)

is a map.

If in the abelian category

A,

is the

are maps and the relation graph of a map, then this map: inverse-composite and is denoted

AO

->-

A

+ 2n l

is called their

-1

-1

fng n ···gl fo:AO-+A2n+l'

Using the Exact Imbedding Theorem and Theorem 2, we obtain: Theorem 3.

(Axiomatic Definition of Inverse Composite).

complete characterization of the inverse-composite -If gl 0

of maps

A

Section 4

54

in an abelian category is given by: (ICl)

In the category

composite

E of abelian groups, the inverse-

is defined if and only if the corres-

ponding set-theoretic composite is a single-valued function defined on all of (IC2)

If

AO;

in which case the two coincide.

F:A~>

B is an exact imbedding of abelian f ng

categories, then the inverse-composite fined in

A

if and only if

is defined in F(fn)F(gn)

-1

B;

F(fn}F(gn}

in which case,

... F(gl)

-1

-1

-1

n

-1

... gl fa

... F(gl)

-1

F(fng n ... f

-1

l

-1

is de-

F(f )

o

fa)

F(f o)'

Combining Theorems 2 and 3, we obtain all the properties of exact imbeddings asserted in the Introduction. Added in proof (October 25, 1960).

The theorem of

§5

can be used to characterize abelian categories: Corollary.

If

A

is an additive category such that kernels,

cokernels and finite direct products exist, then

A

is abelian

iff the following three conditions hold: (1)

with

g

diagram:

Every diagram:

an epimorphism can be imbedded in a commutative

Imbedding of Abelian Categories

with

h

an epimorphism.

(2)

Every diagram:

55

A.{-(----- B

with

g

a monomorphism can be imbedded in a commutative

diagram:

hf+----'

19

A~(-----B

with

h

a monomorphism.

(3)

A map that is both an epimorphism and a monomorphism

is an isomorphism. Proof.

In view of (I) and the theorem of

addi tive imbedding

A .... A',

g .... g'

§5,

is the canonical factorization of a map f =;

Coim ( f ' )

k 'z' = 1m (f' );

in

E;

since

whence

k'z',z'

phisms in, respectively, dual category by (3)

z

A*,

f' =k'z'h',

E,E

we see that

is an isomorphism.

Hence, if of

z

A.

A,

f=kzh

then

we must have

and therefore and

admits an

E preserving kernels

into

and epimorphisms, and hence also coimages.

h'

A

z

are monomor-

Applying (2) to the

is an epimorphism.

Hence Q.E.D.

Section 5 Subquotients

A be an abelian category and let

Let objects. from

Then recall

A

into

A

jects in

B

and

B

be

([I.A.C.]). that an additive relation

is a subobject of

and

A

R:A.,. Band

Ax B.

S:B.,. C

If

A.B.C

R

are ob-

are additive relations.

then see [I.A.C.] for the definition of the composite relation S

0

R:A -+ C;

R-I:B-+A

and we also have the inverse relation of ([LA.C.]).

have the graph of

f.

----

into

Remark.

Let

ject in

A.

iff

of

A.

Let

B B

B

an additive relation from

A

A

be an ob-

be another object in the category Imbeddin~i

is isomorphic to a

A.

Theorem. it is easy to prove subobject of

subobject of a quotient-object

B B

-1

R = r lor 'IT

B

A be an abelian category and let

Let

A.

Then consider the class

is an object in into

a monomorphism

where

then we

This motivates the following definition.

object in

from

A.

is isomorphic to a quotient-object of a

Definition.

where

rf:A-+B.

is a map in

A be an abelian category and let

Then using the Exact

A

f:A-+B

([LA.C]).

B

that:

If

R.

A.

A and

R

SA

is an object in

SA

'IT:C -+B

C

56

R

in

A.

such that

is the class of all pairs

A and

(B.R)

is an additive relation

and an epimorphism

(equivalently:

be an

of all pairs

such that there exists an object

l:C -+A

A

(B. R)

is an additive relation

57

Subquotients

from

A,

B

into

A,

such that there exists an object

an epimorphism

that

R = r-

l

p:A ->- C'

or.). J

p

(B' ,R'),

iff

that

A.

ject in class

Then (1)

R' oR=R" (2)

(B",R")

~

(B' ,R')

the additive relation

such

A

be an ob-

in the pre-ordered such that

R:B"->B'

is uniquely determined by this condition, and

By the Exact Imbedding Theorem,

abelian groups.

A

If

(B' ,R') E SA '

and a monomorphism

TT' :C' ->- B'

such that

A

(B",R") E SA'

Then replacing and of

there exists

and an

l":CII-+A

and that

C"

C'

and

C"

Similarly, since

•

A

in

and a monomorphism such that

and

C'

C"

and that

natural maps.

B'

by

Similarly,

R'

1 "

o

TT': B' ->- C'

and

Then from the relation

{(a' +K',a'):a'E C'}.

by

B' =C'/K',

TT":B"->-C" R'=r

B"

1 '

are the or- l TI

I

'

we de-

-1

r 7f

B'""C'/KerTT', and

C'/Ker TT'

duce

(1)

r

are subgroups of

Then

if necessary, we can assume that ,

R" =

by the set-theoretic images of

are the inclusions.

Replacing

C'

and an epimorphism

'TTII:C" -+BII

epimor~hism

1',1"

B"""C"/Ker TT".

Ell = CIl/K Il

l orr; I

I

we can assume that

1",

C"/Ker TT"

1

is the category of

there exists an object

1': C' ->- A

R'=r

it suffices

[LA.C.l,

to prove the Lemma in the case in which

A,

~

(B" ,R) E SB' .

Proof:

in

by

R: B" ->- B'

be an abelian category and let

Suppose that

SA"

SA'

Then,

A

Let

such

(B",R")

that

there exists an addi ti ve relation

R' °R=R".

Lemma 1.

(B",R")E SA'

in

j : B ->- C'

We have a pre-order on the class

defining, whenever (B' ,R')

and a monomorphism

C'

II •

Section 5

58

(2) If

{(a" +K",a") :a" EC"}.

R"

(B",R").s.(B',R')

in

an additive relation

then by definition there exists

R:B"-+B',

such that

R' °R=R".

(3)

If

SA'

ailE C",

therefore

then

(a" +K",a") ER' oR.

there must exist (0.,13) E R,

(a"+K",a")E R".

suchthat

particular

a

II

Therefore, by equation (1)

(so that

a'E C'

a

II

E C"

a

I

(a' + K I , a') E R'»

and

(a,a') = (a" +K",a").

S=a'+K',

= a' E C'.

By equation (3),

being arbitrary,

In

it follows

that (4)

C" CC ' •

Also, we have a

= (a

II

+ K").

(5) If

(a,S)ER,

= all

Therefore

{(a"+K",a"+K') :a"E C"}c R.

h' E K'

then by equation (1),

in equation (5), we have that equation (3), we have that

(aU+KII,a ll ) ;

II

E K ".

a" E C" a" =h'

h' E K'

Taking

a" = 0

Therefore by

(K",h')E R' oR=R".

i.e., such that

h' = a

(K' ,h') E R'.

(K",K') E R.

tion (2) there therefore exists

Therefore

S=a'+K'=a"+K',

J

But by equa-

such that and such that

(K", h') =

ailE K".

being arbitrary, we have like-

wise that (6)

K'C K".

Next, suppose that

(\!,S)E R,

where

a'E C'.

a" EC"

and

say

a=a"+K",

S=a' +K',

Then by equation (1) we have that

59

Subquotients (a' + K' ,a') E R'.

Therefore from equation (3) we deduce

(a"+K",a')ER'oR'=R". a'EC"

and that

But then by equation (2) we have that

where

(a"+K",a' +K') '= (a' +K",a' +K') C II

a'

is an element of

is an arbitrary element of

(a,5)

Since

•

(a, 5) '=

Therefore

a"+K"'=a'+K".

comparing

R,

with equation (5), we see that R'= {(a" +K",a" +K') :a"E cOIl.

(7)

Therefore, if there exists then indeed

R

R

such that equation (3) holds,

is uniquely determined.

case, the pair

(B", R)

And, when this is the

is indeed an element of

Q.E.D.

SB"

Following the methods of the last Lemma, one can also deduce the following related results. (B' ,R')

If

Corollary 1.1.

(B", R") E SA

and

A,

object in the abelian category

I

where

A

is an

then the following two

conditions are equivalent. (B" ,R") .:. (B' ,R')

Both

2)

There exists an isomorphism

the abelian category R'

0

and

1)

fe'= Ril,

A,

(B', R') .::. (B" ,R") e

from

B"

in

onto

SA·

B'

in

necessarily unique, such that

as subobjects of

B"xA.

This motivates another definition. Definition. object in

A be an abelian category and let

Let

A.

of pairs in

Then a subguotient of SA

where two pairs

I

are eguivalent iff both in

(B' ,R')

~

A

A be an

is an equivalence class

(B' ,R'),

(B", R")

in

(B" ,R")

and

(B", R")

.s.

(B' ,R')

SA·

ExamQle 1. object in

Let

A.

A Let

be an abelian category and let cII,e I

be subobjects of

A

A

be an

such that

C" cC' .

60

Section 5

Let

B

be the quotient- object

inclusion and (B,R)

IT:C'

-+

C'/C"

is an element of where

C' /C".

If

l:C'

-+

A

is the

is the natural map, then the pair

SA '

and therefore defines a sub-1 or . IT

quotient of

A,

quotient of

A

the

subquotient

Corollary 1.2.

Let

A be an abelian category and let

A.

an object in

If

R=r

1

{(B,R)}

there exist unique subobjects COl eC',

and such that

We will call this sub-

C' /C".

is a C'

subquotient of and

C"

of

A

A,

A,

be

then

such that

{(B,R)} = C'/C".

Of course, this Corollary is proved along the same lines as the Lemma. Example 2.

A.

category of

A.

Let

C

be any subobject of

A

Then by Example 1, we have the subquotient

Notice, by the last Corollary, that, if

subobjects

of

(Explicitly, if subquotient of

A,

then, as subquotients of

C A

is a subobject of

A,

(O,R)

A,

Thus, the "object part" of the subquotient are the same even if

and C/C

CiC

C/C are

O

C/c"I CO/CO. C/C

is the

where

0

is

R = 0 x C(cO x A). and

CO/CO

C t- CO' - namely, both are the zero object-

but the "relation parts" of

C/C

and of

CO/CO

are different

C"lCO).

Corollary 1.3.

A.

A,

then

represented by the pair

the zero object in the abelian category

if

in the abelian

Let

Q

and

Let Q O

A

be an object in the abelian category

be subquotients of

Corollary there exist unique subobjects such that Q = CO/ce;. O

C'::JC",

Co::JCo

and such that

A.

Then by the last

C',C", CO,C

O

of

A

Q=C'/C",

Then the following two conditions are equivalent:

Subquotients 2)

COCC'

and

61

Co::::lC".

The proof of this Corollary closely follows that of the Lemma. Example 3. object in

A be an abelian category and let

Let A.

Then. by definition. every subobject of

an equivalence class of pairs \ :C

A

-+

(C.\)

where

C

A

~

\8.

Then by Lemma 1. it follows that:

is a subquotient of

A

A

(literally).

in

A

8

sub object of

Every subobject of

In like fashior. every

is a subquotient of

regarded as a subquotient of

A.

two subobj ects of

A

A.

object) and

C.

Similarly. if

Q

then regarded as a subquotient of of subobjects of

A

If

C and

Co

K

as subquotients of

A

A,

then

BO .:::. B

A

in

K

and

A

(the

quotient-objects) of

COCC

(resp.: of

A

Co':::' C)

A as

iff

A. if

B

quotient-object)

as subquotient of

object (resp,:

A,

the corresponding pair

as quotient-objects)

in the abelian category

subobject (resp.:

(the zero

is the kernel of the natural map:

Also, it follows that, object

A,

are subobjects (resp.:

(resp.:

0

therefore from the last Corollary that:

in the abelian category subobjects

is

then the

is a quotient-object of

by Corollary 1.2 are

whole subobject), where It follows

A.

C

that correspond to this subquotient

under the correspondence of Corollary 1.2 are

Q.

is

such that

Under the correspondence of the Corollary 1.2. if

A -+

A

is a monomorphism. two such pairs

quotient- object of

a

be an

is an object and

being equivalent iff there exists an isomorphism \0

A

A,

then

quotient object) of

is any subquotient of an A, BO B A.

and if there exists a of

A

such that

must also be a sub-

62

Section 5 From the Corollary 1.3, it also follows that:

Corollary 1.4. let

I

object

Let

A

be an object in the abelian category

be a set, and let A.

Write

B , i

B.=C.'/C'.', ~

~

~

i E I, all

A,

be subquotients of the where

i E I,

the uniquely determined subobjects of

A,

all

C'.' c: ~

i E I.

C~

are

~

Then

the following two conditions are equivalent: 1)

There

ex~sts

is an infimum (resp. :

2)

B

of the object B. ,

supremum) of the

(b)

C'

There exists a subobject

C"

(resp. :

C". ,

infimum) of the

that

that is an infimum

Cit

supremum) of the

A

iE I .

~

There exists a subobject

(a)

(resp. : and

a subquotient

i E I. that is a supremum i E I.

~

When these two equivalent conditions hold, then we have that B=C'/C". In terms of

sub quotients, every addi ti ve relation admits

a canonical factorization. Theorem 2.

Let

to the object subquotients

R B

A'

be an additive relation from the object

in the abelian category of

choose representatives

A

and

B'

of

(A",R") E A'

there exists an isomorphism

B,

Then there exist

such that if we

and

8 :A" ~ B",

A.

A

(B",S") E B',

then

such that the diagram

of objects and additive relations:

is commutative.

The subquotients

A'

of

A

and \B'

of

B

Subquotients

63

are uniquely determined; and so is the isomorphism (A" ,R") EA'

selects representatives

and

8

once one

(B", S") E B'.

The proof is trivial and left as an exercise.

An equiva-

lent formulation is: corollary 2.1. A

Let

to the object

be an additive relation from the object

R

B

in the abelian category

exist uniquely determined subobjects and

A"e: A'

Ab

EA',

B', B"

of

and

B6 E B'

B

wi th

BO E B"

A' ,A"

B "e: B'

R

where

of

Then there A

with

such that, if

AOEA",

are any representative elements,

then there exists a unique isomorphism that

A.

8·A'/A"""B'/B" . 0 0 0 0

such

is the composite relation:

1

:Ao

-+

p : B6 -+ B6/Bo

A,

j : BO

-+

B

are the inclusions and

R;

B"

R.

Then

R

R;

and

and images of

R R

is the domain of the

B'

R

R;

A"

is

is the image of the

is the graph of a

determined) map iff the domain of ambiguity of

A'

is the ambiguity of the relation

the kernel of the relation relation

:Ab -+ Ab/Ao'

are the proj ections.

We introduce the terminology: relation

1\

(necessarily uniquely

is all of

A,

and the

(in which case, of course, the kernel

is zero

then coincide with those of the uniquely

determined map of which

R

is the graph).

also define that the additive relation

R

Perhaps, one might is well-defined

(resp.:

everywhere defined) iff the ambiguity of

(resp.:

the domain of

R

is all of

R

is zero

A).

Yet another way of formulating the above theorem and corol-

64

Section 5

lary, is: Corollary 2.2. A

into the object

A'

of

if

(AO,l) E A'

map

Let

A

f :AO

R

be an additive relation from the object

B.

Then there exists a unique subobject

and a unique quotient-object

-+

Q

O

(QO,TI) E Q',

and

such that

Q'

of

B

such that

then there exists a unique

is the composite

R

(r )-1 TI ) B.

A

Thus, essentially, to give an additive relation from to

B

A is equivalent to giving

in the abelian category A;

(2)

(1)

A subobject of

(3)

A map in the category

the quotient-object of

A

A quotient-object of

B;

A from the subobject of

and A

into

B.

Finally, as in section 2, let us make a very minor convention.

Namely, if

object in

A,

A

is an abelian category and

A

is an

then by the (strongest form) of the Axiom of

Choice, see [K.G.)

(which is consistent with the other axioms

of Godel's set theory, if those other axioms are themselves consistent, see [K.G.)} there exists a subclass ordered class

SA

equivalence class.

S'

A

of the pre-

that contains exactly one element from each Make such a brutal choice.

class of all subguotients of class

SA

A

Then by the

we will mean this indicated

(containing exactly one representative from each

subquotient). (As in section 2, the reason why we do this is that, in Godel's set theory, one cannot

speak

of a class

some of the elements of which are proper classes. in Godel's set theory, one cannot use the phrase of all subquotients of

A"

in the literal sense.)

Therefore, "the class

65

Subquotients Remark:

Using the one-to-one correspondence between subobjects

and quotient-objects of a given object in an abelian category, one way to affect such a choice of representatives for each Suppose that,

subquotient, is as follows. in

A,

and also a complete set of representa-

A,

tives for the quotient-objects of AO

A,

in

Given any

follows.

AO

Then, given any specific

can be constructed as

sub:::!uotient

1.2 there exist unique such that

A.

a complete set of representatives for all

of the subquotients of

C"c C'

Q=C'/C".

of

Q

surobj ects

sentative for the subobject

AO'

C'

and

Then, let of

C'

Then the pair

-1

(QO,f ,Of,,) lO

subquotient

Q

of

lTO

AO.

by Corollary COO

of

(CO' l6)

AO '

and let

be the representative for the quotient-object

the

A

we choose both a complete set of representatives for

the subobjects of

object

for every object

with

A

be the repre(QO'lTO)

C'/C"

of

CO.

is a representative element for Then in this way, we obtain a

complete class of representatives for the subquotients of the object

AO

'

for every object

AO

in

C.

Section 6 Left Coherent Rings

This section is of a somewhat more specialized interest than the rest of this chapter.

The main application, is that

condition (2) of Theorem 5 below

~s

suggested by several

theorems, particularly in Part III, of this book; and it is therefore of some interest to study this condition. All our rings have identity elements, but are not necessarily commutative. Definition 1. Then

M is

Let

All modules are unitary. A

be a ring and let

finitely generated iff

lently, iff modules,

an integer

:3

n 2. 0

an integer

n'::'O

AX I + ... + AX n = M.

xl"" ,x n E M such that

elements

3

M be a left A-module. and

Equi va-

and an. epimorphism of left A-

An ·'M.

M is finitely presented iff there exist non-negative integers

nand

such that Lemma 1.

m

and a homomorphism

q, :A

n

-+

Am

of left A-modules

MR> Cok(q,). Let

A

be a ring and let

M be a left A-module.

Then the following two conditions are equivalent. (1)

M

is finitely generated as left A-module.

(2)

For every set I A 0

M-+ MI

I,

the natural homomorphism:

is surjective.

A

Note.

The proof shows, more generally, that if 66

I

is any set,

Left Coherent Rings then the natural homomorphism:

AI

('9

M~ M

67

is an epimorphism

A

iff every left submodule of

M

that can be generated

by~

card (I) elements, is contained in a finitely generated left submodule of

M.

(1)=(2).

Proof: N'VOO I ® N

Let

¢:An-+M

be an epimorphism.

Since

is an addi ti ve functor, we have that

A (1)

AI@An"'(AI®A)n"'(AI)n"'(An)I, A A

so that the natural mapping:

AI ® An -+ (An) I

is an isomorphism.

A

But then from the commutative diagram with exact rows and columns 0

i

~I

(An) I

') MI - - - - - ' ) 0

T

"l

I ) A ® M---~) 0 A

AI ® An A

T 0 we deduce tha t the natural mapping : AI ~ M -+ MI is an epimorphism. A

(2)=(1). left submodules of by inclusion,

Let M.

D

be the set of all finitely generated

Then

D

is a directed set, pre-ordered

and

Since tensor product commutes with direct limits, this implies that

68

Section 6 AI ®M",llm AI ®N, A NED A

(2)

for all sets

I.

Now let I be a set such that the natural mapping I I is surjective. Then, i f IJM:A ® M -+- M (xi)iEI is any family A of elements of M indexed by the set I, then the element is the image under

of some element of

Then by equation (2), there exists some SEAI®N

such that

IJN(B) = (xi)iEI'

NED

and an element

But then from the commuta-

A

tive diagram: IJ M

AI ® M A

T

IJ N

S E: AI ® N A

I >M :3 (Xi)iEI

J

)N

I

I (xi)iEI E N ,

it follows that

i.e. ,

x. E N, ~

Thus, we have shown that every subset of

for all

M of

i E I.

cardinality~

card (I) is contained in a finitely generated submodule. proves the Note.

And the Lemma follows by taking

cardinality~card

M.

Lemma 2.

be a ring and let

Let

A

left A-module.

Let

n

I

This

to be of

M be a finitely generated n ¢:A -+ M

be a non-negative integer, let

be an epimorphism of left A-modules and let

K = Ker ¢.

Then the

following two conditions are equivalent.

(1)

K

(2)

For every set

is finitely generated as left A-module. I,

the natural mapping:

AI®M-+MI A

is an isomorphism. Note: I,

The proof shows, more precisely, that for any fixed set

the natural mapping:

AI®M-+MI

is an isomorphism, iff:

A

Every left submodule of

K

that can be generated by

~

card (I)

69

Left Coherent Rings

elements, is contained in a finitely generated left submodule of

K.

Proof:

Consider the commutative diagram, with exact rows and

columns 0

0

) (Al) I

) KI

0

iI

~r

1

)A I ® An A

AI ® K A

r

M

,>0

II A ®M A

)0

0

Then by the Five Lemma, the mapping iff the mapping: AI ® K -+ A

the map K

AI ® K -+ KI A

AI@M+MI is a monomorphism. But by Lemma 1, A KI is an epimorphism, for all sets r, iff

is finitely generated as left A-module.

mapping:

is an epimorphism

AI®M-+MI

And, since the

is an epimorphism, we have that thatmapping

A

is an isomorphism iff it is a monomorphism.

Q.E.D.

Lemma 2 has some immediate consequences. Corollary 2.1.

Let

A

generated left A-module.

be a ring and let

M

be a finitely

Then, if there is any integer

n.:: 0

and any epimorphism n ¢ :A -+ M of left A-modules such that

Ker ¢

is finitely generated as left A-module, then

the same is true for every other such pair M

(There is such

a pair

n,¢

ProoK:

Condition (2) of Lemma 2 is independent of the choice

of an integer

iff

n,¢.

n.::O

is of finite presentation.)

andofan epimorphism

¢:An+M.

Therefore,

Section 6

70

if condition (1) should hold for one such pair M

n,¢

if

(i.e.,

is of finite presentation), then condition (1) must hold

for every such pair

n,¢.

Corollary 2.2.

A

ule.

Let

Q.E.D.

be a ring and let

M be a left A-mod-

Then the following two conditions are equivalent.

(1)

M

(2)

For every set

is of finite presentation as left A-module. I,

the natural mapping:

AI 0 M -+ MI A

is an isomorphism. Proof:

By Definition 1 and Lemma 1, both conditions (1)

(2) imply that

M is finitely generated.

to prove the Corollary when there exists an integer

M

n ~0

iff

Ker ¢

Therefore it suffices

is finitely generated. and an epimorphism

By Corollary 2.1, we have that

M

is finitely generated.

and

Then

n ¢ :A

-+

M.

is of finite presentation But by Lemma 2 above we

have that this latter is so iff the natural mapping:

AI ® M

-+

MI

A

is an isomorphism. Remark:

Q.E.D.

A careful checking of the proof of Lemma 2, also

shows that:

"If

A

is a ring,

M

is a left A-module and

I

is a fixed set such that there exists a short exact sequence: O-+K-+P-+M-+O of left A-modules with module of

K,

P

projective, and such that every sub-

that can be generated by .s.- card (I) elements, is

contained in a submodule of the natural mapping:

K

that is finitely generated; then

Left Coherent Rings

71

is a monomorphism." Lemma 3.

Let M'

A

be a ring and let

f'

--~)M------7

M"

--~)O

be an exact sequence of left A-modules. (1)

If

M'

and

M"

f'

is a monomorphism, then

Then

are of finite presentation, and if M

is of finite presenta-

tion. (2)

Proof:

If

M'

and

is

M".

M

M'

Suppose that

are of finite presentation,

then so

is of finite presentation.

Let

I

be a set, and consider the commutative diagram with exact rows

(M') I

~~T AI ® M' A Since AI ® M'

M' ->

(f ') I

)M

B I

)A

I

~(M") I ----~> 0

T

y

T

) AI ® M" ----~)o A

®M

A

.

is of finite presentation, by Corollary 2.2 the map

(M') l i s an isomorphism.

Therefore, by the Five Lemma,

A

if

f'

(and therefore also

is an isomorphism, then

S

(f,)I)

if

B

is an isomorphism then

is an isomorphism, proving (2).

Remark:

y

is an isomorphism, proving (1).

And, again by the Five Lemma, y

is a monomorphism and

Q.E.D.

Lemma 3, part (1), can of course be proved alterna-

tively, using results on projective resolutions in [C.E.H.A.j and making use of projective resolutions generated in dimensions zero and one.

p*

that are finitely

And part (2)

easily proved directly from Definition 1.

can be

72

Section 6 The next Lemma is very well known and is included only for

completeness of exposition. Lemma 4.

Let

A

be a ring and let

M be a right A-module.

Then the following two conditions are equivalent. (1)

M is right flat.

(2)

For every finitely generated left ideal we have that the natural mapping:

I

in

A,

M® I -.. M is injecA

tive. (1)~(2).

( 3)

0

-+

Tensoring the short exact sequence

I -.. A -+ A/I

on the right with (2)~(1).

-+

0

M gives

(2).

Every left ideal can be written as a direct

limit, over a directed set, of finitely generated left subideals. ideal

Therefore, condition (2) implies that, for every left I,

the mapping:

long exact sequence of

M® I -.. M is inj ecti ve. A A

Tori (M, ),

sequence (3), it follows that ideals

I

in

left module

A.

i

~

0,

Applying the

to the short exact

A

TorI (M,A/I) = 0,

for all left

In other words, for every simply generated

N , l

(4)

If

N

is a left module generated by

n

elements,

n

~

1,

then there is a short exact sequence

where

Nn _ l

is generated by

by one element.

n-l

elements and

Nl

is generated

Throwing through the half-exact functor

Left Coherent Rings A

Torl(M, ),

73

and using induction, it follows from equation (4)

that A

Tor I (M, N) = 0, for all finitely generated left A-modules

N.

But then, if 0-;.

f'

N'--t

fn

N~N"

-;.

0

is any short exact sequence of left A-modules such that

N"

is

finitely generated, then we have that (5)

M@f'

is a monomorphism.

A

Now, if f:N' -;. H is any monomorphism of left A-modules, then

H

is the direct

limit, over a directed set, of all the left-submodules H

such that

every such

N/N' is finitely generated. N,

M~(inclusionN'

A

IN

the direct limit over

N,

)

of

By equation (5), for

is injective.

it follows that

N

Passing to

M@f

isinjec-

A

Q.E.D.

tive. Theorem 5.

Let

A

be a ring.

Then the following three condi-

tions are equivalent. (1)

Every left ideal in

A

that is finitely generated

as left A-module, is finitely presented as left A-module. (2) flat.

For every set

I,

the right A-module

AI

is right

74

Section 6 (3 )

For every short exact sequence O~M'-+M-+M"-+O

of left A-modules, M'

if

M and

M"

are finitely presented, then

is finitely presented.

Proof:

(3)

=

Let

(l).

J

be a finitely generated left ideal.

Then we have the short exact sequence of left A-modules

o -+ J A (3)

and J

A/J

-+

A -+

AI J

-+

O.

are finitely presented; therefore, by condition

is finitely presented. Let

(l)~(2).

J

be a finitely generated left ideal.

Then we have the short exact sequence

o -+ J A

and

A/J

-+

A -+

AI J

are finitely presented.

finitely presented. set

I,

-+ 0 •

By condition (1)

J

is

Therefore, by Corollary 2.2, for every

in the commutative diagram

o~

Jf-'Ai TJ)

'_;,0

AI ® J _AI ® A --ioAI ® (A/J)~O A A A the vertical mappings are all isomorphisms. mapping:

AI

®

J +A

A

I

®

A ""AI

is a monomorphism.

This being

A

true for every finitely generated left ideal we have that the right A-module (2)~(3).

Therefore, the

Let

diagram with exact rows,

I

AI

J,

by Lemma 4

is right flat.

be a set, Then in the commutative

Left Coherent Rings

since

M and

M"

7S

are finitely presented, we have by Corollary

2.2 that the rightmost and middle vertical maps are isomorphisms. Therefore, so is the leftmost vertical map. Corollary 2.2,

M'

Corollary 5.1.

Let

And therefore, by

is finitely presented. A

be a ring.

Q.E.D.

Then the following condition

is also equivalent to the three equivalent conditions of Theorem 5.

(3.1)

Given a homomorphism

M .... N

of finitely presented

left A-modules, we have that the kernel is finitely presented. Proof.

Condition (3.1) obviously implies condition (3) of the

Theorem. holds.

Conversely, suppose that condition (3) of the Theorem Let

f:M

-+

N

part ( 2), we have that

denote the homomorphism. Cok f

Then by Lerruna 3,

is of finite presentation.

There-

fore, by condition (3) applied to the short exact sequence

o -+ 1m f .... N .... Cok f .... 0, we have that

1m f

is of finite presentation.

And therefore,

by condition (3) applied to the short exact sequence

o .... Ker

f

-+

N -l> 1m f

it follows that

Ker f

Remarks 1.

A

Let

-+

0,

is of finite presentation.

be a ring.

Then another condition

lent to the conditions of Theorem 5 is

Q.E.D. equiva-

76

Section 6 (3.2)

f,

The full subcategory

having for objects all

the finitely presented left A-modules, of the category

A of

all left A-modules, is an abelian category, such that the inclus ion functor: Proof:

Frvv>

A

is exact.

Clearly (3.2) implies (3.1).

part (2), respectively:

A,

cokernel in

A,

kernel in

Since also

f.

3,

by condition (3.1), we have that the

respectively:

is an object in

Conversely, by Lemma

of any map in

is closed under finite di-

f

rect sums, condition (3.2) follows.

Q.E.D.

Yet another equivalent formulation of condition

2.

(3) is (3.3)

A finitely generated submodule of a finitely pre-

sented left A-module is finitely presented. (3.3)~(3).

Proof:

In fact, given a short exact sequence as

in condition (3), choose an integer An ... M of left A-modules. an epimorphism.

Therefore (3.3),

M'

Then the composite:

M",

The image of M'

we have that

Ker(cjJ)

in

is finitely generated.

Let

is

M

Ker(cjJ)

is finitely

is isomorphic to

M'.

Therefore, by condition

M be a finitely presented left A-

module and let

M'

Then

is finitely presented.

='

cjJ :An ... M"" M"

is finitely presented.

(3)~(3.3).

M"

and an epimorphism:

By Corollary 2.1 applied to the finitely

presented left A-module generated.

n.::. 0

M/M'

be a finitely generated left A-submodule. Therefore, by (3),

is finitely presented. Lemma 6.

Let

A

left A-module, let A-modules.

be a ring, let I

M be a finitely presented

be a set and let N , i

i

EI,

be right

Then the natural induced homomorphism of abelian

M'

77

Left Coherent Rings groups ( II N.) ® M ~ II (N. ® M) , iEI ). A iEI). A is an isomorphism. Proof:

If

M= A

M'V'\,> ( II N.) €I M, iEI ). A

the assertion is clear. M'V\,>

II (N. €I M) iEI ). A

Since the functors

are additive, they preserve

finite direct sums.

Therefore, if

tion is again true.

If now

M

n M=A ,

is an arbitrary finitely pre-

sented left A-module, then there exist integers an epimorphism functors

¢ :A

n

~ Am

such that

M'V\,> ( II N.) €I M, iEI ). A

M'VV>

M "" Cok ¢.

II (N. €1M) iE I ). A

1.

n,

m.:::.

°

and

But, since the

are right-exact, it

follows that we also have an isomorphism for Remarks:

the asser-

n'::O,

Q.E.D.

M.

By Corollary 2.2, the conclusion of Lemma 6 is

equivalent to the assertion that

2.

"M

is of finite presentation".

Lemma 6 was really used in establishing equation

(1) in the proof of Lemma 1 above. Corollary 5.2.

Let

A

be a ring.

Then the equivalent condi-

tions of Theorem 5 are also equivalent to: (2.1)

The direct product of right flat, right A-modules,

is right flat. Proof:

Since

A

is flat as right A-module, condition (2.1)

implies condition (2) of Theorem 5.

We give two proofs of

the converse, one complete and one sketched. and let

Ni

Let

I

be a right flat, right A-module, for all

be a set i E I.

Proof 1.

Let

ring

Then by condition (1) of Theorem 5 we have that

A.

J

be a finitely generated left ideal in the

is finitely presented.

Therefore, all of

J

the left modules in

78

Section 6

the short exact sequence

o -+

J

-+

A -+

AI J

-+ 0

are finitely presented. functor:

Therefore, by Lemma 6, if we apply the

M'VV> ( IT N.) eM iEI ~ A

to this short exact sequence, the re-

suIting short sequence is isomorphic to the short exact sequence 0-+

IT

i E:I

(N. @ J) ~ A

IT

-+

i EI

(N. @ A) -+ IT (N. @ (A/J)) -+ O. ~ A i EI ~ A

In particular, we have that the mapping: (ITN.)@J-+ TIN. iEr ~ A iEI 1

is a monomorphism, for all finitely generated left ideals in the ring

A.

Therefore by Lemma 4,

IT N.

iEI Sketch of Proof 2. where

M and

N

J

is right flat.

1

By Lemma 6, given any monomorphism

f: M -+ N,

are left A-modules of finite presentation,

we have that (TIN.)®f

iEI

1

A

is a monomorphism.

But, using condition (3.3) of Remark 2

following Corollary 5.1, it is easy to see that every monomorphism in the category of left A-modules is the direct limit, overa directed set, of such monomorphisms every monomorphism have that (TIN.)@f i EI 1 A

f

f.

Therefore, for

in the category of left A-modules, we

Left Coherent Rings is a monomorphism of abelian groups. IT N. it:I

Remarks.

79

Otherewise stated,

is flat.

l

The elaboration of the sketched "Proof 2" above,

establishes another interesting condition that is equivalent to the conditions of Theorem 5.

Namely, that the ring

A

be

such that (4)

Given any monomorphism

there exists a directed set

D,

f: M -+ N

of left A-modules,

and direct systems indexed by the directed set 0,

of finitely presented left A-modules, and a map -;.(Ni)iED such that

(fi)iED: (Mi)iED

of direct systems indexed by the directed set f. :M. -+N. l

l

D,

is a monomorphism of left A-modules, for

l

lim M.'" M, lim N "; N, and such l iED i iED that under these isomorphisms lim f. corresponds to f. l iED

all

i E D,

and such that

Definition 2. A

Let

A

be a ring.

Then we say that the ring

is left coherent if it obeys the equivalent conditions of

Theorem 5.

(That is, any of the equivalent conditions (1),

(2) ,(3) of Theorem 5: or equivalently (3.1) of Corollary 5.1; or(3.2) of Remark 1 following Corollary 5.1: or (3.3) of Remark 2 following Corollary 5.1; or (2.1) of Corollary 5.2; or (4) of the Remark following Corollary 5.2).

Section 7 Denumerable Direct Product and Denumerable Inverse Limit

A be a category such that, for every sequence

Let i

an integer, of objects in

(as defined in [C.A.]) A

A,

the direct product:

Ai' IT A.

i E6' 1 Then we say that the category

exists.

is such that denumerable direct products of objects exist.

Also, if

A

is any category,

AW

then we let

denote the cateand for maps,

gory having for objects, all sequences (Ai)iE?"~ (B i )iE6"

is a map in

A,

all sequences for all integers

where

(f i )iE6'

f. :A.

~B. 111

i.

(This category can be interpreted as an "exponent category,"

C A ,

see section 3, Example 12,

for an appropriate category

Then, if denumerable direct products exist in the functor gory

A

"denumerable direct product":

A,

C).

then we have

AW'\l\,> A.

If the cate-

is also abelian, then it is easy to see.that this functor

commutes with fin. dir. prods. and kers., and that it is therefore a left-exact functor of abelian categories. functor

Therefore, this

is exact iff the denumerable direct product of a sequencE

of epimorphisms is always an epimorphism. Example 1.

If

A

is the category of abelian groups, then of

course the (denumerable or otherwise) direct product of epimorphisms is always an epimorphism. Example 2.

Let

S(X)

be the category of all sheaves of abelian

groups on a topological space

X, 80

where

X

is such that there

Denumerable Products and Limits exists a sequence of open sets

i

U1."

~ 1,

81

such that

n u,

i~l 1.

is not open (virtually every interesting topological space obeys this extremely mild condition).

Then

Six)

is an abelian

category such that denumerable (and even arbitrary) direct products exist;

but in the category

Six)

the denumerable

direct product of epimorphisms is not in general an epimorphism. (See [R.]).

A is an abelian category such that de-

Now suppose that

numerable direct products of objects exist and such that the functor,

"denumerable direct product":

Then, an inverse system

A

in the category (see [C.A.]), (n+2)

CJ.

,CJ.

ij

),

is exact. 'c._

1. , J '"6'

of objects and maps

j.::.i indexed by the directed set of integers

can be thought of as being a diagram:

CJ.

where

A = (A

i

(n+l)

>An+l

=CJ.

(n+l) CJ.

n+L n

n CJ. (n) )A _ _ _

--+ ...

for all integers

n.

Let us then define cO (A) Then let

dO

=c l

(A)

IT An. nEzr

be the unique map,

such that 'II

where

'II

n

:

0

n

dO

= CJ.

(n+l)

0'11

-

n+l

'II

n

,

is the projection.

Then if we define is a cochain

82

Section 7

complex in the abelian category

o

and

1.

A,

concentrated in dimensions

We sometimes denote this cochain complex more C*( (A ) ~).

specifically as H

O

(C*)

n n...,

~ 11 m

A

nEll canonically.

Then it is easy to see that

n

We define

Then Theorem 1.

Let

A be an abelian category such that denumerable

direct products of objects exist and such that the functor "denumerable direct product": AW'\)\,> A

is an exact functor.

Then given a short exact sequence of inverse systems indexed by the directed set

l',

(1)

there is induced an exact sequence in the category

A of

length six, Ij,m ' fn (2)

0->- (lim 'An)

~l'

nEl'

l (lim
1 lim <'An) nEt'

(liml "An) nEil' Proof:

---l»

11m "f ) (lim An) nEll nEl'

'fn

n

~(lim "An)~ nEil'

11ml "f n nEt' )(liml An) nEil'

)

0 •

Since the functor "denumerable direct product": AW'\)\,> A

Denumerable Products and Limits

83

is exact, we have the short exact sequence (1), from which we deduce a short exact sequence 0-> C* ( • A) -> C* (A) -> C* ("A) -> 0

(3)

A.

of cochain complexes in the abelian category

f- 0,1,

cochain complexes are zero in dimensions

Since these the Fundamental

Theorem of Homological Algebra (section 3, proposition 9) completes the proof.

Q.E.D.

From the exact sequence of six terms of Theorem 1, we deduce Corollary 1.1. Theorem 2.

The functor

Let

11ml nEI'

A be an abelian category obeying all the hypo(B n

theses of Theorem 1, and let

B

gers

n,

S (n))

be an inverse

nE?'

such that there exists an

A, and epimorphisms

in the category

all integers

,

A,

system in the abelian category object

is right exact.

such that

S (n+l)

0

Sn+l '" Sn 00

00'

for all inte-

n. Then

Proof:

Let

An", B,

the identity of

B,

all

n E?',

for all

and let

n E ?'.

be

Then

is an in-

verse system, and we have the epimorphism of inverse systems:

Since by Corollary 1.1, the functor

1

n~;

1

is right exact, to

complete the proof of the Theorem it suffices to prove that

84

Section 7

That is,we are reduced to proving the Theorem for a constant inverse system. dO: cO

->

cl

In fact, let

Then

is by definition the map such that Od O =

lTn

for all integers

a,

(n+1l

n.

olf

n+l

Since

-If

n

a,(n+l) =identity of

B,

this

simplifies to If

I claim that (Proof:

n.

for all integers

n

Replacing

dO A

is an epimorphism. that

A'

by any exact full subcategory

is a set, and is such that the denumerable direct product

A'

a sequence of objects in in

A',

and such that

as sume tha t

A

B

is a set.

A is an object

in the category

is in

A',

of

if necessary, we can

Then the functor

F,

V'VV>

IT Hom (V, ), vEA

A into

is a left-exact additive imbedding from the category

the category of abelian groups that preserves denumerable direct products--although it is not an exact imbedding. functor

F

Since the

is an imbedding, however, to show that

epimorphism it suffices to show that in the category of abelian groups.

F(dO)

(Yn)nE1' E IT An, nE1'

define

is an

is an epimorphism

Therefore, we are reduced

to proving the indicated assertion in the case that category of abelian groups.

dO

Then, given an element

A is the

85

Denumerable Products and Limits

x

I

0 + ••• + Yn-l YO

=

n

' -(Y- + ... +y ), l n

if

n = 0,

if

n > 0,

if

n < 0.

Then we see that

so that

dO

dO: cO (A*)

->

is an epimorphism, as asserted). c

l

(A*)

Thus,

is an epimorphism as asserted.

since the cochain complex

C* (A*)

But then,

vanishes in dimension

~

2,

we therefore have that liml An = HI (C*) = Cok dO = 0, nE;?' completing the proof. Remark.

Q.E.D.

It can be shown that Theorem 1, together with naturality

a

of the coboundary

(i.e., under the hypotheses of Theorem 1,

if we have another short exact sequence instead of

A's,

as in equation (1),

from (I') into (1); lim "An

d

Him

1

d,

'An

nE1

n

=> lim

l

'B

n

nE..,

is commutative), and Theorem 2, characterize the functor and the coboundaries Let

A

B's

together with a mapping

then the induced square about

nEf lim "B nE;?'

(I'), say with

a,

11ml, nE.;r

up to canonical isomorphism.

be an abelian category such that denumerable direct

products exist and such that the functor

"denumerable direct

86

Section 7

product" is exact.

Then it is not immediately clear whether

or not (P.2)

If

(An, C(n,m)

n,m E7 m
is an arbitrary inverse system such that the maps

epimorphisms, for all integers

n,m

with

are m

~n,

then the induced mapping

is an epimorphism. An abelian category such that denumerable direct products of objects exist and such that this condition holds will be said to obey the Eilenberg-Moore Axiom (P.2). Remark.

It is not difficult to show that (P.2)

implies that

the functor "denumerable direct product" is exact. Proof: integers

Suppose that i 2- 0.

all integers

f.:A.-+B. ~

~

~

are epimorphisms, for all

Define

n~O.

Then

n (C )

n2,O that all the maps are epimorphisms.

is an inverse system such By (P.2),

the map

n and CO", ITB. Q.E.D. lim C : ITA n~O n n~O n n~O In most applications ln the next chapter of this Introduc-

is an epimorphism.

But

tion, and in the rest of this book, we will rarely need this stronger axiom (P.2).

However, it will be used in a few re-

suIts of the Introduction, Chapter 2, sections 9 and 10.

87

Denumerable Products and Limits

In the Appendix to this section, we discuss in much more detail (much more than needed in the sequel)

the relationship

between the conditions discussed in this section.

However,

elsewhere in this text, the results of this Appendix will not be used.

Section 7

88

Appendix On The Eilenberg - Moore Axioms

We start by recalling some of the basic definitions in the paper of Eilenberg and Moore, in the first edition of the magazine "Topology", pages 1 - 24.

We refer to this reference as

"[E.M.l" in the Bibliography. We shall make use of the notions of "derived functors" and "satellites", concepts that will rarely be mentioned elsewhere in this book except in Examples and Remarks.

In particular,

the reader who does not have the background for this Appendix may omit it without incurring any difficultly. An abelian category such that denumerable direct products exist is said to obey axiom (P.O.).

If also the denumerable

direct product of epimorphisms is an epimorphism, then the category obeys axiom (P.l). Let

A be an abelian category such that denumerable direct

products exist (axiom (P.O».

Then the following conditions

are equivalent (this is easily deduced by arguments and constructions similar to those in [E.M.l):

(1)

Axiom (P.l) holds.

(2)

If

fi:Ai .... Bi

IIf.: IIA ..... lIB. ~

(3)

are epimorphisms, all

then

is an epimorphism.

~

~

i>O i>O

i>O

"lJ,ml"

the right satellite

i~O

i~O,

(in the sense e.g. of

89

Denumerable Products and Limits universal mapping properties) of the functor

"lim" i>O

exists,

"liml" i>O

" lim" i>O

and

is right-exact, and the two functors

"liml" i>O

together with their connecting homomorphisms are an exact connected sequence of functors. (3')

The right satellites (in the sense e.g. of universal

"lim" -O

mapping properties) of the fun tor

exist and form a non-

negative cohomological exact connected sequence of functors, wi th the second satellite,

~im2 =: O. 1>0

(4)

There exists a right exact functor

gory of inverse systems of objects of

G,

A indexed by the non-

negative integers and maps of such systems, into "lim"

and

{to

G

that the functor

of

0

"lim", i>O

such that

and such

vanishes on constant inverse systems. be an inverse system of objects

A such that there exists an object

and epimorphisms ai+l,i

G

{Ai,aij)i,j~O

Let

and maps

A,

are part of a cohomological exact connected se-

quence of functors of length 2 starting with

(5)

from the cate-

ai:A...,. Ai'

all integers

ai+-1=a i , all integers i.:::.O.

i.:::. 0,

A

of

A

such that

Then, given any short exact

sequence of inverse systems (indexed by the non-negative integers) of objects and maps in

A,

starting with

(A. ,a .. ) . . 1

1)

0 '

1,),:::,

the resulting short sequence (of length three) of inverse limits is exact. When the above five equivalent conditions hold, then the right satellite

"liml" i>O

of

"lim" i>O

exists (e.g., in the sense

of universal mapping properties) and fits together with

"lim" i>O

to make a non-negative exact connected sequence of functors

90

Section 7

(where the functors in spots 2,3,4,5, etc. are taken to be (Ai,aij)i,j~O

zero functors); and, given any inverse system obeying the condi tion (5)

above, we have that

*m

1 · 1 A. = 0 • i>O ~ Notes 1.

G

When the equivalent conditions above hold, then given

as in condition (4), there exists a unique isomorphism (in

fact, a unique homomorphism) of connected sequences of functors (lim, liml) i>O i>O

(
of the funtor

2. (P.l»

~ (lim,

G)

such that

¢ = identity endomorphism

i>O

"lim". i>O

When the equivalent conditions above (i.e., axiom

hold, of course liml(A. ,a .. ) . . 0"" Coker( ITA. ), i>O ~ ~J ~, J~ i>cr i>?

where

7Ti 3.

then let

0

¢ = ai+l,i

0

7T i + l - 7Ti '

When the above conditions (A. ,a. ,). ~

~J

'>0

~,J_

ai:A->A , i

A.

There exists an object all integers

all integers

i>

i > O.

(i.e., axiom (P.l»

hold,

be any inverse system of objects and

maps in the abelian category

Q)

for all integers

i ~ 0,

Suppose that either A

in

A

such that

and epimorphisms ai+l,i

0

a i + l = ai'

o.

For each integer

the map

a. 1 .:A. l->A. ~+,~

~+

~

is a split epimorphism of ob-

There exists an inverse system jects and maps in the category

A

an epimorphism of inverse systems:

®,

obeying condition (¢i).

~£.

O:(B.,e .. ). ~

~J

and

'>0'"

~,J_

Denumerable Products and Limits

91

(A. ,a .. ) . .

1

O. 1) 1,)::,

Then the following two conclusions hold: (a)

For each integer [lim A. 1 ->- A . i>O 1 J

and

(b)

Remark

lim i::,O

1.

1

j 2:. 0,

the natural mapping

is an epimorphism

A . '" O. 1

A obeys condi-

Suppose that the abelian category

tion (P.l).

If

(A.,ex .. ). '>0 is any inverse system of 1 1) 1,]_ objects and maps indexed by the non-negative integers in then it is easy to see that, in Note 3 above, condition for

(A.1 , a" . 0 1) ).1,)::,

are equivalent to conclusion (a) of Note 3. sion (b):

but even if

A

ai+l,i

Ai '" subgroup

and both

(2),

or con-

is an epimorphism, all integers

is the category of abelian groups, then

conclusion (b) of Note 3 does not imply this. AO = l' p I

Q)

(But not to conclu-

CD,

e.g., either of the conditions

elusion (a) imply that i ::. 0,

G),

is equivalent to condition

A,

p

i

• l' p'

i

~

0I

where

(E.g., take p

is any fixed

prime.

Then conclusion (b) of Note 3 above holds, even though

ai+l,i

is a monomorphism not an epimorphism, all integers

0

(Also, of course, condition than the other condi ti ons . g roups,

i

A.1 '" 'l'- /p 'l',

map, all integers

CD

G)

and

Remark

2.

i::. 0,

then

i

_>

0,

a.1+ 1 ,1. '" the natural

(A., a .. ). 1

1)

. 0

l,)~

of Note 3, but not condition

Let

(A., a .. ). . 0 1 1) 1,)2:.

in an abelian category that

A '" category of abe 1 i an

E . g ., if

all integers

of Note 3 is stronger

obeys conditions

(%)

of Note 3.).

be an arbitrary inverse system

A that obeys axiom (P.l).

is an epimorphism, all integers

i >

Suppose

o.

a condition apparently slightly weaker than condition

(This is

@,

or

Section 7

92 equi valently above).

CD,

or equivalently of conclusion (a), of Note 3

Then do (the somewhat stronger, equivalent properties)

condi tion

CD

of Note 3, equivalently condition

equivalently conclusion (a) of Note 3, hold? known in general at the moment.

Q)

of Note 3,

This is not

In [R.M.], an abelian cate-

A that obeys axiom (P.O) and such that this is always

gory

the case ("every inverse system (A .. ,a .. ) . . 0 lJ

lJ

l,J~

a + l, i '

non-negati ve integers, in which the maps

CD

epimorphisms, obeys condition obey axiom (P.2).

indexed by the

i

i

~

are

0,

of Note 3"), is said to

It is not difficult to show that when axiom

(P.l) holds, then axiom (P.2) is also equivalent to: (P.2 ')

Let

(A .. a .. ). l'

lJ

. 0

l,J~

be an inverse system of ob jects

and maps in the abelian category

A

epimorphism, all integers

Then if

lim A.

i>O

t-

i > O.

is an

such that AO t- 0,

we have that

O.

l

Another equivalent form of axiom (P.2) , under the hypothesis that axiom (P.l) holds, is (P.2").

(A. ,a .. ).

Let

l

. 0 lJ l,J'::'

be an inverse system of ob-

jects and maps in the abelian category

A

is an epimorphism, all integers

Then

liml A . l i>O

i > O.

such that

= O.

(For the proof that (p.2)

~

pg. 7 of [E.M.]). (Of course,

(P.2')

~

(P.2"), see Theorem 2.5,

(P.2)=;.(P.I)i

see page 86 above.)

All known abelian categories (e.g., the category of left A-modules, where (P .1) ,

A

is any ring with identity) that obey axiom

are easily shown to obey axiom (P.2) also (one veri-

Denumerable Products and Limits

93

fies, using elements, that any inverse system indexed by the non-negative integers such that the maps are all epimorphisms is such that conclusion (a) of Note 3 above holds). A

is any abelian category with enough projectives - or, weaker,

is such that

AEA,

that

then conditions (P.O),

A = 0 -

equivalent. A

(Also, if

HomA(p,A) =0,

Reason:

allprojectives

P,

implies

(P.l) and (P.2) are all

One first reduces to the case in which

is set-theoretically legitimate.

Such an abelian category

admits an exact imbedding into the category of abelian groups that preserves arbitrary direct products

(even arbitrary in-

verse limits indexed by arbitrary categories), namely the imbedding:

A~>n

Hom(P,A)). However, in most, but not quite PE A, P projective all, applications in this book, the strongest axiom ever needed is (P.'l).

Of course, the most important example, the category

of abelian groups, and also its dual category, obey these

all of

axioms, including (P.2).

Remark 3.

E.g., in the proof of Theorem l' of Chapter 2 of

Part I, below,

(and Theorem 2' in Remark 4 following Theorem

2 of Chapter 2 of Part I, below), axiom (P.l) is used.

Notice

also that we need that in the exact sequence of six terms

con~

structed in the proof of that Theorem (the exact sequence 1.8.1 of [P.P.W.C.]), the fourth object vanishes.

But this object

is (3)

, that ~s, t h e f unc t or

"1'*m, 1" i>O

applied to the inverse system:

94

Section 7

This inverse system clearly obeys condition above (where we take of

Note 3

n A = H (C*) ).

Q) of Note

Therefore, by conclusion (b) A

above, whenever the abelian category

axiom (P.l)

3

obeys

(which is all that we have assumed in, e.g., Theo-

rem I' of Chapter 2 of Part I and in Theorem 2' of Chapter 2 of Part I), then the object (3)

is zero, as required in the

proof. Remark 4. quired.

In Chapter 1 of Part I, even axiom (P.O) is not reAll that we need is that: "Denumerable suprema and in-

fima of sub-objects exist."

And, in fact, the barest minimum

condition is that "For the fixed integer

n,

the supremum of of

the increasing sequence of subobjects:

r ~ 0,

exists; and the infimum of the decreasing sequence of subobjects:

of

exists" .

It is easy to see,

using the exact sequences established in Lemma 1 and in Corollary 1.1 of Chapter 1 of Part I, generalized to arbitrary abelian categories, that the following condition is equivalent to this "bare minimum":

"For the fixed integer

creasing sequence of sub-objects of n H (C*)),

cise ti-torsion part of

i

~ 0,

Hn+l(C*))O (Im(restriction of

sion part of i

~

0, exists

Hn + l (C*))

-+

has a supremum; and (precise t-torsion

ti):

(precise ti+l-tor-

(precise t-torsion part of

Hn + l (C*))) ,

(so that one can speak of the t-divisible part of

the precise t-torsion part of Remark 5.

the in-

Hn(C*): t · Hn(C*) + (pre-

the decreasing sequence of subobjects, part of

n,

Hn+l(C*))".

The reader who is not congnizant with, or does not

like, abelian categories, may (totally) ignore this Appendix, and may read all Theorems (including e.g., Chapter 2 of this

Denumerable Products and Limits

95

Introduction and Parts I, II and III of this book), instead of, as occasionally being stated, for an arbitrary abelian category A

obeying specified axioms, only for the special case in which

A

is the category of all left modules over some ring with iden-

tity

A.

In fact, to date, I think that all special cases "of

practical use" (a term difficult to define, and a contention that some might debate, since some regard all abelian categories as being of interest - and certainly future applications at the "abelian category theoretic level" are even probable) of the Theorems, Propositions, etc. of this book, occur in this special case.

(However, it is perhaps instructive to understand things

at the more general level).

This Page Intentionally Left Blank

CHAPTER 2 THEORY OF SPECTRAL SEQUENCES Section 1 Spectral Sequences in the Ungraded Case

Definition 1.

A is an abelian category, then a spectral

If

A is:

sequence in the abelian category 1)

An integer

2)

A sequence where

phism of

E

r

Er in

rot; 71. indexed by the integers (Er,dr)r>r' - 0 is an endomoris an object in A and d r

A,

such that

d

r

0

d

r

all integers

'" 0,

A sequence (Tr)r>r' where Tr:H(Er,d r ) ~Er+l is an - 0 isomorphism in A, where H(Er,d ) =Ker(dr)/Im(d ), all inter r 3)

gers

r.:.. r o.

Such a spectral sequence is also called a spectral

sequence starting with the integer

rOo

sequence starting with the integer

rO

Clearly, every spectral can, by a translation in

indexing, be made to start with any other integer. is a spectral sequence starting (Er,dr,Tr)r>r - 0 is a specwith the integer r ' then clearly (Er,dr,Tr)r>r O

Also, if

-

tral sequence starting with the integer

r

l

,

I

for each integer

r l ,:::, rOo

Remark: category

Thus, intuitively, a spectral sequence in the abelian

A is simply a sequence of differential objects in

A,

such that each is isomorphic to the homology of its predecessor. be a spectral sequence in (Er,dr,Tr)r>r - 0 the abelian category A starting with Then we define

Definition 2.

Let

97

98

Section 1

for each integer two sequences

A,

~rO'

Zi (E ), r

each integer gory

r

r

~

by induction on the integer Bi (E ) r

r 0'

i

~

0,

identity of

E , r

Bl (E r ) = Im(d ) r composite:

from

all integers

First, define

E , r

~O,

for

and also a sequence of maps in the ca te-

"i-fold image",

integers

of sub-objects of

i

E + , i r

for all

r 2 r O.

Zo (E ) = E , r r

BO (E ) = 0, r

for all integers and

into

Zi (E ) r

"O-fold image" =

r2rO'

and

Zl(E ) =Ker(d ), r r

"I-fold image"

Zl (E ) -+ Er+l to be the r natural Ker(d r ) T Zl(E r ) =Ker(d r ) map >Im(d) ""r >E r + l , for r

all integers defined Ej+r

r~rO.

Zj (Erl,

If

B j (Erl

for all integers

into

Ei+r

we have defined the subobject

i

is an integer

and j

Zl (E ) r

of

such that

for all integers Zi (E r )

into

r

~rO'

Er+i

We call

Zi (E ) r

r.:: r 0'

j < i,

Bi -

Er'

and the map

-1

and all integers

By the inductive assumption, l

(E ), r

subobjects of

Er'

(Zi-l (Er+l))'

and define

"i-fold image" from

Zi-l (E r + l )

" (i-I) -fold image"

E , r

> Er ,

completing the inductive construction.

the i-fold cycles in

i-fold boundaries in r..::,rO·

~

into

Zj (Erl

to be the composite:

Z. (E )" I-fold image" l r " for all integers

0

and

(I-fold image)

and if we have

"j-fold image" from

as follows:

Zi_l (E r )

>2,

all integers

E , r i

and ~

0,

Bi (E ) r

the

all integers

Ungraded Case Remark 1.

99

Let

(Er,dr,Tr)r>r be a spectral sequence starting - 0 with the integer rO in the abelian category A. Fix an exact imbedding from

A (or, if

A that is a set and that contains

abelian subcategory of all

r> r ) - 0

and let cycle? u?

r

A is not a set, from some full exact

u E Er .

If

~

i

0,

then when is

u

an i-fold

And, when this is so, what is the i-fold image

Explicitly,

d r (u) = 0,

u

r ,

A~>A',

into the category of abelian groups, say

~rO'

E

U. l

of

is a O-fold cycle always; a I-fold cycle iff

in which case the I-fold image of

u

is the image

under the composite: an i-fold cycle (1)

u

is an (i-I)-fold cycle, and (2) if

fold image of

u

in

Er + i - l ,

then

in which case the i-fold image

U. l

d r + i - l (u) = 0 of

I-fold image of the (i-I)-fold image of case, then

u

u.

ui_l=dr+i_l(v) in E r + i - l for some vEE r + i _ l Let

the abelian category

(Er,dr,Tr)r>r - 0

integer (2)

j

~

0,

the map and that

E . r

is the (i-l)in

Er +i - , l

If this is the u E Bi (E » r

iff

(i.e., iff ui=O in Erri ).

be a spectral sequence in

A.

Then for every integer

as subobjects of

iff

is defined to be the

u

is an i-fold boundary (Le.,

Proposition 1.

u _ i l

(i ~ 0)

r >r

-

0

we have that

Also, for every integer

and every

we have that "j-fold image"

Zj (E ) .... E + r r j

is an epimorphism,

100

(3)

Section 1 (j-fOld image) (Z. (E » = Z. +' (E ) -1 1 r 1 ] r

!

(j-fold image)

all integers integer

-1

i,j,r

j.::. 0

(B i (E r » with

1

i,j :::,0,

and every integer

as subobjects of

'

= Bi+j (E r )

r :::.r ' O r.::. r 0'

Moreover, for every there is induced a

specific isomorphism: (4)

Z. (E ) /B . (E )". E

r

1

Proof:

]

r

r +"]

A is the category of abelian groups,

In the case that

the proof is by induction on bedding Theorem

[~J,

i

and is easy.

By the Exact Em-

the theorem follows for every abelian

category. Remark 2.

Let

(E r ,d r ,lr)r>r

with the integer

for each integer

we have that

Er

E

r+i

(Er,d r ) , E

r :::. rO

Remark 3.

A"

A'

C

r+l

Then since

= Ker

d r /1m d r '

is a subquotient

and each integer

is in a natural way a subquotient of

Therefore there exist unique subobjects with

A.

category

it follows that

r :::.rO'

Therefore for each integer

Er ·

i :::. 0, Er .

in

is isomorphic to the homology of

E r+l

of

rO

be a spectral sequence starting

-° the abelian

such that

Er+i

A'

= A"

•

A'

and

Explicitly

A" A'

of is

Let

be a spectral sequence starting (E r ,d r ,lr)r>r - 0 with the integer rO in the abelian category A. Then, for each integer

r :::. rO

in Remark 1 that definition of call it

and each integer

Er+i

is a subquotient of

'~ubquotien~'we

R . , r,l

from

i :::. 0,

Er+i

of the sequence of relations:

we have observed

Er .

Therefore by

have the natural additive relation, into

We can form the composite

Ungraded Case R

E

101

.

~> E

r+i

to obtain an additive relation from

E

r into itself.

r

It is

fairly easy to show that the domain of this relation is that the ambiguity is relation is Bi+l(E r ), Remark

Bi(E r ),

Zi+l(E r ),

that the kernel of this additive

and that the image of this relation is i~.O,

all integers

4.

all integers

r:=:.rO.

A be an abelian category, let

Let

Zi(E r ),

rO

be an inte-

ger and let

(Er,dr"r)r>r be a spectral sequence starting - 0 with the integer rO in the abelian category A. Then we have

AO

the dual category tral sequence in

O

A

in

",

A,

in

AO "

(E

Er

and

certain specific subobjects of

r~rO.

all integers O

A

in

object of

Er

r :=:.rO.

in

in

r

E

in

r

0

A

-

equiva-

A - for all integers

Explicitly, the "i-fold cycles of

and the "i-fold boundaries of

gers

E

is the quotient-object of

"

-1)

r' r " r

and the "i-fold boundaries of

lently, certain quotient-objects of i~O,

d

is a specr>r - 0 Therefore, by Definition 2, we have the

AO •

"i-fold cycles of

of

E

A: Er/Zi (E r ),

r

in

E

r

AO"

in

A:

is the quotient-

all integers

i:=:. 0,

all inte-

Otherwise stat;ed, under the operation of passing

to the dual category, and then replacing a quotient-object by the corresponding subobject, the constructions, Bi (E ), interchange. r daries in

Er

Zi(E ) r

and

(So, roughly speaking, the i-fold boun-

can be thought of as being the dual construction

of the i-fold cycles in

E

•

r'

all integers

We will say that an abelian category

i,r,

i:=:. 0,

A is closed under

denumerable suprema (respectively: infima) of subobjects iff whenever

A

is an obj ect in

A

and

Ai'

i :=:. 1,

are subobj ects

102 of

Section 1 A

then there exists a supremum (resp.:

ordered class of all subobjects of

A

ject in the abelian category jects of

A,

L A.

i>l

A.

in

and if

then the supremum (resp.:

exists, will be denoted Lemma 2.

A

(resp. :

infimum) in the A

i .::.1,

Ai'

is an obare subob-

infimum), when it

n

A.).

i>l

1.

If

1.

A be an abelian category such that denumerable

Let

direct sums of objects exist.

Then

A is closed under denumer-

able suprema of subobjects. Proof:

The reader will quickly verify that

e

1m «

Ai)

+

A)

is the required supremum.

i>l Corollary 2.1.

A

Let

be an abelian category such that denum-

erable direct products of objects exist.

A is closed un-

Then

der denumerable infima of subobjects. Proof:

By Lemma 2 applied to

AO,A o

is closed under denumerable

suprema of subobjects. Definition 3.

Let

A be an abelian category and let

(Er,dr,Tr)r>r be a spectral sequence in the abelian category - 0 A starting with the integer r O' Then for every integer r'::' r 0' we define (1)

r

co

subobjects of (2)

n Zi (E r ) if this denumerable infimum of i>O Er exists, and

Z (E )

B

(E co

subobjects of

r

=

) =

E

r

LB.1. (E r ),

i>O

if this denumerable supremum of

exists.

is called the permanent cycles of

E

•

r'

is called the permanent boundaries of

E

•

r'

all integers

Ungraded Case Proposition 3.

103

Let

(Er,dr,Tr)r>r be a spectral sequence in - 0 the abelian category A starting with the integer rOo Then for every integer

r

~rO'

we have that: exists, in which case,

exists iff

(1)

Zoo(E

r

under the epimorphism,

is the pre-image of

)

o

"(r - rO)-fold image": (2)

Boo(Er)

Z (E) r-r O rO

E

-+

B (E

exists iff

rO Boo(Er)

r )

exists, in which case,

00

Boo(Er)

is the pre-image of

under the epimorphism,

o "(r-ro)-fold image": (3)

If both

Zoo

and

Boo

exist as in conclusions (1) and

(2) above, then Er = Zo (E r ) Boo (E ) r

::::>

::::> ••• ::::>

for all integers every

r ':'rO'

Zl (E r )

r.

::::> ••• ::::>

Bi + (E r l

Zi (E r )

)::::>

And, when

the epimorphism

::::>

Bi (E r ) Zoo

Zi+l (E r ) ::::> ••• ::::>

and

Boo

::::> ••• ::::>

Bl (E r

)::::>

Zoo (E r )

::::>

BO (E r ) = 0,

both exist, for

"(r - rO)-fold image" induces an

isomorphism: Z (E

(4 )

00

rO

) /B (E 00

rO

)::;' Z (E ) /B (E ). 00 r 00 r

Proof:

By the Third Isomorphism Theorem applied to the epimor-

phism

¢ =" (r - rO)-fold image",

Z (E) .... E, r-r O rO r

isomorphism of ordered classes, from that contain

Ker ¢}

onto

we have an

{subobjects of

{subobjects of

E }. r

By the second

equation in conclusion (3) of Proposition 1, in the case

i = 0,

and the fact (from conclusion (1) of Proposition 1) that BO(Er) =0,

we have that

Ker ¢= B (E) . r-r O rO

¢-l(O) =B

r-r O

(E),

rO

Le.,

By equation (1) of Definition 3, Z (E 00

rO

) exists

104

Section 1

n

iff

Z.

i>O l+r-r O

(E)

rO

exists.

These are all subobjects of

(E) = Ker rp, and by the first r-r O rO part of conclusion (3) of Proposition 1, the subobject of Er Zr-r (EO)

o

that contain

B

corresponding to

n

Z.

i>O l+r-r O

is

rO

rp -1

clusion (1).

r

l

n

iff

(E)

which case

Z. (E )

(n

Z. (E

i>O l

)) =

r

E

C

Z. (E

i> 0

l

r

r

II Z. + (E) . i>O l r-r O rO

Therefore

•

) C E

exists, in

r

This proves con-

Conclusion (2) is proved similarly.

Conclusion

(4) then follows from the Third Isomorphism Theorem applied to the epimorphism:

( E ) -+ E. Finally, r-r O rO r conclusion (3) follows from conclusion (1) of Proposition 1 and

the fact that

"(r - r 0) -fold image": Z

Z (E ) =

'"

r

n

Z. (E ),

i>O l

r

B

'"

(E

r

)

=

lB.

i>O l

(E ).

r

be a spectral sequence in (Er,dr,Tr)r>r - 0 Then the abelian category A starting with the integer

Definition 4.

Let

Z (E ) and B",(E r ), '" rO 0 in the sense of Definition 3, both exist, in which case we de-

we say that the abutment

E",

exists iff

fine

From Proposition 3, conclusions (1), (2) and (4) we immediately deduce: Corollary 3.1.

Let

(Er,dr,Tr)r>r be a spectral sequence in - 0 the abelian category A. Let r be any fixed integer ':J o.

Then the abutment

E",

exists iff both

Z",(E ) r

and

B",(E ) r

exist, and when this is the case there is induced a canonical isomorphism:

Ungraded Case Remark 1.

Let

(1)

(E r ,d r

starting with the integer (E

(2)

d 1- 1 ) r' r' r r~rO

integer

rO

105

be a spectral sequence r )r>r - 0 rO in the abelian category A. Then ,1

is a spectral sequence starting with the O A .

in the dual category

From Remark 4 following

Proposition I, it follows that: (3)

Zoo(Ero)

Boo(Er)

"Ff

exists for (2) in the dual category

o

o

(E

00

in

rO O A ,

exists for the spectral sequence (1) in A iff O A

(call this

0

)")

in which case

'

BA (E 00

rO

i.e., quotient-object of

) E

is the sub-object of

rO

in

A,

Also (4) Z

(E 00

gory

B

rO

(E

)

exists for the spectral sequence (1) in

iff

rO

)

exists for the spectral sequence (2) in the dual cate-

O A

O

"z A

(call this

O

(E

00

the subobject of

rO

)")

'

E00

ZA (E

in which case

00

rO

)

is

i.e. , the quotient-object of

in

o zA (E ) = E /B (E ). in A, E 00 rO rO 00 rO rO tion 4, therefore imply that: (5)

A

00

(3) and (4), and Defini-

for the spectral sequence (1) exists in

A

iff

for the spectral sequence (2) exists in the dual category

E 00

AO,

in which case they coincide. Roughly speaking, it follows that the abutment

E

00

of a

spectral sequence in an abelian category is a "self-dual" concept, both in its existence and value; and that, in passing to the dual category and replacing quotient-objects by their corresponding subobjects, integers Remark 2.

Z

(E 00

r

)

interchange (all

r':' r 0) . Let

abelian category

be a spectral sequence in the r )r>r - 0 starting with the integer Then we

(E r ,d r A

,1

Section 1

106

have observed, in Remark 2 following Proposition 1, that a subquotient of of

E

r

,

Clearly

all integers

o

r

~

r

o.

Er

~Er+l

E

is

r

as subobjects

Then notice that, by Introductiol

Chapter 1, section 5, Corollary 1.4, we have that the subquotients ment

admit an infimum, iff the abut-

Er' Eoo

abutment

exists in the sense of Definition 4; in which case the Eoo

is that infimum.

Of course, this gives another

equivalent definition of the abutment

in terms of subquo-

E 00

tients. Definition 5.

Let

A

be an abelian category, let

rO

be an

(E r ,d r ,l r ' r >r

integer and let

and ('Er,'dr,'lr'r>r be 0 - 0 spectral sequences in the abelian category A starting with the same integer

rOo

Then a map of spectral sequences from

is a sequences into ('E r , 'dr' 'l r )r>r (E r ,d r ,l r )r>r (fr)r>r - 0 - 0 - 0 indexed by the integers r ~ r 0' where f is a map in the cater

gory

A

from

Er

into

'E r ,

for each integer

r

~

r 0'

such

that (MI) fr

0

d ; r (M2)

For every integer

r

~rO'

we have that

r

~

the diagram:

'd

r

0

fr

=

and such that For every integer

:~rrI--

r-+-l-'-'

r 0'

:rr.·drl

f Er+l-------------->'E r + l in the category Remark 1.

A

is commutative.

Given axiom (MIl, axiom (M2l is equivalent to the

107

Ungraded Case

statement that, for every integer

r

fr

by passing to the subquotients.

by

f

~rO'

fr+l

is induced by

Hence, if

(f r ) r>r is a - 0 is induced map of spectral sequences, then it follows that f r

Therefore a

by passing to the subquotients,

rO map of spectral sequences in the abelian category with the integer

A

starting

rO:

is completely determined by the initial map,

f

rO Otherwise stated, an alternative, equivalent, definition of a map of spectral sequences in the abelian category with the same integer ('Er,'dr"'r)r>r

-

exist maps

f

r

:E

->- 'E

conditions (MI) and Remark 2.

O

'

is:

0 r

r

into

from

such that there

"A map

for all integers

r

A starting

r .:::.rO + I

such that

(M2) above hold."

Suppose that

(Er,dr"r)r>r

o

and

are spectral sequences in the abelian category Suppose that

the same integer

A,

the abelian category

('E r , 'dr' "r)r>r

A

- 0 starting with is a map in

such that, for every integer

r .:::.rO'

induces a map fr:E r ->- 'Er by passing to the subquotients. rO (Z.(E »c:Z.('E ) (It is equivalent to say, "such that f rO 1 rO 1 r0

f

and

fr

(B i (E r » c: Bi ('E r

),

for all integers

i.:::. 0").

Then is

000 the sequence

(fr)r>r

-

necessarily a map of spectral sequences?

0

It is easy to see that the answer is, in general, "no".

Necessary

and sufficient conditions for such a constructed sequence (fr)r>r - 0 integers

of maps r .:::.rO'

(i.e., a sequence of maps, such that

f

rO

induces

fr

fr:Er->-'E

r

,all

by passing to

108

Section 1

the sUbquotients, all integers

r .:.r )' to be a map of spectral O sequences, is that Axiom (Ml) above hold.

A be an abelian category and let

Let

rO

be an integer.

Then we have the category Spec.Seq.

(A), having for objects rO all spectral sequences starting with the integer rO in the

A,

abelian category

and for maps all maps of such spectral

sequences as defined in Definition 5 above.

It is easy to see

that Spec. Seq.

(A) is an additive category, but in general is rO not abelian. (Of course, Spec. Seq. (A) is an additive subcaterO gory of the additive category having the same objects, and for

maps all sequences of maps

(fr)r>r as in Remark 2. This - 0 larger additive category, which is also in general not abelian,

is not very interesting, however).

i.:. 0,

For each integer

and

B. l

are covariant add i-

(A) into the category A. I f rO (A) (resp. : F' ,F") are the full subcategories of Spec.Seq. F rO generated by those spectral sequences such that Z"" (E ) (resp. : r

Spec. Seq.

tive functors from

o

(E

);

into

A.

B

r 0 Boo,E",,) 00

both

Z (E ) and Boa(Er» exists, then Zoo (resp.: "" rO 0 is a covariant, additive functor from (resp.: F',F") F

Proposition 4.

A be an abelian category, let

Let

rO

be an

integer, let

(Er,dr,Tr'r>r' ('Er,'dr,'Tr)r>r be spectral 0 - 0 sequences in the abelian category A starting with the integer be a map of spectral sequences from f=(fr)r>r - 0 (Er,dr'Tr)r~ro into ('E r , 'dr' 'Tr)r.:.ro· Then the following

rO

and let

two conditions are equivalent: 1)

2)

f

: E rO rO (fr)r>r - 0

is an isomorphism in the category A. rO is an isomorphism of spectral sequences.

->-'

E

Ungraded Case Corollary 4.1.

109

Under the hypotheses of Proposition 4, suppose

that the two equivalent conditions (1) and (2) of Proposition 4 Then

hold. If

Z (E 00

Zoo(E r

r 0

r

: E

-+ 'E

r

exists

)

and

)

o

f

r

is an isomorphism, all integers

(resp.:

If

o

)-+B ('E ); rO 00 rO an isomorphism in the category A. 00

by induction on

that

r

=

r 0'

fr

if

Zoo(f): Zoo(Er ) -+ Zoo{IEr ) o 0 resp. : Eoo(f): Eoo -+ 'Eoo) is

00

Proof of Proposition 4:

For

o

both exist), then

Boo(Er )

B (f): B (E

(resp. :

exists; resp.:

Boo(Er)

r,:: rO •

r,

r

(2)=> (1) is clear.

~ro'

we claim that

this is condition (l).

is an isomorphism.

isomorphism

H(f ). r

implies that

fr+l

Assume (1). fr

Then

is an isomorphism.

Suppose that

r,:: r 0

and

Then by Axiom (Ml), we have the

Then the commutative diagram in Axiom (M2) Q.E.D.

is an isomorphism.

Proof of Corollary 4.1.

The assignment:

(Er,dr,Tr)r>r ~>Er is a functor from Spec.Seq. (A) into A - 0 rO and therefore carries isomorphisms into isomorphisms. Therefore fr

is an isomorphism, all integers

r.

Also, since

is isomorphic to in {Er,dr,Tr)r>r - 0 Spec.Seq. (A), it follows readily that Z (E ) exists iff 00 r 0 rO If that is the case, then since Zoo ( I Er) exists.

o

(Er,dr,Tr)r>r ~>Zoo(Er ) is a functor on the full subcategory - 0 0 is an isomorphism F of Spec.Seq. (A), it follows that rO Q.E.D. in A. The other assertions are proved similarly. Remark 1.

In the statement of Proposition 4, suppose we weaken

the hypothesis, that

"(fr)r>r is a map of spectral sequences" - 0 to the weaker assertion, that " (fr)r,::ro obeys the hypotheses of Remark 2 following Definition 5."

Then the Proposition and

110

Section 1

"(f) " a s in r r~rO Remark 2 following Definition 5 induces a map from Zoo(E ) r

Corollary remain valid.

(Notice that an

o

into

Z ('E 00

and

E ). 00

rO

)

whenever both are defined; similarly for

B

00

Section 2 The Spectral Sequence of An Exact Couple, Ungraded Case

Definition 1. object in itself.

A

A be an abelian category, let

Let

and let

t:V -+ V

be a map in

V,

V,

V) = r

(Im(t ))r>O

a decreasing sequence of subobjects of

V.

then it is called the t-divisible part of

= n

of

V,

then it is called

Similarly, the sequence of subobjects,

V)

into

If a supremum exists

(t-torsion part of

(t-divisible part of

V

(Ker t )r>O

is an increasing sequence of subobjects.

the t-torsion part of

from

be an

r

Then the sequence of subobjects,

in the collection of all subobjects of

A

V

L Ker (t r ) . r>O of

V

is

If an infimum exists, V,

r Im (t ) .

r>O Definition 2.

On the other hand, we can consider the inverse

system, indexed by the positive integers, such that

i~l,

and such that

i>1.

If an inverse limit

t(i+l): v(i+l) -+v(i) lim(v(i),t(i))

V(i) =V,

is the map

t,

exists, then we

itl have the natural map:

The image of that map, a subobject of t-divisible part of

V.

V,

Finally, consider the direct system 1n-

dexed by the posi ti ve integers, such that t(i):V (i)

-+

V (i+l)

is the infinitely

is the map

t, 111

i> 1.

V (i)

= V,

i

~

I,

and

I f the direct limit of

Section 2

112

system should exist, then we have the natural map:

this direct

The kernel of this latter map is the infinite t-torsion part of V. Example 1.

If in the abelian category

A,

denumerable direct

products of objects exist, then both denumerable inverse limits and denumerable infima of subobjects exist. case, if part of in

t:V .... V V,

A.

is any map in

A,

Therefore, in this

then both the t-divisible

and the infinitely t-divisible part of

Similarly, if the abelian category

A

V, exist

is such that de-

numerable direct sums of objects exist, then denumerable direct limits and denumerable sups of subobjects exist, and therefore, t:v .... V

whenever

A,

is any map in

and the infinite t-torsion part of Example 2. in

AO,

object in

V

t:V .... V

be a map in

t:v .... V

and

V

exist.

A be an abelian category, let

Let

A and let

the t-torsion part of

A.

Then

V V

be an object

is also an

AO •

is also a map in

Therefore

we know what we mean by the t-divisible part, t-torsion part, infinitely t-divisible part, and infinite t-torsion part, of considered in

O

A

•

These are subobjects of

quotient-objects of

V

in the given category

V

O

in A.

A

,

V

i.e.,

It is easy to

see that, under the usual 1-1 correspondence between subobjects of

V

in

A

and quotient-objects of

the t-torsion part of

V

part of

A,

V

exists in

in

O

A

V

in

A,

that:

(1)

exists iff the t-divisible

in which case they correspond (under

the usual 1-1 correspondence between subobjects and quotientobjects of

V

in

A);

(2)

the t-divisible part of

V

in

AO

Exact Couple, Ungraded Case exists iff the t-torsion part of case they correspond:

V

113

exists in

A,

in which

(3) the infinite t-torsion part of

V

in

AO

exists iff the infinitely t-divisible part of

A,

in which case they correspond; and (4) the infinitely t-divi-

sible part of of

V

V

exists in

O

in

A

V

exists in

exists iff the infinite t-torsion part

A, in which case they correspond.

otherwise stated, if we replace quotient objects by their corresponding subobjects throughout, then in passing to the dual category, the notion of nt-divisible part" and nt-torsion part" interchange, and similarly "infinitely t-divisible part" and "infinite t-torsion part" interchange. Example 3.

If

t:V--V

is a map in the abelian category

and if both the t-divisible part of divisible part of

V

V

A,

and the infinitely t-

exist, then we always have

(infinitely t-divisible part of

V)c (t-divisible part

This is because, the infinitely t-divisible part of

V

of~.

is by

definition the image of the composite map:

V

V

and therefore is contained in

n Im(t ),

all integers

therefore the infinitely t-divisible part of in

n 1m (t n )

= (t-divisible part of

V

n':: 0:

is contained

V).

n>O Example 4.

If

t:V .... V

in the abelian category

is a map from the object A,

V

into itself

then Example 3 applied to the dual

category, using Example 2, tells us that, if the t-torsion part of

V

and the infinite t-torsion part of

V

both exist, then

Section 2

114 necessarily (t-torsion part of

V)

C

(infinite t-torsion part of

V).

Example 5. Proposition 1.

Let

A

be an abelian category such that denum-

erable direct sums exist and such that denumerable direct limit is exact.

Then if

t:V + V

is any map from any object

itself, the t-torsion part and infinite t-torsion part V

V

into

of

both exist, and (t-torsion part of

Proof.

V)

= (infinite

t-torsion part of

By Example 1, they both exist.

nite t-torsion part of

V).

By definition of "infi-

V", this latter is equal to

Ker (V (1) + it~ (V (i) , t (i) ) ) .

Since we are assuming that denumer-

able direct-limit is exact, we have that den. direct limit commutes with kernels. H~ Ker (V (1) + V (i) ) .

V(l)+V(i)

i l t - .

is

Therefore, this latter is equal to But

and the map

Therefore in this case (infinite t-tor-

r r V) = l-im Ker (t ). Since Ker (t ) are an increasing r>l sequence of subobjects of V, and since the denumerable direct

sion part of

limit of monomorphisms is a monomorphism in

A

(since denumerable

r l-im Ker(t ) is a subr>l r Ker(t ), r :: 1; and this

direct limit is exact) , i t follows that object of

V,

namely the sup of

latter is by definition the (t-torsion part of

Example 6.

Suppose that

and let

be an object in

V

Then explicitly:

A

V) .

Q.E.D.

is the category of abelian groups, A

and

t:V+V

be a map in

A.

Exact Couple, Ungraded Case (1)

(t-divisible part of

V)

=0

there exists an element (2)

n

EV

For every integer such that

(infinitely t-divisible part of a sequence

(3)

v

{v E V:

v n E V,

n

(t-torsion part of that

~

V)

0,

=0

115

V)

=0

t

n

(v )

=0

n

{v E V:

n, v}.

There exists

such that

{v E V:

3

an integer

n > 0

such

tn(v)=oO}.

These are, of course, all familiar concepts.

There are well-

known examples of cases in which the infinitely t-divisible part of

V

can be strictly smaller than the t-divisible part-

examples also appear, e.g., in Chapter 4, of this book. course, in the case of the category

A

Uf

of abelian groups, the

hypotheses of Proposition 1 in Example 5 hold, so that in this case (infinite t-torsion part of

V) = (t-torsion part of V).

(Of course, since we have seen in Example 2 that, in flipping to the dual category, and replacing quotient-objects by subobjects, that the concepts of nt-torsion" and nt-divisible" interchange, and also "infinite t-torsion" and "infinitely

t-divisibl~

interchange, it follows that, e.g., in the dual category of the category of abelian groups, that there exists an object

V

and

a map

V

is

t:V .... V

such that the infinite t-torsion part of

strictly bigger than the t-torsion part). Definition 3.

Let

A

be an abelian category.

graded) exact couple in the abelian category

Then an (unA

is a diagram

Section 2

116

(1)

in the category

A,

three corners. with maps

such that we have exactness at each of the

That is, it is a pair of objects, V,E,

t:V->-V,

h:V->-E

and

k:E->-V,

such that

together

Imt=Kerh,

1m h =Ker k, 1m k =Ker t. Proposition 2. gory V.

A, Let

d 2 = (h

0

If (1) is an exact couple in the abelian cate-

then let d =h k)

0

k,

0

(h

(tV)

0

denote

1m (t:V ->- V),

an endomorphism of

k) = h

(k

0

0

h)

is a differential object in

0

k =h

A.

0

E. 0

0

a subobject of

Then

k = 0,

so tha t

(E , d )

El = H (E, d) = Ker (d) 11m (d) .

Let

Then there is induced an exact couple, called the derived couple of the exact couple (1),

where the maps map

t,

"t",

l "ht- ",

and

"k" -1

the additive relation

rho r t

are induced from the ' and the map

k, respec-

tively, by passing to the subquotients. Proof. Let

tV

"t"

is a subobject of

and

t

maps it into itself.

denote the induced endomorphism of

is an additive relation from where defined. that

V,

dh = hkh = h -1

1m (r dh 0 r t ) = O.

Since 0

0

= O.

d =h

0

tV k,

into

E.

and since

tV.

r

0

h

(r

t

)-1

Clearly it is everyk

0

h = 0,

Therefore we have that

Therefore the addi ti ve relation:

we have

d (Imth

0

r~l»=

117

Exact Couple, Ungraded Case -1

rho r t

actually maps into the subobject

ker d

of

E.

There-

fore we have the everywhere defined relation induced by -1

rho r t '

from

tv

(Ker dllm d) = E . l

into

By diagram chasing,

I claim that the ambiguity of this additive relation from into

El

is also zero, and therefore that that relation comes

from a uniquely defined map:

tV -+ E . l

by using the Exact Imbedding Theorem category of abelian groups.

(In fact, we can assume, [~~J

that we are in the

The most general image of

under the relation is obtained as follows: such that in

t (v) = O.

Ker (d) 11m (d) = El

v=k(e),

dee) E Im(d), quired).

But, since

for some

l "ht- "

Therefore, Next,

is a map,

kl:Ker(d)-+V.

1m d c 1m h,

vanishes on the quotient,

eEE.

so the image of

k:E-+V

exactness of (1),

k

Choose any

t (v) = 0,

But then h(v)

h (v)

in

by exactness of

h(v) =hk{e) = El

is zero, as re-

is a well-defined morphism:

is a map; the restriction to

Ker(d)

Since

By

d=hok,

vanishes on

Therefore

k : El -+V. 2

into the subobject

v EV

and the image of

Imh.

Imdclmh. Therefore

and therefore the restriction

1m d.

0 E tV

is the most general image of zero under

(tV) -+E . l

on

h (v) E Ker d,

Then

the additive relation. (1),

tV

tV

of

kl

k

kl

of

vanishes k

also

defines a map by passing to

I claim that this map actually maps V

(In fact, we can assume, by

using the Exact Imbedding Theorem, that we are in the category of abelian groups. Ker(d),

then

keel

k(e) EKerh=Imt=tV,

Then we must show that, if is in

tV.

But

as required).

e EE

hk(e)=d(e)=O. Therefore

k

passing to the subquotients, a uniquely defined map as asserted.

is in Therefore

defines, by "k": El-+(tV) ,

Section 2

118

Therefore we have the diagram (2) of objects and maps in the abelian category three corners.

A.

It remains to show exactness at the

By the Exact Imbedding Theorem, this reduces to

the case in which

A

is the category of abelian groups.

Then,

it is proved by easy diagram chasing, which we leave as an exercise. Definition 4.

Let

A

be an abelian category, and let t

V

)V

\f

(1)

E

be an exact couple in the abelian category integer A,

r >0

A.

Then for each

we define an exact couple in the abelian category

which we call the r'th derived couple of the exact couple

(1), and denote: t

each integer

r >

o.

r

The construction is by induction on

r.

The zero'th derived couple of (1) is defined to be the exact couple (1). teger

r.:. 0,

Having defined the r'th derived couple for any indefine the (r + 1) 'st derived couple of (1) to be

the derived couple, as defined in Proposition 2, of the r'th derived couple of (1). Thus, explicitly, e.g.,

VO=V,

EO=E,

V =tV, l

Exact Couple, Ungraded Case El=(Kerd)/(Imd), is induced by by

k,

t,

where hI

d=hok,

119

to=t, hO=h, ht- l , and kl

is induced by

by passing to the subquotients.

kO=k.

tl

is induced

By induction on

r,

we

deduce Corollary 2.1.

A.

gory

Let (1) be an exact couple in the abelian cate-

Then for each integer

r

~

0,

the r I th derived couple

(lr) of (1) is such that

v r = try ' Er

a subobJ' ect of

is a subquotient of

and such that the maps

V, E, and

tr:Vr -+ Vr '

are induced, respectively, by the map,

rh

relation,

all integers

-r

0

r t :V-+E,

r > O.

d r = hrk r ,

where

Also,

t:V-+V,

and the map, k:E-+V EO = E,

and is such that

and d

r

0

k:E r r

-+-

V r

the additive respectively,

Er+l = Ker (d r ) 11m (d r ) , d = 0, all integers r

r > O.

The proof of this Corollary is immediate from the Proposition. Remark:

We might also picture the r'th derived couple of (1)

as:

where

II

til,

"h

0

t -r

ll

and

"k"

denote the unique maps induced

by passing to the subquotients by the map lation

and the map

k,

t,

the additive re-

respectively.

Also, worth

120

Section 2

recording by induction on

r,

Corollary 2.2. gory

Let (1) be an exact couple in the abelian cate-

and let

A,

(lr)

for each integer ( *) r

0

-<-

is

r > 0

be the r'th derived couple, r

~

Then

O.

we have the short exact sequence

[(Ker t) n (trV) )

~E r

.;x'[

r (V/Ker t ) ) t .(V/Ker t r )

in which the maps are the unique maps induced by

-<-

0

,

hand

k, re-

spectively, by passing to the subquotients. Proof.

From the r'th derived couple, as written in e.g. the

last Remark, we deduce the short exact sequence (2)

where

"k"

is deduced from

Again,

~rom

exactness of the r'th derived couple,

Cok (tr:V r -+ Vr) . fore

But

Vr

=

k

trV

by passing to the subquotients.

and

Cok (t :V -+ V ) = trV/tr+lv. r

r

tr

1m h "'-' r

is induced by

t.

There-

Therefore

r

(3) But, considering the map

tr:v-+v,

we see that

trV"'-'V/Ker tr.

Combining this with equations (2) and (3) gives the desired short exact sequence (*r). Remark:

The short exact sequence

(*r) of Corollary 2.2 is such

that, the objects at the right and left depend only on the endomorphism E . r of

t

of

V;

V

and

yet the object in the middle is

Therefore the sequence (*r) gives "a sort of computation" Er

in terms of

V

and

t.

(Of course, this computation is

not perfect, since it is only a short exact sequence.) Definition 5.

Let

Exact Couple, Ungraded Case

121

t

(1)

be an exact couple in the abelian category

A.

Then we asso-

ciate with the exact couple (1) a spectral sequence (Er,dr,Tr)r>O' category

A,

starting with the integer

as follows.

Let

Er

0,

in the abelian

be the object in the r'th

derived couple of (1) as defined in Definition 4, dr

=

hr

0

dr

0

dr

=

define

kr , 0,

r> O.

r> O.

Let

Then as noted in Corollary 2.1, we have that

and that

Er+l

=

Ker (d ) lIm (d r ). r

to be the identity map of

Er+l

Therefore, if we onto itself, then

is an (ordinary, ungraded) spectral sequence in the abelian category (Er,dr,Tr)r>O Remark:

A

starting with the integer

is the spectral

O.

sequence of the exact couple (1).

We have, for simplicity, constructed the spectral se-

quence to start with th·:! integer

O.

Of course, an (ordinary,

ungraded) spectral sequence can always be re-indexed to start at any other fixed integer

rO

if desired, videlicit

(E r - r ,d r - r ,T r - r )r>r o 0 0 - 0 The spectral sequence of an exact couple was first introduced in [E}::J. We now turn to computing the r-fold cycles, r-fold boundaries, permanent cycles and permanent boundaries, and eventually Eoo'

in the spectral sequence of the exact couple (1), in terms

of the given couple (1). quence starts with

First, notice that the spectral se-

EO = E,

ginal exact couple (1).

the "bottommost

object" in the ori-

Therefore, as in every spectral se-

Section 2

122

quence,

Er

is a subquotient of

Er = Zr (EO) /Br (EO)' and

Br(EO)

where

EO = E,

Zr (EO)

and as always,

is the r-fold cycles in

is the r-fold boundaries in

read off exactly which subquotient of

EO'

Therefore, to

E ( = E)

is

o

suffices to be able to determine the subobjects Br(EO)

of

E( =EO)

explicitly.

EO

E,

it

r

Zr(E ) O

and

The next theorem will do this.

First, we prove a lemma.

(1)

B,

of

f:A-+B

Let

Lemma.

Then

B , i E I, be an indexed collection of subobjects i l and let f- (B.) denote the pre-images, which are subLet

~

objects of of

A.

be a map in an abelian category

A,

for all

i E 1.

i <: I,

B , i

I f the subojbects

n B., then the subobjects iEI ~ admit an infimum, and in fact

admit an infimum, call it

B

i E I,

of

A

f-l( n B.) = n (f-l(B.)). iEI ~ iEI ~ Let

(2)

Ai'

i

E;;

I,

denote the images under i E 1.

I f the

Proof:

I

f (A.), ~

f (

2 A.)

(1)

Let

f (D) c B . i

iff

A,

iEI

iEI ~

f(D)C::B , i

all

f,

i E I,

Ai'

the subobjects of supremum,

be subobjects of

'i

iEI

Therefore, iEI,

n

iEI

f

-1

iff (B. l. ~

Dcf- l (

B, all

2 A. , among iEI ~ f (Ai)' i E I, admit a B,

and

f (A. ) • ~

be any subobject of

0

f (Ai)

admit a supremum,

then the images

iff

That is, among subobjects iEI,

which are subobjects of

among the subobjects of

=

A, and let

n B.l. itI ~

Dcf

-1

(B

i

),

A.

DCf-l(B.)

Then

for all

~

i E I,

f(D )c n B. iff iEI ~ of A, we have 0

Dcf

Otherwise stated,

f-l(

-1

iff

(nB.).

-1 iEI ~ Dc:: f (B ), all i

n

B.l

iEI ~

Exact Couple, Ungraded Case (2)

Let

A.ef-l(D).

0

be any subobject of

Therefore,

1.

A. e f- l (D)

o

Therefore, for all subobjects for all f (

i (;; I

iff

I

f(

f(Ai)eD

i E I,

A. e f-l(D)

iEI of

A.)eD.

iEI

t A.). iEI

I

iff

Then

for all

f (Ai) e 0,

i E I,

for all

1.

B.

123 iff

iff

iff

I

f (

A. leD.

iEI 1. we have that f (Ai)e 0

1.

B,

t

Otherwise stated,

f (A.) =

iEI

1.

1.

Q.E.D.

1.

Theorem 3.

A be an abelian category.

Let

Let

t

\}V

(1)

E

A.

be an exact couple in the abelian category

A

bedding from

(or from some "sufficiently large" full exact

A that is a set, and that contains the

abelian subcategory of

objects in (1), and the (t-divisible part of and the

(t-torsion part of

gory of abelian groups.

V),

Let

(1)

E = EO

(Er,dr,Tr)r>O

is in

Zoo(EO) (lr)

(2)

is in

Boo (EO)

be the spectral EO = E.

Let

e

be

Then:

exists, then the e E EO

is in the (t-divisible part of V). r'::' 0,

(t-torsion part of

manent boundaries, e E EO

k(e)

V)

an element

is an r-fold cycle, iff

I f the

into the cate-

exists, and in fact an element

For each integer i.e. ,

Zr (EO) ,

Zoo(EO)

iff

if it exist&

(using the exact imbedding).

If the (t-divisible part of

permanent cycles

V),

if it exists)

sequence of the exact couple (1), so that any element of

Fix an exact im-

Boo(EO),

V)

e E EO

is in

k(e) E trV. exists, then the per-

exists, and in fact an element

iff there exists an element

v

of the

124

Section 2

(t-torsion part of (2r) Br(E

o)'

ment

such that

For each integer i.e.,

v EV

Proof:

V)

r

~

h (v) = e.

0,

an element

e E EO

is in

is an r-fold boundary, iff there exists an ele-

such that

t

r

(v) = 0

and such that

h(v) = e.

We first prove (lr) and (2 ), for each integer r

r> O.

First, consider the short exact sequence (*r) of Corollary 2.2. We have that

Br(EO)CZr(EO)CEO=E,

The map

of (*r) is induced by

"k"

But the map

subquotients. (Ker t) if

n

(trV).

eE Zr(E )' O

e EE

Er

e EE

we must have that

Zr(EO)/Br(E O) =E r •

k:E +V

"k" maps

Therefore, if

must show that if

and

by passing to the

= Zr

onto

(EO) /B r (EO)

is an r-fold cycle, i.e.,

k(e) E trV.

is such that

Conversely, we

k(e) E trV,

then

e

is

an r-fold cycle. First, observe from the short exact sequence (*r)' and the fact that

"h"

is induced by

that it follows that such that

k (e) E trV.

h

Im(h) C Zr(E

by passing to the subquotient,

o).

To show that

Now suppose that e E Zr (EO)'

e EE

is

In fact, from

exactness of the upper left corner in the exact couple (1), we have that

t(k(e»=O.

Therefore,

k(e)E(kert)n(trV).

the exact sequence (*r)' and the fact that k

"k"

From

is induced by

by passing to the subquotient, it follows that there exists

e'EZr{E ) O

such that

k{e')=k(e).

Then

k(e-e')=O,

soby

exactness of the bottom corner in the exact couple (1), we have that

e - e' = h(v),

Therefore,

:3 II ~

V.

But recall that

e=e' +h(v) E Zr(E )' O

Im(h) C Zr(E )' O

as required.

Next, considering the short exact sequence (*r)' and noting that the map tient and that

"h"

is induced by

h

by passing to the subquo-

Er = Zr (EO) /Br (EO)' from the fact that

"h"

is

125

Exact Couple, Ungraded Case r (V /Ker t ) ] [ t· (V/Ker t r ) r must map Ker t

a map from

into

that

into

h

Since

"h"

Br (EO)'

all integers

r> O.

is a monomorphism, it follows more strongly that

Since by exactness of the upper right corner of the exact couple (1), we have of (2 r ),

it suffices to show that

e E Br (EO)'

fore

h (tV) = (ht) (V) = 0,

Then

e E Zr (EO) ,

of

e

makes sense.

of

e

is zero.

e E Ker k

=

to complete the proof

Br(EO)C 1m h.

In fact, let

and therefore the r-fold image

Since

the r-fold image

But then

1m h.

There-

Therefore

Br (EO) C 1m h

completing the

proof of (2 ). r Before proving (1) and (2) and thereby completing the proof of Theorem 3, we note that Theorem 3 can of course be written without reference to the Exact Imbedding Theorem.

So

written, it is Corollary 3.1.

Let

A

be an abelian category.

Let

t V

~

V

~/"

(1)

E

be an exact couple in the abelian category (Er,dr"r)r>O

A.

Let

be the spectral sequence of the exact couple (1).

Then (1)

If the (t-divisible part of

manent cycles

Zoo(EO)

V)

exists, then the per-

exists, and we have that

126

Section 2 Zoo (EO) = k -1 (t-divisib1e part of (lr)

(2)

For each integer

r

~

0,

we have that

If the (t-torsion part of

nent boundaries

Boo(EO)

V)

For each integer

r

~

0,

(1) and (2) of Corollary 3.1.

Theorem 3, take and (1)

A = E,

B = V,

Bi = the subobject

f = k, V,

of Lemma 1, if

exist in

V).

we have that

Completion of the proof of Theorem 3: forms

exists, then the perma-

exists, and we have that

Boo (EO) = h (t-torsion part of (2r)

V).

We prove the equivalent In the Lemma preceding I = {non-negative integers}, all

V,

i > O.

Then by part

i.e. , i f the t-divi-

n k- 1 (t i V) exists in E. But sible part of V exists, then i~l_l i by (lr) for r = i, we have that k (t V) = Zi (EO)' Therefore n k -1 (t i V) = n z. (EO) = Zoo(E O)' Therefore the permanent cycles i>l i>l 1 n k- 1 (t i V) =k- 1 ( n tiV), existsin Eo' -Also, by the Lemma, i>l i>l i.e., Zoo (EO) = k- 1 (t-divisib1e part of V). This proves (1) of the Corollary. Theorem 3, take A. = Ker ti, 1

Next, in the second part of the Lemma preceding A = V,

i> O.

B = E,

f = h.

I = {non-negative integers},

Then by the second part of the Lemma pre-

I Ker t i exists, ie., if i>O the (t-torsion part of V) exists, then I h(Ker til exists. i>O i But by (2r) of the Corollary with r = i we-have that h (Ker t )= ceding Theorem 3, we have that, if

i > O.

Therefore

Exact Couple, Ungraded Case Therefore the permanent boundaries exist in

Z h(Ker

Lemma, part of

l

til = h(

i>O

Ker t

i

),

Le.,

127

EO'

Also by the

Boo (EO) = h (t-torsion

i>O

V),

Note.

which proves

(2) of the Corollary.

Q.E.D.

In the course of proving Theorem 3 and Corollary 3.1,

we have actually proved some additional facts, which are worth recording. Corollary 3.2.

Under the hypotheses of Corollary 3.1, we have

that (lr)

k (Zr (EO»

n Ker

= (trV)

If the (t-divisible part of (1)

k (Zoo (EO»

V)

t,

all integers

r > O.

exists, then also

= (t-divisible part of

V)

n Ker

t.

Always, we have h -1 (B (EO» r

(2r) I f the

(t-torsion part of

(2)

h

-1

(Boo (EO»

Corollary 3.3.

Remark 1.

1 + tV.

= [Ker (tr:V ->- V)

V)

exists, then also

= (t-torsion part of

Under the hypotheses of Corollary 3.1, if both

The unusual thing in this is

thing between

V) + tV.

Boo(EO)

and

Zoo(E )' O

"Ker k = 1m h",

some-

a phenomenon that does not

occur in an "abstract" spectral sequence (as opposed to one that comes from an exact couple). trivial examples in which and

Boo (EO);

It is not difficult to give non-

lm(h)

is strictly between

and also examples in which

also examples in which

1m (h) = Boo (EO).

Zoo(EO)

1m (h) = Zoo (EO);

and

(Of course, if either

Section 2

128

Boo(EO)

or

Zoo(EO)

or both do not exist, then Corollary 3.3 re-

mains valid, if we simply delete the occurrence of or

"Zoo(E )" O

Remark 2.

"Boo(E )" O

or both, whichever or both do not exist.)

In Theorem 3, conclusion (1), and in Corollary 3.1,

conclusion (1), a sufficient condition for

"Z

is given, namely that the (t-divisible part of This result can be improved. conditions for

V,

0

V)

exists.

Namely, necessary and sufficient

to exist is that the decreasing sequence

Zoo(EO)

of subobjects of

to exist

(E )" 00

(Ker t)

And, when this is the case,

r ~ 0,

n (trV) , Zoo (EO)

have an infimum.

is always the pre-image

n «(Ker t) n (trV». A similar imr>O provement of conclusion (2) of Theorem 3 and of Corollary 3.1,

under

k

of that infimum,

is: Necessary and sufficient conditions for

Boo(EO)

that the increasing sequence of subobjects of r (tV) + [Ker (t : V -+ V) 1, is the case,

Boo (%)

r ~ 0,

to exist is

V:

have a supremum.

is always the image under

And, when this h

of this

L (tV) + [Ker (t r : V -+ V) 1) ) • r>O We take the next-theorem seriously.

supremum:

Theorem 4.

Boo

lEd

=

h(

Let t

(1)

be an exact couple in the abelian category (t-divisible part of exist.

V)

and the (t-torsion part of

Then, for the spectral sequence

exact couple (1), we have that short exact sequence:

A such that the

Eoo

(Er,dr"r)r>O

exists.

V)

both of the

And we have a

Exact Couple, Ungraded Case

n

0-<- [(Ker t)

(*)

(t-divisible part of

129

~E ~

V)]

'"

VAt-torsion part of V) ] 0 , ( t (VI (t-torsion part of V)} -
"h"

and

"k"

are induced by

hand

k

by

passing to the subquotients. Proof:

From conclusion (1) of Corollary 3.1 and conclusion (I)

of Corollary 3.2 we deduce the short exact sequence 0-<- [(Ker t)

(I)

n

Z",(E ) }inclusion} O where

"k"

"k II

(t-divisible part of

V}] <:--

Ker k -<- 0,

is deduced from

k

By Corollary 3.3 we have that

by restricting to subobjects. Boo(EO}clm h'=Ker k.

Therefore,

modding out the last two terms in the sequence (1) by the subobject (2)

h

Boo(EO},

we obtain the short exact sequence "k"

0-<- [(Ker t) n (t-divisible in

induces an

epimorphism from

3.1, conclusion (2),

V

Boo(EO}C 1m h.

sion (2), the pre-image of V) + tV.

V) ] <:---

B",(E O}

onto

E

j-nclusion '"

1m h.

By Corollary

By Corollary 3.2, concluunder

h

is (t-torsion in

Therefore, by the Third Isomorphism Theorem,

h

in-

duces a canonical isomorphism (by passing to the quotients) : (3)

[1m hlB", (EO)]

"h"

<: ""

(VI [(t-torsion in

V) + tV]} .

The latter object is canonically isomorphic to: V (t-torsion in V) ] ( t(V (t-torsion in V)} .

Substituting the isomorphism (3)

and this last isomorphism into the last term in equation (2) gives the short exact sequence (*).

Q.E.D.

130

Section 2

Remark 1.

We will call the rightmost group in the short exact

sequence (*), which is a subobject of

00

will be called the left

,

or sometimes the inverse part of

part of

the right part of part of

E

00

is a quotient of

is a subobject of

E 00

Remark 2. A,

E

The leftmost group

E 00 .

or sometimes the" direct part of in (*), which is a quotient of

the right part of

Eoo'

Notice that

E 00 •

and that the left

V,

V.

Let (1) be an exact couple in the abelian category

as in Theorem 4.

Then (1) can also be regarded as being an O

exact couple in the dual category

A

.

(Note that in re-inter-

preting the exact couple as an exact couple in the dual category, V,t,

and

E

change.)

retain their significance, but

hand

k

inter-

It is an easy exercise to see that the derived couple

AO,

induced by (1) in the dual category phic to the derived couple of (1)

is canonically isomor-

in the category

this latter is regarded as an exact couple in

A,

AO.

when

I.e., the

process of passing to the derived couple is a self-dual process. Therefore, likewise, the r'th derived couple of (1) considered in

O

A

(1) in

AO.

is canonically isomorphic to the r'th derived couple of A,

when this latter is regarded as an exact couple in

In particular, it follows that the construction of the

spectral sequence: A

(E r ,d r , l )r>O r

of the derived couple (1) in

is also self-dual; in the precise sense that, if one cons-

tructs the spectral sequence in couple in

AO,

O

A

of (1) regarded as an exact

then the resulting spectral sequence is canoni-

cally isomorphic to the spectral sequence in

A,

in

AO.

(E ,d r r

,l

r

)r>O

of (1)

when this latter is reinterpreted as a spectral sequence Since we have observed in the last section that the

131

Exact Couple, Ungraded Case construction of

Eoo

of a spectral sequence is also "self-dual"

in this sense, it follows that,

Eoo

the exact couple (1) in the category O

A

structed in

A from

constructed in A,

exists iff

Eco

con-

AO

from the exact couple induced by (1) in

exists, in which case they are canonically isomorphic. The reader will notice, most interestingly, that under the hypotheses of Theorem 4, in passing to the dual category the short exact sequence (*) is self-dual - that is, passes into an isomorphic copy of itself. have observed, is self-dual. left part) of

Eoo'

The middle

ter~

Eoo'

as we

However, the right part (resp.:

when re-interpreted in the dual category,

becomes the left part (resp.: right part) of sequence "turns around" under duality.

E . 00

That is, the

Thus, under the hypo-

theses of Theorem 4, the dual concept of the "right part" of Eoo

is the "left part" of

Remark 3.

Eoo'

and conversely.

The right part of

can be thought of as being

E 00

[V/(t-torsion)]/(mod t). For example, if

A is the category of abelian groups, or more

generally the category of right modules over some fixed ring then the right part of

Eoo

[V/(t-torsion)] where as

V

making

t

can be written as: 0 R, R[tJ

is regarded as right

t:V .... V,

and where

R

act as zero. [VI (t-torsion)]

R,

R[tJ-module by making

is regarded as left

Nt"

act

R[tj-algebra by

Another equivalent designation is Q

71.

Ji'[tj In the special case that

A is the category of all right mod-

132

Section 2

ules over a ring ter of

R,

element

R,

and if we have an element

such that the map

t:V'" V

t

in the cen-

is multiplication by the

t E R, then in this case the right part of

E",

can be

written as: [Vi (t-torsion)

J

~

(R/tR) •

R

Remark 4.

In general, if

an object in

A

and if

A

is an abelian category, if

t:V ... V

V

is

is a map, then for each inte-

. by the precise t r -torslon Eart 0 f V we will mean r.::. 0, r Ker (t ). Thus, by definition, the (t-torsion part of V) exists

ger

iff the increasing sequence of subobjects (precise tr-torsion part of

V),

r.::. 0,

has a supremum, in which case these coin-

cide. Notice, that, under the hypotheses of Theorem 4, the left part of V)

n

Em

can be

described as (precise t-torsion part of

(t-divisible part of

latter,

V).

{precise t-torsion in

Occasionally, we will call this V

that is t-divisible}.

From

the short exact sequence (*), we deduce corollary 4.1.

Under the hypotheses of Theorem 4, the following

several conditions are equivalent. (1)

(precise t-torsion part of

V)

(t-di visible part of

V) = {O} ,

(2)

The left part of

is zero,

(3)

E ~ [ vi (t-torsion) "':;. t· (Vi (t-torsion»

Em

]

n

,

the inverse of the isomorphisrr being induced by

"h"

by passing

to the sUbquotients. It should be noted that, e.g., if

A

is the category of

all modules over some Noetherian commutative ring

R,

and if

Exact Couple, Ungraded Case t

is an element in the center of

icle of

R,

then if

V

and if the map

t:V

V

and in the Jacobsen radis multiplication by

is finitely generated as R-module,

any non-zero divisible elements. conditions

->-

R

133

V

t,

cannot have

Therefore in this case the

of Corollary 4.1 hold, so that in this case the

right part of

is all of

E 00

E.

(Many textbooks on spectral

00

sequences make enough hypotheses to force conclusions (3) of corollary 4.1 to hold, as it does in a great many practical spectral sequences that one comes across.

However, in view of

the fact, observed in Remark 2, that the right and left parts of

Eoo

are dual to each other, this can never describe the

general situation.

(E.g., if under the hypotheses of Theorem

4, we have that the conclusions of Corollary 4.1 hold, then the right part of

E 00

in

AO

must vanish for the corresponding

exact couple to (1) in the dual category

AO,

and therefore

this dual couple cannot possibly obey the conclusions of Corollary 4.1, unless

E 00 = 0

(which is usually not the case)).

In

fact, as we shall see later, there is an interesting and important spectral sequence (although a graded one, see section 5) in algebraic topology, known as the Adam's spectral sequence, in which the right part of Eoo

Eco

vanishes, so that in this case

is isomorphic to its left part.

Remark 5.

In Remark 1 following Corollary 3.3, we noted that,

the unusual aspect in the conclusions of Corollary 3.3, was the existence of a canonical object,

Ker k = 1m h,

between

(As we noted, this unusual feature is a property only of couples.)

those spectral sequences that corne from exact

Since

Eoo= Zco(EO)/Boo(E )' O

this extra data is equiva-

134

Section 2

lent to giving a subobject of

Eoo'

Considering that the

rightmost map in the short exact sequence (*) is induced by it follows that the subobject of Im(h) ,

E

h,

induced by the subobject

00

is simply the right part of

E 00 .

(Therefore the short exact sequence (*) can be interpreted as containing all of Corollary 3.3). Remark 6.

In Theorem 3, Corollary 3.1, Corollary 3.2, and Cor-

ollary 3.3, we have assumed that the (t-divisible part of and that the (t-torsion part of

V)

both exist.

V)

All the re-

suIts of these three theorems, and also of Remarks 1-5 above, can be made to go through, if we assume the two weaker conditions listed in Remark 2 following Corollary 3.3, which, as noted in that Remark, are necessary and sufficient for both

Zoo(EO)

Boo(EO)

and

to exist (and therefore those two condiE '"

tions together are necessary and sufficient for

to exjst).

To recall the conditions:

Zoo(EO)

(C ) l objects of

V:

(Ker t)

Boo(EO)

(C 2 ) objects of

exists iff the decreasing sequence of sub-

n

(trV),

r ~ 0,

has an infimum; and

exists iff the increasing sequence of sub-

V: tV + Ker tr,

r ~ 0,

has a supremum.

When these

two conditions hold, then the short exact sequence (*) of Theorem 4 is replaced by the short exact sequence: (*)

o+[n

[vI[

r>O Whenever condition (C ) l

I

(tV+Ker tr)]]+o.

r>O (respectively:

(C » 2

fine the left part (resp.: right part) of

holds, one can de-

Eoo

to be the object

on the left (resp.: right) in equation (*) above. Thus: tion (C ) holds iff the left part of l

E 00

Condi-

(as just defined)

Exact Couple, Ungraded Case exists iff

Zoo(EO)

right part of

135

eixsts, and Condition (C ) holds iff the 2 (as just defined) exists iff

E 00

Moreover, Conditions (C ) l

and (C ) both hold iff 2

iff both the left and right parts of in the short exact sequence (*)

E 00

Boo(EO) E

exists. exists

'"

exist, iff every term

(just defined above) exists, in

which case we have the short exact sequence (*) above.

(Of

course, if the stronger hypotheses of Theorem 4 hold, then the "right part of

E",", "left part of

E","' and "the short exact

sequence (*)" as just defined above coincide with the

cor~es-

ponding constructions in Theorem 4.) Definition 5.

Let

A

be an abelian category, and let

(1)

and t' (2)

be exact couples in the abelian category exact couples from (1) and

f: E

->-

E'

If

A

into (2) is a pair

are maps in the category

t'og=got,

A.

h'og=foh,

A,

Then a map of

(g,f)

A

g:V-+V'

such that

k'of=gok.

is an abelian category, then the category

exact couples in

where

EC(A)

of all

is the category having for objects all

Section 2

136

exact couples in the abelian category of exact couples. abelia~

EC(A)

A

and for maps all maps

is an additive, but in general not

category.

If (1) and (2) are exact couples in the abelian category A,

and if

EC(A),

(g,f)

is a map of exact couples, i.e., a map in

from (1) into (2), then

couples.

Therefore

ditive functor from a map

(g,f)

(g,f) induces a map on derived

"derived couple" may be regarded as an adEC(A)

into itself.

By induction on

of exact couples also induces a map

the r'th derived couples,

r > O.

Therefore, if

r,

(gr,f ) r (g,f)

on

is a

map from the exact couple (1) into the exact couple (2) in the abelian category

A,

then there is induced a map from the spec-

tral sequence of (1) into the spectral sequence of (2).

Other-

wise stated, the assignment which to every exact couple (1) in EC(A),

associates the spectral sequence of the exact couple

(1), is an additive functor from

EC(A)

into Spec.Seq. (Al.

Section 3 Graded Categories

Definition 1.

Let

D

be an additive abelian group, the ele-

ments of which we will call degrees. ~

with degrees in the abelian

graded category) is a category (1) in

deg: Home (A,B) e,

deg (f

0

and

D

(or, more briefly, a D-

together with functions,

defined for every pair of objects

such that, whenever

g E Home (A,B) (2)

D,

-+

e,

Then a graded category

A,B,C

f E Home (B,C),

g) = deg (f) + deg (g)

are objects in

A,B

e,

we have that in the addi ti ve group

D,

and

such that (3)

For every object

exists an object lld :Ad

-+

A

e

in

e,

such that there exists a map

e

e,

and every degree

such that

and such that

lld

there

is an isomorphism

deg(lld) =d

"A D

with degrees shifted by

in

D.

dO.

is an additive abelian group, and if

category with degrees in the abelian group Definition L then for every map we mean

dE D,

Ad' in Axiom 3 of Definition 1, is, intuitively

speaking, If

in

in the category

in the category Remark:

Ad

A

f

in

e

D

(e,deg)

in the sense of

by the degree of

deg(f) ED.

An equivalent definition is as follows: Definition 1'.

Let

D

be an additive abelian group, the

elements of which we will call degrees. 137

is a

Then by a graded

f

138

Section 3

category

e

(1) of

graded

~

the additive abelian

~

0

we mean

A class, the elements of which are called the objects

e, For every

(2)

dE 0,

and every pair d Horne (A, B) ,

the giving' of a set, denoted

e,

d Horn (A, B),

briefly of degree

d

(3) pair

in

e

of objects of

or sometimes more

the elements of which are called the maps from

A

For every triple

d,d'

A,B

of degrees in

into

A,B,C

B, of objects in

e,

and every

the giving of a function:

0,

d d' Horne (B,C) x Horne (A,B)

-+

d+d' Horne (A,C),

called the composition. If

f

is a map of degree

a map of degree pair

(g,f)

d' from

B

d

from

into

C,

A

into

Band

A

under the function "composition" is denoted

into

is

then the image of the

(and is therefore by datum (3) above a map of degree from

g

go f

d + d'

C).

In addition, the data (1), (2), (3) must obey the following three axioms. (Existence of identity elements).

(4) A

e,

in

there exists a map of degree zero,

from the object are objects,

A

dE 0

map of

into itself.

e).

d

and

f oe A = f,

eA

e

o

A

E Horne (A,A) ,

into itself, such that, whenever

have that A

For every object

e

f E Horne (A,B) , A

0

g = g.

(e

A

d

g E Horne (C ,A),

Band

C

we

is called the identity

It is easy to verify from Axiom 4 that

is uniquely determined by Axiom

4, for all objects

(Also, if one dropped, in Axiom 4, the condition that

A

in e

A

is of degree zero, then one could deduce this property from the

139

Graded Categories rest of Axiom 4). ~

(5)

ssociative law of composition).

objects in

e

and

d, d' ,d" ED

g E Home (B, C)

and

d" hE Home (C,D),

d'

If

A,B,C,D

are d

are degrees and if

f E Home (A,B),

then ll

(h

g)

0

f = h

0

0

(g

0

f)

(Shift axiom).

(6)

in

Homg+d'+d (A,D). A

If

is an object in

dE D

is a degree, then there exists an object

maps

TJ

d

E Home (B,A)

d

and

, -d TJ_dEHOm e (A,B) where

ty maps of

B

into itself and of

B

and if in

e,

and

such that

and A

e

are the identi-

into itself, respectively,

as defined in Axiom 4. Definitionsl and l' are essentially equivalent: if

(e,deg)

group

is a graded category with degrees in the abelian

as defined in Definition 1, then for every pair of

D

objects

A,B

in

e,

d

e,

dE D,

and for every degree

define

Then the class of objects

Home (A,B) = {f E Home (A,B): deg (f) = d}. of

Namely,

together with the definition just posed of

for all objects

A, B

in

e

and all degrees

composition induced from the category

e,

dE D,

d Home (A,B)

and the

satisfy the condi-

tions of Definition I', and therefore are a graded category graded by

D

in the sense of Definition 1'.

Conversely, if abelian group

D

e

is a graded category graded by the

in the sense of Definition 1', that obeys the

additional axiom: (7) of degrees

For every pair of objects d,d' ED

such that

d d' Home (A,B) n HOP!e (A,B)

A,B

drid', =~,

in

e

and every pair

we have that

Section 3

140

then the graded category

C

graded by

in the sense of Defi-

0

nition I' comes, by the procedure of the last paragraph, from a uniquely determined graded category with degrees in sense of Definition 1.

Explicitly, given

C

in the

0

as in Definition

Hom (A, B) = U HO~ (A,B), for all C d(;D Then the class of objects of C, together

I' that obeys Axiom 7, define

objects

A,B

in

C.

with this definition of induced from

C,

HomC(A,B),

and with the composition

and with the obvious definition of for all

deg: HomC(A,B) +0,

A,B

objects in

C,

is the unique

D-graded category in the sense of Definition 1 that yields

C

by the procedure of the last paragraph. Remark:

Of course, technically, Definition I' is slightly

more general than Definition 1. abelian group

Namely, given a fixed additive

a graded category with degrees in

0,

in the

0

sense of Definition 1 is actually exactly equivalent to a graded category graded by

0

in the sense of Definition I' that obeys

the additional Axiom 7 above. graded category graded by

0

However, of course if

Cdisj ,

graded by

D

C

have the same objects as

c,

in the sense of DefiniNamely, let

and define

d d Hom d' . (A,B) = {d} x HomC(A, B) , all objects A,B in C l.S] degrees d ED. And take the obvious composition for

duced from

C.

Then of course

C

C

by another

tion I', that does obey the additional Axiom 7. disj C

is a

as in Definition I' above, and

fails to obey Axiom 7, then one can replace graded category,

C

and

disj C

c,

all

dis C j

in-

are canonically

isomorphic (for the obvious definition of isomorphism of "graded categories graded by the fixed additive abelian group Definition I').

Therefore every graded category

C

0"

as in

graded by

Graded Categories

o

141

in the sense of Definition l' is canonically isomorphic to

that obeys the additional Axiom 7, and therefore one Cdisj comes from a uniquely determined graded category with

0

degrees in

in the sense of Definition 1.

Hence, the two

concepts are essentially equivalent, and we shall often switch freely between Definition 1 and Definition 1'. Example 1.

Let

abelian group.

A

Then we define

0

by the abelian group lows.

The objects of

jects of

A

0

be any category and let AD,

be any additive

a graded category graded

in the sense of Definition 1', as folAD

d

are all families:

indexed by the set

D.

(A ) dED

of obe Given two objects (A ) eED

e

and (B) eED' and a degree dE 0, the maps of degree d from e e e (A leED into (B )eED are the indexed families: (f leED e e d+e ) , all e ED. We define the obvious where f E HomA (A , B composition using the composition in

A.

Then

category graded by the additive abelian group of Definition l' above.

We call the objects of

objects in the category

A

AD 0,

is a graded in the sense

AD

the graded

graded by the additive abelian group

0, or more simply the D-graded objects in the category we call category

AD A.

construction (1)

If

Three especially important special cases of the AD

are:

0 = (Z ,+), A.

then we have

AZ,

the category of

A z-graded object in

a singly graded object:

therefore,

AZ

A

0 = (Z

x

Z, +) ,

then we have

Z-

is also called

is also called the

graded category of all singly graded objects in If

and

the graded category of all D-graded objects in the

graded objects in

(2)

A;

AZxZ,

A. which we call

similarly the graded category of all bigraded objects in the

142

Section 3

category

A. More generally, if

(3)

n

~

then we have similarly

0,

n

A (~ '+),

the graded category of all n-graded objects in

A,

a graded category graded by the additive abelian group D =

n

(~n,+).

A,

graded objects in If

A (~,+)

(We sometimes call

D

if

n

the category of multi-

is understood.)

is an additive abelian group, then we will some-

times use the more brief term D-graded category in lieu of "graded category graded by the abelian group Example 2.

Let

Then define all

A

~n+d+l

0

gers

n.}

Then

abelian group (~-indexed)

(A*,B*) = {(fn)nE?: fn

An

into

fn = f n + l

0B*

0

Co(A) (~,+).

n

dA*

Bn+d,

all integers

A

gree

d"

n,

and such

for all inte-

is a graded category, graded by the Co(A)

is the ?-graded category of all

cochain complexes in the additive category d,

and for every pair

cochain complexes of objects and maps of

Hom~o(A)

A,

is a map in the

Hom (An,B n + d + l )

i~

(Notice that, for each integer

of

(~,+).

cochain complexes of objects and maps in

A from

that

D=

the ?-graded category having for objects

Hom~O(A)

and such that category

be an additive category and let

Co(A),

(~-indexed)

D".

(A*,B*)

A,

A. A*,B*

of

that the elements

are the "maps of cochain complexes of de-

in the usual sense).

Notice, 'in this case, that i f

A* E Co (A)

and

is a degree, then an example of a cochain complex

dE 7! A*

that sat-

d

isfies the conclusions of Axiom 6 of Definition I' is the cochain complex

A*

the coboundaries

d

such that dn * = d n+d A*, Ad

with degrees shifted by

all all

n E ~.

n E ~,

and with

Clearly, this is

d", perhaps justifying the Re-

"A*

Graded Categories

143

mark following Definition 1. Example 3.

If

A

is an additive category, then we define a

(~ x~,+)-graded

category, which we denote

(~x~,+)-graded

category of double complexes of objects of

The objects of maps in

A:

D(A)

and call the A.

are all double complexes of objects and

that is, all doubly indexed families:

(AP,q'd~i:o),a~O:l)) (p,q)E~x~ dP,q

D(A),

. AP,q -+-AP+l,q

and

(1,0)'

where

aP,q

AP,q

is an object in

. AP,q -+-AP,q+l

(0,1)'

A,

are maps in

p,q _ 0 "p,q+lo "p,q _ such that 0(1,0) 0 d(l,O) - , 0(0,1) 0(0,1)-0 and p+ 1 I q "p, q _ p, q+ 1 "p I q d(O,l) 0 O(l,O)-a(l,O) 0(0,1)' all integers p,q. If

A

,

"p+l,q

0

and

B**

are two such double complexes, then for every pair of

integers

d,d',

A** into

B**

be:

A**

{(fP,q)

: fP,q AP,q -+- BP+d,q+d' (p,q)Eilxil' ,

A,

category

we define the maps of bidegree (d,d') from (d,d' ) (Le., the elements of HomD(A) (A**,B**}) to

all integers

p,q,

is a map in the

such that in

Horn (AP,q BP+d+l,q+d') A' ,

in

Horn (AP,q,BP+d,q+d'+I),

and such that dP+d,q+d'

0

fP,q

(0,1)

all integers D(A)

= fP,q+1

p,q}.

0

aP,q (0,1)

Then, with the composition induced from

is a graded category, graded by

~ x~.

It is the

~x~-

graded category of double complexes in the additive category Notation:

In a

(~x~)-graded

A,

A.

category, by the bidegree of a map

we mean simply the degree of the map, in the sense of Definition 1.

Example 4, integer

> O.

Let

A

be an additive category and let

n

be an

Then an n-ary complex in the additive category

is an indexed family:

A

Section 3

144

for every

n

(PI, ... ,Pd~zr,

and for every PI,···,p a. n

Pl'···'p A n

(PI' .•. , P n ) E: zrn

and every integer

is a map in the category

1

Pl,···,p· I,P.+I,p·+I,···,p A 11 1 n, Pl,···,p· I,P.+I,p·+I'···'P a. 11 1 n

is an object in

A

i,

A, I 2 i 2 n,

PI,···,p A n

from

into

such that 0

PI'···'P a. n= 0

1

in

1

HomA(A

PI'···'P

(PI' .•. 'P n ) Ezr

n,A n

Pl,···,p· I,P.+2,p·+ I ,···,p 11 1 n) and all integers

i,

12i2n,

for all and such that

a.PI'··· ,Pn 1

PI'··· ,P

a.

J

n

in

Pl,···,p Pl,···,p· 1,P.+l,p·+l,···,p· 1,P.+I,p·+l,···p HomA(A n,A 11 1 ]J J n), If

whenever B*···*

are n-ary complexes in the additive category n

(e l , ... ,en) E zr, A*···* (f

A*···*

into

then a map of multidegree

B*···*

A,

(e ,.·· ,en) l

and and if from

is an indexed family:

PI' ... 'P n )

n, indexed by zrn where (PI'··· ,Pn)Ezr PI,···,p PI,···,p Pl+el, .•• ,p +e f n A n .... B n n is a map in A, all Pl+el,···,p +e Pl,···,p (PI' ... ,P ) Ezrn, such that d n no f n n i PI,···,Pi-I,Pi+I,Pi+I,···,Pno Pl,···,Pn n f a i ' all (PI' ... , P n ) Ezr , all integers i, I ~ i ~ n. Then, for each fixed positive integer n,

the class

category (el, .•• e n )

A,

Mn(A)

of all n-ary complexes in the additive

together with the maps of multi-degree just defined for all

vious composition induced from

A,

is a graded category with

Graded Categories n (7 ,+)

degrees in the abelian group

This is the

I' .

~

n

in the sense of Definition

-graded category of all n-ary complexes in

the additive category Notation:

145

A,

Given a map in a

which we denote by ~

n

M (A).

n

-graded category in the sense of

Definition I, by the multidegree of the map we mean the degree of the map as defined in Definition 1. (Of course, Examples 2 and 3 above are the special cases of Example 4, in which

n =1

or

2

respectively; where, e. g. ,

in identifying Example 3 with the special case

n =2

of Example

4, we are making the notational identification: ~p,q

01

"p,q = "p,q °(0,1) °2 '

'

Example 5.

Let

D

a

11' t

1n egers

p,q.

)

be the trivial additive group

D = {O}.

Then of course the notion of "graded category graded by

{OlIO

is equivalent to that of "category" in the usual senSe([C.A.

n. This

example shows that the notion of "graded category" can be interpreted as a generalization of that of "category". Remark:

In general, given a graded category graded by the

additive abelian group n

D = ~ ,+);

(~xZ ,+);

or

~n ,+)

where

is a non-negative integer, we will sometimes call such a

graded category a singlY graded category; a biqr!ded category; or an n-graded category, respectively.

(Thus, in Example 5

above, it is noted that the notion "O-graded category" and the usual notion of "category" essentially coincide). D=

~/2Z,+),

Example 6.

Let

order two.

Then certain D-graded categories are occasionally

considered in mathematics. we have gory.

A (~/21)

the additive cyclic group of

E.g., if

A is any category, then

as defined in Example 1, a

(~/2~)-graded cate-

A special case of this is the one in which A = {abelian

146

Section 3

groups}.

If

X

is a base pointed topological space, then the

usual complex K-functor evaluated at two.

Therefore

K

X

is periodic of period

can be regarded as a contravariant functor

from the category of base pointed topological spaces and homotopy classes of maps into the underlying category (in the sense of Definition 1) of the (;;?/2iZ")-graded category tor spaces} ;;? /2:i.

Of course, f or every con t'lnuous b ase pOln ' t-

preserving function degree zero.

f,

e.g., if

cal space such that

t-

0,

we have that

K

1

(X)

t-

then" cupping with

in the

K(f)

is a map of

C

of

However, there are interesting maps of degree one

in complex K-theory:

u

C={complex vec-

(~/2:i)-graded

X

and i f

0,

u"

is any base pointed topologiis such that

is a non- zero map of degree +1

category

C

from

K(X)

into itself.

The notion of "graded category with degrees in an additive abelian group" as in Definition 1 above, and also Definition l' above, are original to Birger Iversen of the University of Aarhus. Let group

C

D.

"deg",

be a graded category with degrees in the abelian

Then, by Definition I, if we ignore the function

C

is also an (ordinary) category (without grading).

The category

C

has a natural subcategory, which we denote

CO'

and call the category of maps of degree zero of the D-graded category

C:

Namely, take for

objects the objects of zero in C.

(CO

C,

Co

the category having for

and for maps, all maps of degree

o

(A,B) = Hom (A,B), all objects A,B in C C o is of course an ordinary category, without grading).

C.

Thus,

Proposition 1. belian group

Let D

Hom

C

and let

be a graded category graded by the aA

and

B

be objects in

C.

Let

147

Graded Categories dE 0,

and choose

n_ : A_ -+A d d

an isomorphism in

(by Axiom 3 of Definition 1).

-d

e

of degree

Then there is induced a spe-

cific 1-1 correspondence between the set of all maps of degree d A_

from

A

into

into

d

Proof:

B,

and the set of all maps of degree

0

n_d

g -+ g

0

(n_

d 1-1 0 Hom (A,B) ---t~ Home(A_d,B). e on 0

B:

is such a bijection, with the inverse function: )

d

-1

•

In general, if belian group degree

dE D,

Definition 1.

e

with range

d,

We call

A,

e

and every

Ad'

and an iso-

as in Axiom 3 of

call it Ad'

in

the object

A

fixed isomorphism

with degrees shifted

nd: Ad .... A

the shift isomorphism (of degree

all objects

Definition 2.

A

let us fix an object, call it

and the

d,

is a graded category graded by the a-

then for every object

0,

morphism of degree

A),

from

In fact, by Axioms 2 and 3 of Definition 1, the function:

f .... f

2Y

0

A

Let

in 0

e,

all degrees

of degree d

with range

dE D.

be a fixed additive abelian group.

an additive graded category indexed

2Y

d

Then

the abelian group

0,

or

more briefly, an additive D-graded category, is

(1)

D,

A graded category

e

indexed by the abelian group

together with (2)

The structure of additive abelian group on

for all objects

A

and

B

in

e

and all degrees

d

Home(A/B) dED ,

such

that (3)

Whenever

A,B,C/D

are objects in

e,

d,d' Id" E 0

are degrees and f and g E Hom~ (A, B) are maps of the same degd' d" and k E Home (D/A) then we have that ree I h E Hom e (B I C) h

0

(f + g) = h

0

f + hog

in

d+d' HOme (A/C)

148

Section 3

and (f

+ g)

If

C

k = f

0

0

k +g

Hom

in

k

0

d+d" C

(D,B) .

is an additive graded category with degrees in the

abelian group

D,

C,

ree zero of

Co

and if

then clearly

is the category of maps of deg-

Co

is an (ordinary, ungraded)

additive category. In addition, by Axiom 3 of Definition 2 (the distributive laws) we trivially have that

(*)

If

A

category dE D

and

B

if

C,

are objects in the additive D-graded

f,g: A-+B

are maps of degree

if

is any degree and if

the shift isomorphisms of degree

B

0,

d

with ranges

A

and

respectively, then in

A converse of this observation is true, which can be used to characterize additive D-graded categories the graded category structure on

C

Co

Co

(ignoring the

Namely,

Proposition 2.

C

in terms of

(ignoring the addition)

and of the additive category structure on grading).

C

Let

D

be an additive abelian group and let

be a graded category graded by the abelian group be the category of maps of degree zero of

an (ordinary) category (without grading). given additive binary compositions all objects

A,B

Co

Then

Let

Co

is

Suppose that we are on

in the (ordinary) category

(ordinary) category

C.

D.

Hom

Co

Co

(A,B)

for

such that the

becomes an (ordinary, ungraded) addi-

Graded Categories

149

tive category, and such that Whenever

(*)

and

d

A,B

is a degree in

B)-l ( Tld

0

(f + g )

0

D

f,g E Home (A,B)

CO'

are objects in

o

we have that

A = '(ndB)-l Tld 1

0

f

0

A + (B)-l T]d Tld

g

0

A T]d

0

Then there exists a unique way of introducing

in

d

additive binary compositions on and all degrees

dE D,

A,B

for all

Homc(A,B)

c

in

such that the D-graded category

C

be-

comes an additive D-graded category, and such that the thereby induced structure of additive category on one,

Co

is the prescribed

C }.

o

Proof:

First, let us prove uniqueness.

If we have any collecd

tion of additive binary compositions on the that the graded category

C

with degrees in

ditive graded category with degrees in induced addition on (+AB)ABEC ,

,

0

'

Co

then:

Hom (A,B) 's

D,

D

such

becomes an ad-

and such that the

yields the given additive structure If

dED,

A,BEC

and

d

f,gEHomC(A,B),

by the distributive law (Axiom 3 of Definition 2) we must have (1) However, on the right side of equation (I), since

pends only on the additive structure of in

A

and d are of degree zero, the right side of equation (1) de-

A gO T]_d

C.

Co

f

0

Tl_

and the composition

This proves uniqueness.

Next, let us notice that, although the hypothesis (*) depends

~

priori on the choice of elements

an isomorphism of degree all degrees

dE D,

d

with range

A Tld A,

such that for all

A Tld

A E A,

that actually condition (*) will hold for

is and

150

Section 3

one such set of choices iff it holds for any other. claim that if (*) holds, if

A,B t;; C,

f,g E Home

resp.:

degree

o

d

(Proof:

and if

(A,B),

with range

A

-1

L

-1

(f+g}n=L

B -1

[(nd)

A,

w,

dE D

and if

is any isomorphism of

B,

respectively

then

There exist unique isomorphisms of degree zero

such that n = nd w

n,

if

I.e., we

-1

0

e,L

Then

e,

B -1 A -1 B -1 A B -1 A (n d ) (f+g)nde=L rInd) fnd+(n ) gn ]8= d d

A

fndJe+l

-1

B -1

rInd}

A

gndJe=w

the last equality holds since

Co

-1

fn+w

-1

(The next-to-

g6.

is an additive category; the

equality before that fo11ows'from (*}}). Next, let us prove existence of the indicated additive structure on

C.

If

A,Bt;;C,

d

and

dE D

f,g E HOme (A,B) ,

then

define (2)

Then by Proposition 1,

d

Home (A,B)

becomes an additive abelian

group with this definition of addition.

To complete the proof

of the Proposition, it is left to verify Axiom 3 of Definition 2. First, notice that, by the hypothesis

(*), the definition

of sum, equation (2), is identical to the definition of sum: (3)

(Proof:

By (*), applied to the maps,

and to the isomorphisms of degree have that

-d,

B -1

(nd)

A n-d

f

and

and

B -1

(n d )

B -1 (nd) ,

g, we

Graded Categories

151

B [ (nBd) -1 0 f + ( B) -1 ] A B (B) -If A B (B) -1 A lld 0 'I lld 0 9 0 ll_d = fld 0 lld fl_ d + lld 0 fld 9fl_ d · The latter is clearly equal to

A

Composing on the right with side of equation

(ll_d)

-1

,

we obtain that the right

(3) equals the right side of equation (2), as

required) .

o

Next, suppose that where (4)

A,B,D E e,

d

t:;

D.

d

and

f, g E Horne (A, B)

hE Horne (D,A) ,

'rhen we claim that

(f+g) oh=foh+goh.

In fact,

8 = (ll~d)-l

let

morphism in such that

e

of degree

h = hO 0 8.

A

and

hO = h 0 ll_d. and

hO

[(f + g)

0

d

Then

8

is an iso-

is a map of degree zero

And

(f+g) oh= (f+g) 0 (hOo8)

ho]

8

0

But, by equation (2), f 0 h + 9 0 h = [f 0 h8

-1

+ 9 0 h8

-1

] 0 8 = [fh

o + 9h O]

These two equations imply equation (4), when both of degree zero.

d

Next, let

f,gEHome(A,B),

and

dE Dare arbitrary, and let

and

d' E D

tha t

f 0 8

is arbitrary.

f

0 8 and

are

where

A,BEe

d'

hE Horne (E ,A), where

Then choose an isomorphism

is of degree zero.

9

Then also

g

zero, so by equation (4) applied to the maps

0

8

8

such

is of degree

f 0 8, 9 0 8,

have: (f + g) 0 h = (f 0 (88- 1 ) + go (88- 1 »

EEe

0h =

[(f8) 08- 1 + (98) 08- 1 ] 0 h= [(f8 +g8) 08- 1 ] 0 h=

we

152

Section 3 [fe + ge) [f

0

0

(ee- l )

(6- 1 h) = [(fe) 0

h) + [g

0

0

(e-lh)) + [(ge)

(ee- l )

0

0

(e-lh))

h) = fh + gh.

This proves one of the two distributive laws; the other is proved similarly. Remark:

Q.E.D.

Proposition 2 implies that, if

C

belian group, and if additive abelian group so that

C

0

is an additive a-

is a graded category graded by the 0,

then to introduce an addition on

C

becomes an additive D-graded category in the sense

of Definition 2 is exactly equivalent to introducing an addition on the

Co

(ordinary, ungraded) category

(consisting of

all objects, and of all maps of degree zero, in

Co

C),

such that

becomes an (ordinary, ungraded) additive category in the

usual sense, and such that condition (*) above holds.

There-

fore, an alternative equivalent definition of "additive graded category" is: Definition 2'.

Let

0

be an additive abelian group.

additive graded category with degrees in

0

A graded category

(2)

An additive binary composition on all objects

A,B

is:

C with degrees in

(1)

in

ungraded) category Co

CO'

Then an

Hom

0,

Co

(A,B)

for

such that the (ordinary,

together with this addition

becomes an (ordinary, ungraded) additive category, and such that (3)

Condition (*) above holds.

Notice also that, in the course of proving Proposition 2, we have verified that when we have data (1) and (2) of Definition

153

Graded Categories

2' above, then the verity or falsity of condition (*) above, ~

which

A (lld)A,d for A where lld is an

priori depends on the choice of a family

all objects

e

in

A

and all degrees d

isomorphism of degree of degree

d

with range

with range

A"),

dE D, A

(the "shift isomorphism

is actually independent of the

choice of such a family.

e

Let group

D.

be a graded category with degrees in the abelian

Then if we regard

e

as a category as in Definition

I, it is not difficult to see that, unless

D

is the zero group,

then virtually never will direct products of two objects in the

e

category

exist.

e

Also, if

is an additive graded cate-

gory graded by the abelian group

e

group, then if we regard

e

then

D,

and if

one of degree

d

A

into

for each

B

dE 0)

nel (or cokernel) of a map in

e"

(since there is no

for each

A

and

B,

but

so that the concept of "kerdoesn't even make sense, if

e

we use just the category structure of ignoring the grading.

is not the zero

as a category as in Definition 1,

is not even a pointed category,

unique "zero map" from

D

as in Definition 1,

It is better to think in terms of Defi-

nition 1', and to pose some "graded" definitions. Definition 3.

Let

D

be an additive abelian group and let

be a graded category with degrees in (1)

Let

I

be a set and let

family of objects in is, an obj ect

A

in

e. e,

ree zero, such that, if ree in ree

d,

D,

and

D.

Then:

(Ai)iEI

be an indexed

Then a direct product of the together with maps B

f i: B -> Ai

is any object in are maps in

then there exists a unique map

e

e

1T i:

A -> Ai

e,

d

(Ai)iEI of deg-

is any deg-

all of the same deg-

f: B ->- A

(necessarily

Section 3

154

of degree

d)

such that

direct product of

TIi

(Ai)iEI

0

f = fi'

all

i E I.

Clearly, if a

exists, then the usual argument

shows that it is unique up to canonical isomorphisms. (2) D,

If

and if

is an additive graded category with degrees in

f: A -->- B

a kernel of and where

C

f

is a map in

is a pair

l:K .... A

C

(K,l)

of arbitrary degree, then

where

is a map such that

K f

whenever

j: L .... A

f

then there exists a unique map

0

j = 0,

0

is an object in

1 =

0

C,

and such that,

is any map (of any degree) such that

Clearly, if a kernel

(K,l)

of

f

h: L .... K

such that

exists, then the usual

argument shows that it is unique up to canonical isomorphisms. Also, i f a kernel of f

f

(K, l)

of

(K, l)

is any kernel of

where

d

=

exists, then there exists a kernel

such that

deg (l) ,

is of degree zero f

deg(l) t- 0,

and

is such a kernel of

f.

-

in fact, if

then

K (K_d,l on_d)'

Therefore, since i t

is no restriction on generality of existence, we henceforth require, for convenience, that

~.his

the definition of a "kernel

(K,l)

additional condition hold in of a map

f

of arbitrary

degree" - namely, we insist henceforth, as we may, that deg (l) = O. (3)

If

C

is a graded (respectively:

category with degrees in

D,

then there exists a unique struc-

ture of D-graded (respectively: the dual category in

C,

CO

of

additive D-graded) category on such that for every map

we have that the degree of

map in the dual categroy f: A .... B

C,

CO

additive graded)

f: B .... A

f: A .... B

considered as a

is the same as the degree of

considered as a map in the D-graded category

C

(and

Graded Categories

155

in the additive case, such that, in addition, if f +g

then the sum

of

f

and

sum considered as a map in

g

C.)

in

f, g E Hom

d

C

(A,B) ,

coincides with their

CO

Therefore, from (1) and (2),

taken in the dual category, we know how to define direct sums of objects in a D-graded category, or cokernel of a map in an additive D-graded category, when these exist. Proposition 3.

(1)

Let

{Ai)iEI

jects in the graded category abelian group

with degrees in the additive

Then

D.

graded category

C

be an indexed family of ob-

Ell A. (resp.: IT A.) exists in the iEI l iEI l in the sense of Definition 3 iff the direct

C

sum (resp.: product) of

Co

graded) category

{Ai)iEI

exists in the (ordinary, un-

of maps of degree

0,

in which case they

coincide. (2)

Let

f: A-+ B

D-graded category

C.

be a map of degree Then

Ker f

the additive graded category Ker (T]~d

iff

0

f)

(resp.:

C

(resp.:

in the additive Coker f)

exists in

in the sense of Definition 3

Coker (f

T]~d))

0

Co

nary, ungraded) additive category 0,

d

exists in the (ordi-

C of degree

of maps of

in which case they coincide.

IT A. exists in CO' with projeciEI l i E 1. Then i f f. : B+A. are maps of the same degtions 'IT i' l l B then fi 0 T]_d: B_ d + Ai are maps in CO' ree d, all i E I, E.g. , suppose that

Proof:

whence there exists a unique map 'IT

i

0

with

f = fi

B

0

T]-d'

(T]~d)-l,

of degree for all

it follows that

i E I.

uct of the

Ai'

in

Co

such that

But then composing this equation on the right

from

d

f: B_ d-+ Ai

B

into

A

Conversely, if i E I,

f

0

(T]~d)-l

is the unique map

such that A, 'ITi'

i E I,

is a direct prod-

in the D-graded category

C

in the sense

Section 3

156

of Definition 3, then considering the universal mapping property d = 0,

in (1) of Definition 3 in the special case that

i E I,

A,TT i'

it follows

obey the universal mapping property condi-

tion for a direct product of the

iE I,

Ai'

in the (ordinary)

CO.

category

The other assertions are proved similarly. Remark:

Perhaps a stronger observation that Proposition 3 is

the fOllowing:

Let

D

V be a category that is a set.

be a D-graded category, and let Let

F

into

be a contravariant, resp.:

CO.

Then

V

covariant, functor from

there exists an object 1i:F(i)->-L,

L

in

C,

CO'

of degree zero for every object f: i ->- j

resp. :

and such that, whenever

resp.: d

C,

and

d

i E 1. Remark:

V

we have

D,

i

F(f) B

in

V,

OTT.=1T., J 1

is any ob13i:B ->- F (il,

and

13i:F (i) ..,. B, are any set of maps all of the same degree

0 F (f) = B. , J 1 sarily also of degree

13:L"" B,

in

is any fixed degree in

such that for every map

resp. :

iff

together with maps TTi:L->-F(i),

such that for every map

ject in

V

has an (ordinary, ungraded) inverse, resp.:

direct, limit in the (ordinary, ungraded) category

resp.:

C

be an additive abelian group, let

B.

such that

TTi

f: i ..,. j

in

we have

V

then there exists a unique map (necesd) in the category 0

13 = 13 , i

resp.:

13

C, 0

13:B..,.L,

1i = 13 , i

resp.:

for all

(The proof is the same as that of Proposition 3). The substance of Proposition 3 is to tell us that uni-

versal mapping property constructions in a D-graded category

C,

that depend on the D-graded structure in the sense of Definition 3 or the Remark following Proposition 3, are closely related to the corresponding usual (ordinary, ungraded) category-theoretic

Graded Categories

157

constructions in the (ordinary, ungraded) category of degree zero in

C.

Co

of maps

Notice however, that such constructions

are very different from the corresponding (ordinary, ungraded) category-theoretic constructions on the underlying category of

C,

ignoring degrees.

These latter category-theoretic construc-

tions rarely exist if D 1 {O}, and are not of much interest. There-

Co'

fore,

C,

not the underlying category of

is the (ordinary,

ungraded) category most closely connected with D-graded univer-

C.

sal mapping property constructions in the D-graded category If

D

tive graded category with degrees in in

C,

then we define

Coim(n~d the f,

1m f) 0

f)

D,

and if

Coim f = Coker Ker f,

f

is a map

1m f = Ker Coker f, Coim f

C

exists in the additive D-graded category (resp.:

Im(f

0

n~d»

A=domain f,

d=deg f.

If both

(If

f.

d = deg f,

then it is easy to see that D-gra d e d category

C

f,

l'ff

CO'

Coim f

then there is induced a natural map from of the same degree as that of

where

and

Coim f

B = range,

1m f into

exist, 1m f

which we call the factorizaB = range f, and

Coim f

(nBd)-l of

and

1m f

A = domain f,

exist in the

has a Coim and 1m in the

usual sense in the (ordinary, ungraded) additive category has a

Coim

and

1m

CO.

three equivalent conditions hold for a map and

a"

And, when these f

in

C,

are the respective factorization maps of A

f on_

d

Co

in the usual sense in the

(ordinary, ungraded) additive category

a,a'

iff

exists in the usual sense in

(ordinary, ungraded) additive category

tion map of

is an addi-

Then by Proposition 3,

whenever these are defined. (resp.:

C

is an additive abelian group and if

if f,

of

in, respectively, the D-graded additive

Section 3

158

C,

category

and the (ordinary, ungraded) additive category

e

exist unique isomorphisms d = deg (f) a = a"

0

Co

the (ordinary, ungraded) additive category

and

p

in the graded category

CO'

then there

of the same degree C,

such that

a =

e

0

a

I,

p.)

Proposition 4.

Let

D

be an additive abelian group and let

be a non-empty additive D-graded category.

Let

Co

C

be the

(ordinary) additive category of maps of degree zero in the graded category

C.

Then the following two conditions,

(1) and

(2) below, are equivalent: (1)

(a)

If

A

and

exists a direct sum of (b)

and degree in B

into

A

A

and

and B

f:A ... B

and i f

f (cl

B

C,

then there

in the sense of Definition 3.

are objects in

C,

is a map of degree

C,

if

d

d

is a

from

A

then the kernel and co-

exist in the sense of Definition 3. The hypotheses as in (b) , i f

is the factorization map of

f,

d = deg (f»

(necessarily of degree (2)

are objects in

in the graded category

kernel of and

D

If

B

then

Ct

a: Coim f "'Im f

is an isomorphism

.

The (ordinary, ungraded) additive category

Co

is an

abelian category. Proof:

Follows immediately from Proposition 3, and from the

observations about factorization maps immediately preceding this Proposition. Definition 4.

Let

D

be an additive abelian group.

Then an

abelian graded category (or a graded abelian category) graded by the abelian group gory

C

D

is a non-empty additive graded cate-

graded by the abelian group

D

such that either of the

Graded Categories

159

two equivalent conditions of Proposition 4 hold. We sometimes use the phrases abelian D-graded category, or D-graded abelian category, for "abelian category graded by D". Remark 1.

Notice, therefore, that if

group, and if

A

D

is a non-zero abelian

is a D-graded abelian category, then the un-

derlying category of

A

in the sense of Definition 1 is not an

abelian category - since in fact

A

is not even pointed.

How-

ever, of course, the category

AD

(of all objects of

A

and

of all maps of degree zero in

A)

is an (ordinary, ungraded)

abelian category. One way of interpreting some of the above material is that Definition l' is perhaps superior to Definition li since e.g. Definition I introduces "the underlying category of the D-graded category

C",

which indeed is a "white elephant":

The in-

teresting (ordinary, ungraded) category associated to aD-graded category

C is

minology:

If

CO. D

We therefore introduce the following ter-

C

is an additive abelian group, and if

is

a D-graded category, then we call the underlying category of

C

in the sense of Definition 1, the white elephant category of

C.

Thus, if

D

is an additive abelian group and

category graded by

D,

C,

of

is a graded

then there are associated two natural

(ordinary, ungraded) categories to gQ£y

C

and the category

all mapsof degree zero of

C.

Co

C:

the white elephant cate-

of all objects of

C

and

It is clear from the terminology

which is the more important one. Remark 2.

By the Remark following Proposition 2, it follows

that:

D

If

is an additive abelian group and if

C is a

Section 3

160

graded category graded by the abelian group

C

D,

then to make

into an abelian graded category graded by the group

is equivalent to:

D

it

Make the (ordinary, ungraded) category

Co

C

of maps of degree zero of

into an (ordinary, ungraded) a-

belian category (in the usual sense), in such a way that condition (*) of Proposition 2 holds.

Since the addition

on an a-

belian category is determined by the category structure, from this last observation it therefore follows that:

D,

d

e

(for all

Home (A, B) Is

e

bel ian group and if

dED)

such

If

D

is any additive a-

A

in

c,

D,

the direct product

exists in the sense of Definition 3, then there exists at

e

most one way of making Example 7. (1)

into an additive D-graded category.

In Examples 1-6 given earlier, In Example 1, if

the D-graded category

AD

is any category, then

A

is an additive category, then

is in a natural way additive. AD

is abelian iff (AD)

In Example 1, in all cases, the category ree zero of the D-graded category ponent category" from

all

is any graded category graded by

such that, for every object

A

A,BEe,

becomes an abelian D-graded category.

This observation generalizes:

A x A

is any

then there is at most one way of introducing an addi-

tion in the that

D

C is any graded category graded

additive abelian group and if by

If

D

into

A,

AD,

AD

If

A

is abelian.

o

of maps of deg-

is identical to the "ex-

consisting of all covariant functors

where

D

is regarded as a category, by

taking for objects the elements of

D,

and for maps only the

identity maps. (2)

In Example 2, for every category

A

we have that

Graded Categories the

~-graded

category

ColA)

defined in Example 2 is in a

~-graded

natural wayan additive

161

category.

ColA)

z-graded category iff the additive category Remark:

A

is an abelian

is abelian.

We construct a peculiar specific additive category,

which we call "Coch".

The objects are the integers (positive,

negative, and zero). = HomCoch(n,m)

j

If

n,m E ~

then define

the addi t i ve abelian group {o} ,

if

m=n or m=n+l, otherwise.

Then there exists a unique composition of maps in "Coch" such that "Coch" becomes an additive category, and such that, for each integer

n E ~,

the element

tity map in Coch from Then if

A

n

IE Hom

Coch

(n,n)

is the iden-

into itself.

is any additive category, then the (ordinary,

ungraded) additive category

Co(Al

O

of objects and maps of

degree zero in the singly graded additive category

ColA)

can

be characterized as follows: (Co (A» 0 = A (Coch) , the category of all additive functors from Coch into

A,

and

of all natural transformations of such functors. (3)

In Example 3, for every additive category

have that the bigraded category additive bigraded category.

D(A)

A,

we

is in a natural wayan

The bigraded category

D(A)

is an

abelian bigraded category iff the (ordinary, ungraded) category A

is an (ordinary, ungraded) abelian category.

Remark.

It is an easy exercise, which we leave for the reader,

to construct an additive category, "Doub", the objects of which are the elements of

~ x~,

such that, for every additive cate-

Section 3

162

gory

A,

if we consider the (ordinary, ungraded) additive cate-

gory

D(A}O

of objects and maps of bidegree zero of the addi-

tive bigraded category

D(A),

then

D (A) 0= A (Doub) as (ordinary, ungraded) additive categories, where

A (Daub)

is the additive category having for objects all additive functors from Doub into

A

and for maps all natural transformations

of such functors. (4)

In Example 4, for every additive category

every integer

n

complexes in category. ger

n

~

A,

Mn(A)

~

0,

the n-graded category

Mn (Al

A

and

of all n-ary

is in a natural wayan additive n-graded is abelian iff

A

is abelian.

For each inte-

it is not difficult to construct an additive cate-

0,

gory

the objects of which are the elements of

such that, for every additive category

A,

(ordinary, ungraded) additive category

(Mn(A»O

all objects in

Mn(A)

71

,

we have that the consisting of

(i.e., of all n-ary complexes in

of all maps of multidegree

n

Al

and

(0, ... ,0), is such that

(M (A» = A (Multn ) nO' where

A (Multnl

Mult

into

n

E.g.,

Multo

such that Mult

2

A

=

=

denotes the category of additive functors from and all natural transformations of such functora

(the additive category having one object

HomMulto (0,0) =71

as ring),

Mul tl = Coch

0,

and

and

Doub.

Exact Imbedding Theorem for Abelian Graded Categories. One can define the notion of functor of graded categories:

Graded Categories Let

D

be a fixed additive abelian group and let

D-graded categories.

0

into

163 C

0

and

A functor of D-graded categories from

be C

is:

(1)

A function

F

0,

the class of objects of

d

HomO (F (A) , F (B) )

C

into

together with d

A function

(2)

from the class of objects of

F(=FA,B)

d

for each degree

into

HomC(A,B)

from dE D,

A,B

C,

objects in

such that (3)

F (idA) = id F (A)'

(4 )

Whenever

all objects

A

in

C,

and such

that

A,B,C F(g

0

d

f E Hom (A, B) C

are objects in f) =F(g)

tor of

0

F(f)

C in

and

and

d,d'

g E Hom

are degrees in

d+d' Homv (A,C) .

D-graded categories and if

d' (B,C), C

CO,V

If

O

FO

of

F

to

Co

C

and

then

is a func-

are the (ordinary, C

Co

V,

and of

is an (ordinary)

tor of (ordinary, ungraded) categories from If

D,

F :C-vv> V

ungraded) categories of maps of degree zero of then the restriction

where

into

func-

Vo'

V are additive D-graded categories and if

F: C + V is a functor of D-graded categories, then the functor d f,g,= Hom (A,B), C

is additive iff whenever jects in

C

and

d

F(f+g) =F(f) +F(g)

is a degree in in

d

Hom (A,B). V

D-graded categories and if

F:C~>V

where

A,B

are ob-

D,

then we have that

If

C

and

V

F

are additive

is a functor of D-graded

categories, then it is easily seen that

F

is additive con-

sidered as functor of additive D-graded categories iff the (ordinary) functor

FO:CO~VO

of (ordinary, ungraded) additive

categories is additive in the usual sense. If

C

and

0

are D-graded abelian categories, and if

164

Section 3

F:C~>V

is an additive functor of additive D-graded categories,

then

is left exact; resp.:

iff

F F

preserves kernels; resp.:

nels and cokernels; resp.: O-+A' F (A")

!

right exact; exact; half exact

A <J A"-+O

cokernels; resp.:

has the property that, whenever

is exact in C,

V.

is exact in

both ker-

then

I f F : C~> V

is a functor of D-graded

FO:CO~>VO

abelian categories, and if

F{A,).l:..i!4F(A)...!:.J.s4

is the induced ordinary

functor on the categories of maps of degree zero, then it is easy to see that

F

is additive; left exact; right exact; ex-

act; or half exact iff the (ordinary) functor of (ordinary, ungraded) abelian categories,

FO:CO~>Vo'

has the corresponding

property. A functor

F:C~>V

of D-graded categories is an imbedding

iff it is injective on the objects, and for every in

C and every degree d

HomV(A,B)

is injective.

d,

the function:

d

objects

d

FA,B:HomC(A,B)-+

Since a functor of D-graded categories

preserves isomorphisms, it is easy to see that, if a functor of D-graded categories, then FO:CO~~>VO

A,B

F

F:C~~

V is

is an imbedding iff

is an imbedding of (ordinary, ungraded) categories. F: C~> V

In particular , it follows that, if D-graded abelian categories, then

F

is a functor of

is an exact imbedding of

D-graded abelian categories iff the induced functor of ordinary, ungraded abelian categories on the categories of maps of degree zero,

FO:CO~>VO

is an exact imbedding in the usual sense

([lAC.]) of abelian categories.

If

F:C'VU> V

is an exact imbed-

ding of D-graded abelian categories, then all the familiar properties of exact imbeddings of (ordinary, ungraded) abelian categories hold, see [I.A.C., Introduction], including about exact se-

165

Graded Categories quences, inverse-composites, subobjects, etc. If

A

is a D-graded abelian category, then let

AO

be the

(ordinary, ungraded) abelian category of maps of degree zero of A.

Then we have constructed, in Example 1 above, the category

(Ao)D

of D-graded objects in the abelian category

AO'

which

is also a D-graded abelian category. Theorem 5.

There exists a canonical exact imbedding of D-graded

abelian categories from

A

into

Proof:

A

in

For each obj ect

ject in and

If

are objects in

A,B

n~)dED '

A let F (A)

e f E Hom (A,B), where A

A,

verify that

F

Corollary 5.1. Categories.)

(Ao)D.

e

an ob-

0

is a degree in

e

then define

which is a map of degree

in the D-graded category

(Ad)dED'

F(f)=«n

from

B e+

F(A)

d)

-1

into

0

f

0

F(B)

We leave it as an exercise to

is a functor, and is an exact imbedding. (Exact Imbedding Theorem for D-Graded Abelian Let

be an additive abelian group and let

0

be a D-graded abelian category that is

'Cl

set. Then there exists

an exact imbedding of D-graded abelian categories from (the D-graded abelian category),

A

{abelian groups}D,

A

into

the cate-

gory of D-graded abelian groups. Proof:

AO

is a set.

is an (ordinary, ungraded) abelian category that Therefore by the Exact Imbedding Theorem

there exists an exact imbedding groups}.

G

grom

AO

into

([~l),

{abelian

This clearly defines naturally an exact imbedding of

D-graded abelian categories, call it

o

{abelian groups}.

from

into

By Theorem 5 above, we have a canonical

exact imbedding of abelian D-graded categories, The composite functor, the proof.

GO

0

F:A'V'v> {abelian groups}D,

completes Q.E.D.

Section 3

166

Of course, the last

c~rollary,

the Exact Imbedding Theorem

for O-Graded Abelian Categories, reduces many constructions and proofs for arbitrary O-graded abelian categories, to the special ~he

case of

category~

O-graded

{abelian groups}O, of D-graded

abelian groups. Of course, given any O-graded abelian category that is not

A,

a set

if

S

there exists a O-graded subcategory

S

O-graded subcategory (i.e., such that for all objects that

S

A,

is any subset of the objects of

A,B

in

S

of

A,

then

that is a full

d d HornS (A,B) = HomA (A,B)

and all degrees

d

0),

in

such

is an abelian O-graded category, such that the inclu-

sion functor:

[3'V'v>

A

is exact, and such that

Then the last corollary applies to

S.

S

is a set.

In this way, the appli-

cations of the Exact Imbedding Theorem for Abelian O-Graded Categories that are sets, can be used also to prove results even for those that are not sets, just as in the ungraded case. Exercise 1.

V

and

0

Let

be an additive abelian group and let

be D-graded categories.

ural equivalence from Exercise 2. Let

-+

E

into

Then define the term a O-nat-

V.

(Change of Grading Group, Non-Additive Case) . be an additive abelian group and let

0

graded category.

1>: 0

C

Let

E

be an epimorphism of abelian groups.

Ccrude

are objects in

C

e Hom crude(A,B) C E

and i f U

dEO 1> (d)=e

be a 0-

Then one can asCcrude E

are the same as the objects in

E

C

be another abelian group and let

sociate an E-graded category which we denote jects in

C

e E E,

then define

d ({d} x Home (A, B) )

e.

•

The obIf

A,B

Graded Categories

167

Horn e d (A,B) is the disjoint union of the sets: ecru e E for all dE D such that ¢ (d) = e). Then with the

(that is, d

Home(A,B)

ecrude

obvious composition, one verifies that graded category. (1)

If

e

becomes an E-

E

Some special cases: is an additive D-graded category, and if the

epimorphism of additive abelian groups

¢:D+E ecrude

morphism, then the E-graded category

E

is not an isois not additive

(since the disjoint union of more than one group has no natural In fact,

structure as group. even prove that

ecrude

in these circumstances, one can

is not even an E-graded pointed cate-

E

gory) . If

(2) E=;Z

and

(n,m) E;Z

D=;Z x;Z,

¢:D+E

X;Z,

the additive group of pairs of integers,

is the sum map,

¢(n,m) =n+m,

all

then a "D-graded category" is a bigraded category,

and an "E-graded category" is a singly graded category.

There-

fore every bigraded category defines a singly graded category, by the "crude" procedure. More generally, if

(3)

p:{l, ... ,n}+{l, ... ,r} epimorphism quiring that f

l

,· .. , f

r

,

p

l

r-graded category,

¢p'

are integers, and if

P

of additive abelian groups, by rewhere

(.)' l

is the canonical

Then, by means of

r
is any onto function, then define an

¢=¢ :;Zn .... ;Zr ¢ (e.) = f

1 <

;Y-basis of

resp. :

n

;Z ,

resp. :

every n-graded category

ecrude ' ;Z r

e

of

;Z

r

.

defines an

by the "crude" construction of Ex-

ercise 2. (4)

In the special case

there is only one function ¢p

r = 1

of special case (3) above,

p:{l, ... ,n}+{l}.

as in special case (3) above.

This

p

defines

Therefore every n-graded

168

Section 3

category defines, by the "crude" procedure, a singly graded category. (5)

If

¢:D+ {oJ

D

is any additive abelian group then let

be the unique homomorphism to the trivial group.

every D-graded category

c{~}de -

C

defines a

Then

{OJ-graded category

that is, an ordinary, ungraded category by the "crude"

construction.

Of course,

Ccrude {O }

"white elephant" category of

C

is simply the underlying in the sense of Definition 1.

Special cases (1) and (5) above show the problems of the "crude" way of passing from a D-graded category to an E-graded category.

E.g., in (5), we have previously observed how dif-

ferent are the categorical properties of

C

and of

Ccrude {O} -

and (1) above shows just "how crude" the construction is. Exercise 3. D

(Change of Grading Group, Additive Case).

be an additive abelian group and let C be an additive D-

graded category. ¢:D+E C ' E

Let

e EE

and

Home (A,B) C E

Then we define

C E

are the same as the objects in

A, B

are objects in Ell

C,

C.

we define

d HomC(A,B).

dED ¢(d)=e

f= (fd')d'ED :A+B and g= (gd")d"ED :B+C are maps ¢(d')=e' ¢(d")=e" C of degrees e' and e" respectively then define E

L" gd"ofd')dED a map of degree e' + e" from d ,d ED (d) , " d'+d"=d ¢ =e +e into C in CEo Then CE ' together with the indicated no-

go f=(, A

be another abelian group and let

an associated additive E-graded category.

Whenever

in

E

be an epimorphism of additive groups.

The objects in

If

Let

Graded Categories

169

tion of "map" and "composition", is an additive E-graded cate(Notice that, if we had taken the direct product of Horn's

gory.

instead of the direct sum of Horn's in defining "maps" in

CE ,

that we would not have been able to define composition of maps). (1)

If the additive D-graded category

C

is an abelian

D-graded category, then in general, the additive E-graded category

CE

is not an abelian E-graded category.

CE

by Proposition 4,

(Notice that,

is abelian iff the (ordinary, ungraded)

category: (2) and let let K

Let

A

C = AD,

¢: D -+ E

be an (ordinary, ungraded) additive category the category of D-graded objects in

A,

and

be an epimorphism of addi ti ve abelian groups.

be the cardinality of

nary) category

A

Ker ¢,

Let

and suppose that the (ordi-

is closed under direct sums of

< K objects.

Then the assignment:

defines an additive functor (AD)

E

into

AE.

The functor

I

of I

E-graded categories from of E-graded categories is

easily seen to be an imbedding(*) of E-graded categories.

Every

(*) Actually, technically speaking, the functor I of Egraded categories as constructed above is only a faithful functor (that is, a functor that is injective on the Horn's but not necessarily on the objects) rather than an imbedding. However, faithful functors of E-graded categories share all of the important nice properties of imbeddings. (In fact, if F: ~ is a faithful functor of E-graded categories, and if C has "enough objects in each isomorphism class", then F is isomorphic to an imbedding. Therefore, in (2) if we replace A by a D-equivalent category, having enough objects in every isomorphism class, then the functor I constructed in special case (2) above can be taken to be an imbedding.)

Section 3

170

object of of

(AD)

E

(3)

AE

is isomorphic in

under

AE

to the image of some object

1.

In special case (2) above, if the kernel of

¢:D

->-

E

is finite, then the additive imbedding - more precisely, additive faithful functor -

I

of special case (2) is a full im-

bedding - more precisely, full faithful functor - and such that every object of the range category of object in the image of

I.

I

is isomorphic to an

Therefore, in this case,

I

is an

equivalence of E-graded categories, and therefore abelian E-graded category iff

AE

is an abelian E-graded cate-

gory; and of course, this latter occurs iff

A

is an abelian

category. (4) where

If

D

A

is an arbitrary additive D-graded category,

is an addi ti ve abelian group, then let

the unique homomorphism to the trivial group. A{O}'

¢p:~

Let

nand

p: {I, ... ,n}

n

be

Then we have

an additive (ordinary, ungraded) category. (5)

let

¢: D -+ {O}

-+~

r

means of

-+

r

be integers such that

{I, ... ,r}

be an onto function.

constructed in (3) of Exercise 2. ¢p'

1 < r < nand

Therefore, by

every n-graded additive category

r-graded additive category.

C

sider the additive n-graded category

A,

then

we have ~

category.

If

A

C = A~ r'

A

Then if we con-

n

of all n-graded

an additive r-graded

is abelian, then as we have seen in Example

A~

7, part (1), the n-graded additive category abelian category.

defines an

As a further special case, let

be any (ordinary, ungraded) additive category.

objects in

Then we have

However, if

r
n

is an n-graded

then it is easy to see

that the additive r-graded category ~

r

is not, in general,

Graded Categories abelian; A

a counterexample is

171

A = category of abelian groups.

If

is an additive category such that denumerable direct sums of

objects exist, then we have the functor

I

of special case (2),

a canonical additive imbedding - or, more precisely, a canonical additive faithful functor - from the r-graded additive category

n (A~)

into the r-graded additive category

~r

graded objects in (6) Then

A~

r of all r-

A.

A special case of (5) is the one in which

p

is necessarily "constant function one".

for every additive n-graded category singly graded category

C~.

C

r = {lJ,

Therefore,

we have an associated

As a further special case, for

every (ordinary, ungraded) additive category

A,

such that de-

numerable direct sums exist, we have an additive imbedding - or, more precisely, an additive faithful functor - I of additive n (A~ ) singly graded categories from into A~ . The image, ~

of an n-graded object

I (A),

A

in

A

the associated singly graded object (7)

Let

and let

A

n,r,p

and

¢p

of

is called

A.

be as in special case (5) above

A.

Then we have

the category of n-ary, resp.:

abelian category category.

I,

be an additive category such that denumerable

rect sums of objects exist in Mr(A),

under

A,

Mn(A),

diresp.:

r-ary, complexes in the

an n-graded, resp.:

r-graded, abelian

Then, by setting up appropriate sign conventions for

coboundaries, it is not difficult, as in special case (5) above, to define an additive functor

that is an additive imbedding - more precisely, an additive faithful functor- of r-graded additive categories.

If

A

is

172

Section 3

an n-ary complex in the abelian category

A,

then

I(A)

is

called the r-ary complex obtained from the n-ary complex by contracting indices according to (8)

A

p.

A special case of (7) above is the one in which

Then necessarily Mr (A) = Co (A) .

r = 1.

is the constant function one, and

p

In this case, for every n-ary complex

the cochain complex

A E Mn (A) ,

thus obtained is called the

I (A) E Co (A)

associated singly graded complex of

A.

Special cases (6) and (8) of this Exercise, and also special case (4) of Exercise 2 above, motivate the following terminology:

If

f

n-graded category

le t

D

C,

then by the total degree of

Let and

A E

be addi ti ve abel ian groups and let

full subcategory d

many

object in

C

of

AD

such that A.

C

¢ : D ->- E

Then one can consider the

¢ (d) = e

e E E,

there exist only finitely

and such that

Ad

is not a zero

is an abelian, D-graded subcategory of

and the inclusion:

C~>AD

be

generated by those objects

such that, for every

dED

we mean

be an (ordinary, ungraded) abelian category,

an epimorphism of additive groups.

(A )dED

f

in an

d = d l + •.• + d . n

the integer (9)

(dl, ... ,d ) n

is a map of multidegree

is exact.

generality, we can construct a functor cial case (2) above on objects of

C,

AD,

Then, at this level of "I" and

exactly as in speI

is a full imbed-

ding - or, more precisely, a full faithful functor - of E-graded categories from

C E

into

AE;

moreover every object of

isomorphic to an object in the image of graded additive categories I

I.

AE

Therefore the E-

C and AE are E-equivalent with E being one half of the E-equivalence; and therefore C is E

is

Graded Categories

173

an abelian E-graded category. Let

(10)

A

be an abelian category and let

n

be a non-

negative integer. Then we have the n-graded abelian categories n A71 Then, of course, we have the "stripping funcand M (A). n n tor" from

into

Mn(A)

A71,

which to each n-ary complex asso-

ciates the corresponding n-graded object in

A.

This "stripping

functor" is an exact imbedding of n-graded abelian categories (but of course is not a full imbedding unless

A

is equivalent

to the zero abelian category).

there exists an integer Pl' ... 'Pn gory

B

n

~ -N.

Fn(A)

and

Bn(A)

n=O

unless

are both n-graded abelian categories.

Fn (A)

(resp.:

of

objects (resp.: n-ary complexes) in . A

Bn (A) )

the n-graded

that are bounded be-

The stripping functor is an exact imbedding of n-graded

abelian categories from

Bn(A)

For every positive integer

tion

such that

under the "stripping functor" is in

We call the obj ects of

low.

gen-

generated by all n-ary complexes such

that the image in Fn(A).

,p

n

Similarly, we have the full n-graded subcate-

of

(A)

Pl"" A

such that

N

A71

of

We have the full n-graded subcategory Fn(A) Pl" .. ,p (A n) n era ted by all objects (Pl"" ,Pn)E71

p : { 1, ... , n} ... {l, ... , r} ,

into

Fn(A).

r
and every onto func-
p

structed as in special case (5) above. tained in the full subcategory

C

of

: (;r

n ,+) ...

Then n

A71

(il

r ,+)

Fn(A)

be con-

is con-

constructed in

special case (9) above, and is a full subcategory of

C.

There-

fore, from special case (9), we see that the restriction of the functor

I

of special case (9) is

r-graded abelian categories:

h~lf

of an equivalence of

Section 3

174 F

n

(A)

'J'

r

'VV>F (A). r

The image of an n-graded object

bounded below under this

functor

I

in

from

~

contracting indices following the function

A

F

A

Similarly,

is called the r-graded object obtained

(A)

r

n,r

and

p

p.

as above, we obtain, by the con-

struction in special case (9) above, an additive functor of rgraded categories I :B

(A)

n

'VV> B

('J'r ,+)

(A).

r

One can show in this case that the r-graded additive category B (A) n

tor

is an r-graded abelian category, and that the func('J'r,+)

is an exact imbedding - or, more

I: B (A) 'V'u> B (A) n ('J'r ,+) r

precisely, an exact faithful functor - of r-graded abelian categories.

The image of an n-ary complex

category

A

A

in the abelian

that is bounded below under this functor

called the r-ary complex associated to dices following the function

A

~

I

is

contracting in-

p.

(11) A further special case of special case (10) above is the one in which tion one.

Then

r = 1, I

so that necessarily

p = constant func-

for n-graded objects bounded below is

one half of an equivalence of singly graded abelian categories: F

n

(A)

The image

'J'

'V'u>

F 1 (A) •

I(A) of an n-graded object

low in the singly graded abelian category functor

I

A

that is bounded be-

Fl (A)

under this

is called the associated singly graded object of

And, similarly, when

r = 1,

I constructed for n-ary com-

A.

Graded Categories

175

plexes bounded below in special case (10) above is then an exact imbedding - more precisely, an exact faithful functor -of singly graded abelian categories, 1:

The image gory

A

Bn (Al( ;z , + )CV\,> Bl(A).

I(Al

of an n-ary complex

A

in the abelian cate-

that is bounded below under this functor

I

is called

the associated singly graded complex, or the associated cochain complex, of

A.

It is an object in

dexed cochain complex in

¢:O

+

E

that is bounded below.

(Change of Grading Group, other direction).

Exercise 4. Let

A,

and

0

E

be additive abelian groups and let

be a homomorphism of additive groups.

E-graded category

A,

the same as the objects of every pair of objects A

of degree

into ¢ (d)

A.

A,B,

DA.

The objects of

F.or every degree

DA

dE D,

in the E-graded category

oA A.

are

and

we define the maps of degree

in the O-graded category

B

Then to every

we associate a O-graded category, as in

Definition 1', which we denote by

from

CoCA) =Ml(A), Le., a ;z-in-

d

to be the maps That is,

d ¢(d) Hom A (A,B) = Hom (A,B). A o

The composition in

OA

is defined in the obvious way.

one verifies easily that "maps of degree

d"

oA,

together with this notion of

and composition, is a O-graded category.

It is immediate that, if the E-graded category ditive, respectively: gory

DA.

Then,

A

is ad-

abelian, then so is the O-graded cate-

Similarly, if the D-graded category

A

is closed

under, e.g., denumerable direct sums (in the sense of Definition 3); or under arbitrary direct products (in the sense of

Section 3

176

Definition 3); or under any analogous universal mapping property construction; then the D-graded category responding property.

DA

inherits the cor-

The reason for the assertions of this

paragraph being true, is that the E-graded category D-graded category

DA

A

and the

have the same category of maps of degree

zero:

and, as we have seen above, all the properties listed in this paragraph depend only on the category of maps of degree zero. Therefore, the construction in this Exercise, of "changing the grading group in the other direction", is indeed a smoother operation than those discussed in Exercises 2 or 3. Remark:

The reader should note that the construction of Exer-

cise 4 above is in general not an inverse (in fact, not in general either a left inverse or a right inverse)

of either the

construction of Exercise 2 or that of Exercise 3. ?ropos±tion 6. A

Let

D

be an additive abelian group and let

be a D-graded abelian category.

Suppose that the D-graded category products of system

i

2card (I)

(A,a

ij

).

'cI

~,J"

objects (*).

Let A

I

be a directed

set~~.

is closed under direct Then, given any inverse

of objects and maps in the underlying

white elephant category of

A

(as defined in Definition 1),

(~)It is equivalent to say, suppose that the (ordinary, ungraded) category A of all objects, and of all maps of degree zero, of A, is cl sed under direct products of < card (I) ob-

g

jects.

177

Graded Categories

ij lim (Ai,a ) in the (ordinary, ungraded) iEI white elephant category of A exists. the inverse limit

Proof:

Recall that, for every

definition

dE D,

A Ad "".... A nd:

d.

of degree

finite, the Proposition is trivial. is infinite.

Then let

~

i

i O}.

iO

be an element of

Then since

by

I

I,

), all

is

I

I,

I

and let I,

exists, then this

call it

i E I,

Then for each iO

and

i ij (A,a ). ·cI. l.,)"

Therefore,

if necessary, we can assume that there

J

exists an element in

d =- deg (a i

Ad

is a corinal subset of

J

latter is also an inverse limit of replacing

If the set

by

Therefore assume that

it is easy to see that, if

i E 1.

A E A,

of "D-graded category", we have an object

an isomorphism

J = {i E I:

and every

i E I.

such that

0,

we have the map

0

~

i,

all

aiO: Ai..,. AO.

Let

Define

i P = IT (A ) d ' the dir. product in the sense of Def. 3. iEI i

Then we have ree zero.

7T

i

:p .... (Ai}d.'.

the i'th projection, a map of degl.A l. i I':l i nd.:(A }d.""A, an isomorphism of l. l.

We also have

Ai .... A. is a map of Pi = n d . 0 7T i. Then p.:P l. l. iO l. = -deg (a ) . Whenever j~i are elements of I,

degree

di ·

degree

d

i

Let

have that ·0

a l.

..

0

·0

a) l. = a ) ,

·0

whence

..

·0

deg(al. } +deg(a)l.} =deg(a)}. deg(a

jO

} =-d .. )

Hence, whenever

Therefore

j > i

in

I,

But

deg(a

ji

deg(a

iO

) =-d , i

} =d. -d .. l. J

we have that

we

Section 3

178

(1)

j::.-.i

in

I

Therefore, since

implies

deg (a j i

a ji

and

p

are both maps from

A,

gory Let

K"

~J

Then

0

into

we can form =Ker(a

ji

0

pj

AO

category

A,

zero of

i

Ai ,

pj _pi),

P.

p j ) = deg (p i ). have the same degree,

all

whenever i,j E I

j:,i.

i,jEI,

j > i.

such that

is an indexed family of

Since the (ordinary, ungraded) abelian

is by hypothesis closed under direct products of

.s. card (I),

the proof of Corollary 2.1 of section

1 applied to the (ordinary, ungraded) abelian category

AO

is closed under forming infima of

objects of an object. of the subobjects: P,

exists.

A ' O

2. card (I)

In particular, the infimum, call it

Koo, ~J

all

(i,j)E{(i,j)ElxI: j::...i},

It is easy to see that

limit of the inverse system Remark 1.


consisting of all objects and all maps of degree

collections of

srows that

and

in the additive D-graded cate-

ajiopj_pi,

(Kij){(i,j)EIXI: j::...i}

(I) subobjects of

p

0

D

subD, of

is the desired inverse

i ij (A,a ),

Q.E.D.

1,J'EI.

A consequence of the construction of the last propo-

sition, is Corollary 6.1.

Under the hypotheses of Proposition 6, we have

also that the (ordinary, ungraded) abelian category

AO

is

closed under inverse limits of inverse systems of objects and

Ao

maps in

indexed by the directed set

verse system, the inverse limit in

AO

I.

For any such in-

is also an inverse limit

in the (ordinary, ungraded) white elephant category of Remark 2. spect.

A.

Theorem 6 and Corollary 6.1 are unusual in one re-

Namely, it is a case when a universal mapping property

construction (namely, the inverse limit over a directed set) turns out to exist in the "white elephant" category iff it

Graded Categories

179

exists in the category of maps of degree zero.

For most other

common universal mapping properties (e.g., the direct product of objects, kernel of a map, etc.), it usually

turns out that

the "white elephant" category is pathological for the construction. Remark 3.

Of course, Proposition 6, Corollary 6.1 and Remark 2

have corresponding statements about direct limits, which immediately follow by applying these results to the dual D-graded O

abelian category Definition 5.

A

Let

•

D

be an additive abelian group, let

a D-graded abelian category, and let Then by a subobject (resp.: the object

A

A

A.

graded abelian category is a subobject of

of

B,

f-l(T),

f (S),

A

quotient-object; subquotient) of

maps of degree zero of

A,

to be

If

A,

A.

f:A

->-

B

1).

If

A

in the

is any map in the D-

of any degree, and if

0

we mean a

of all objects and

AO

then define the image of 1m (f

T=

then define the inverse image of

{

Sunder

(D, j) } T

S = { (C, i) }

under

Namely, if

d = deg f,

same as the fiber product of the diagram:

f,

is a subobject f,

denoted

to be the fiber product of the diagram:

(This always exists:

be

quotient-object; subquotient) of

(ordinary, ungraded) abelian category

denoted

be an object in

in the D-graded abelian category

subobject (resp.:

A

then it is the

180

Section 3

in the (ordinary, ungraded) abelian category degree zero of

A).

AO

of maps of

Then the Lemma immediately preceding Theo-

rem 3 of section 2 generalizes to D-graded abelian categories. That is, in the statement of that Lemma, replace the phrase "an abelian category

A"

with "a D-graded abelian category

A".

Then the resulting Lemma is also valid. Let

Proof:

d = deg f.

Lemma is true for

(n~)-l

0

f

is an isomorphism, the

Then since

(n~)-l

iff it is true for

f

0

But

f.

is a map of degree zero, and is therefore a map in

the (ordinary, ungraded) abelian category

AO'

(n~) -1

verity of the generalized Lemma for

0

f

Therefore, the follows from

the verity of the Lemma preceding Theorem 3 of section 2 for the (ordinary, ungraded) abelian category

AO

(n~) -1 of: A..,. Bd

AO·

in the abelian category

Recall that, if AEA where

and every Ad

A

dE D,

A

we have fixed a specific pair A

such that

and

in

A,

where

Definition 6. B

"0"

Let

A

and

~

additive relation

A

deg (nd) = d.

AX B . d

AO

is an isomorphism

We can and do in-

= A, n~ = idA'

is the zero element of

all objects

D.

be a D-graded abelian category.

be obj ects in

subobject of

A

nd :Ad ..,. A A

sist that the choice is such that A

Q.E.D.

is a D-graded category, then for every

is an object in

in the category

and the map

R

A, from

and let A

into

d ED B

Let

be a degree.

of degree

(An identical definition is:

d

Then

is a

An additive

Graded Categories relation from

A

into

A,

an category

B

A,

lation from

A

jects in So R,

A

from

B and

of degree

C

d,e E D,

First, let

d,

is an additive re-

R

If

Bd - )

of degree

B

into

into

S

and e,

is an additive

where

A,B,C

are ob-

then we define the composite relation

an additive relation from

as follows:

in the D-graded abeli-

of all objects and maps of deg-

AO'

A

into

relation from

d,

is an additive relation in the (ordinary, un-

graded) abelian category ree zero of

of degree

181

n :C + d e

-+-

A

into

C e

C

of degree

d + e,

be the isomorphism of deg-

n~+e (nC)-l ---" C e " C ) _ Then e e SC: B x C , so that (n~ x n) -1 (S) c: Bd x Cd +e Also Rc: A x B e d B -1 Therefore Rand (n x n) (S) can be thought of as being d (ordinary, ungraded) relations in the (ordinary, ungraded) aree

d,

belian category A,

from

(C d +

the composite:

A

of all objects and maps of degree zero of

AO

into

B , d

C + , respectively_ d e B -1 [(nd x n) (S) loR in the

and from

Bd

Therefore the composite relation:

into

(ordinary, ungraded) abelian category AX Cd +e -

a subobject of

But then

AO

(n

B d

makes sense, and is

x n)

-1

(S) 1 oR

can

equally well be thought of as being an additive relation of degree

d +e

from

posite relation_

A

into

C,

That is,

and we define this to be the comS

B

R = (nd x n)

-1 (S))

0 R, where C -1 C n=(n e ) on d +e - Also, if R is an additive relation from into B of degree d in the D-graded abelian category A, 0

then we define the inverse relation from

B

gory

A_

into

A

Namely,

of degree R

-d

R-1 ,

an isomorphism of degree

an additive relation

in the D-graded abelian cate-

is a subobject of

the "twist" isomorphism:

A

Bd x A~ A x B d -d:

A x Bd _ Also

Let

T denote B -1 A (n ) x n_ is d d Then define R- l

Section 3

182

to be the inverse image under the composite isomorphism To [(1'1

B x A_

d

B -1

d

)

,

A

""

-1

xn_dl:BxA_d-+AxB . Then R is a subobjectof d l i. e., Ris an additive relation from B into A

is a map of degree

d

A,

in

B -1 (1'1 ) d

0

f:A .... B d

R

.

d

from

f:A .... B

If

B,

into

A

f,

~of

then we define the

r f , an additive relation of degree follows.

-1

completing the definition of

of degree -d,

as

is a map of degree zero, and there-

AO'

fore is a map in the (ordinary, ungraded) abelian category

A.

of all objects and maps of degree zero of

r (n~) -1

the graph

0

f

of the map

AO'

ungraded) abelian category from

A

into

B , d

(1'1~) -1

Therefore we have

f

0

in the (ordinary,

AO

an additive relation in

or eql,livalent1y, a subobject of

Ax B . d

But this is also an additive relation in the D-graded abelian

A from

category

A

to be the graph of fined to be

r

f.

(1'1 B )-l

is the graph of

and

f

That is, the

d,

and we define this

rf

~

of

then that map is uniquely determined: and

f .... r f

resp.:

S,

resp.:

resp.:

C,

into

resp.:

e,

resp.:

of degree

if

d +e +f

d, e, f E 0 T, B,

I.e., if

are degrees in

is a relation from resp.:

f, from

then A

go f

g

is de-

Similarly, it is easy to see that

composition of relations is associative. A,

(I.e., i f

preserves composi tes.

r go f=r g or f ·)

objects in

is de-

f

are maps of arbitrary degrees such that

fined, then

f

Of course, if an additive relation 0

R map,

the correspondence:

B of degree

into

C,

resp.:

A, 0,

0

and if resp.:

of degree

(ToS) oR=To (SoR), into

A,B,C,O

are

R,

a, d,

relations

o.

Finally, we pose two more related definitions, that will be useful later.

Namely, let

A

be a O-graded abelian

categor~

183

Graded Categories let

A

and

B

be objects in

quotient of

A

and let

Let

dE D,

and let

B

of degree

d

B

of degree

d).

A,

let

T

into

V

let

f

to

as V

T

U

(resp.:

d. V

deduced from

fJ. T

T

(resp.:

If the additive relation from (resp.:

f)

f) ,

from the map

f)

as

T

(resp. :

U

in-

by passing to the subquo-

the same degree

d

Qy

fJ

Notice that it has the same deg-

U

V

into

We call this latter, the additive

the resulting uniquely determined map:

into

A

an additive relation from

into

deduced from

U

into

be a map from

tients should happen to be the graph of a map, say

from

A

B.

Then we have the additive relation

passing to the sUbquotients. ree

be a subquotient of

be an additive relation from

(respectively:

of degree

relation from

be a sub-

l

V = { (B , Sl)} l

(resp. : U

U = { (Al,R )}

as

-+

then

fg'

necessarily of

V,

is called the map

deduced from the additive relation

T

(or

Qy Eassins to the subguotients.

The proofs of all the assertions following Definition 6 are as follows.

If the D-graded abelian category

A

is a set,

then by Corollary 5.1 there exists an exact imbedding of Dgraded abelian categories from abelian groups.

A

into the category of D-graded

Therefore in this case the assertions reduce

to the case in which the D-graded abelian category

A

is the

category of D-graded abelian groups, where the assertions are trivial by set-theoretic methods. If

A

is not a set, then let

S

be a set of objects con-

taining all the objects under consideration. an exact, full D-graded abelian subcategory taining all the objects in

S,

such that

Then there exists A'

A'

of

A

is a set.

conThen

184

Section 3

to prove each given assertion for for

A',

A,

it suffices to prove it

which has been done in the last paragraph.

Section 4 Spectral Sequences in a Graded Abelian Category

Definition 1.

Let

D

be an additive abelian group, and let

be a D-graded abelian category.

Then by a spectral seguence in

the D-graded abelian category

A

gether with an indexed family

(Er,dr,Tr)r>O'

object in from

A,

Er

all integers

r

~

we mean, an integer

0,

and

dr:E r

r

where -+

Er

' O

Er

tois an

is a map

into itself (of any possible degree; we do not even in-

sist that the dr's have the same degree for different such that T : r

A

d

r

0

d

r

= 0,

all integers

[Ker(d )/Im(d )] ~E +1 r r r

If

D

~

A,

all

rO) ,

and where

is an isomorphism of degree

the D-graded abelian category Example 1.

r,

r

r

~

0

in

r O.

is the zero-group, then of course the notion

of "spectral sequence il a D-graded abelian category" reduces to that of "(ordinary, ungraded) spectral sequence in an (ordinary, ungraded) abelian category", as defined in section 1. Example 2.

If

D

is any additive group, and if we consider

the D-graded abelian category

AD,

ungraded) abelian category, then

AD

objects in the abelian category

3.

A

where

A,

A

is any (ordinary,

is the category of D-graded discussed in section

spectral sequence in the D-graded category

AD

will

be called a D-graded spectral sequence in the jordinary, ungraded) abelian category

A.

Thus, explicitly, if

additive abelian group, and if

A

185

D

is any

is any (ordinary, ungraded)

Section 4

186

abelian category, then a D-graded spectral sequence in the abelian category

A

is, an integer

r

' O

together with an in-

dexed family:

where (1)

En r

and where

is an object in

d:: E:

+

E:

A,

all

nED,

is a map in the D-graded category

AD.

That is, (2)

for each integer

r

~

r 0'

we have a degree

a

r

ED,

such that (3 )

d

n r

is a map in the abelian category all degrees nED,

to

A

from

En

in-

r

all integers

In addition, we must have that (4)

all

nED,

all

r~rO.

And finally, (5)

Tn r

is an isomorphism in the abelian category

(1)-(5) above give

A,

an equivalent definition of aD-graded spec-

tral sequence in the (ordinary, ungraded) abelian category

A.

Three special cases of Example 2 are worthy of special mention. Example 3.

The special case of Example 2 in which

the additive group of integers.

Then if

A

D = (;Z, +) ,

is any (ordinary,

ungraded) abelian category, a ;z-graded spectral sequence in (i.e., a spectral sequence in the

;z-graded abelian category

A

Graded Abelian Category A7)

187

is called a singly graded spectral sequence in the (ordi-

nary, ungraded) abelian category

A.

Thus, explicitly, if

A

is an (ordinary, ungraded) abelian category, then a singly graded spectral sequence in

A

is:

an integer

(E~,d~,T~)nE7

with an indexed family

r

where

' O

together En

(1)

is an

r

r~rO

object in r

~rO'

A,

all

nt:7,

we have an integer and every integer

Cl n,

~

r 0i

t:7,

and for every integer

all integers

and where

(4)

r

n,r

A:

integers

r

such that

[Ker(dn)/Im(dn-Clr)]':';.En+I' r r r

Cl ( = deg d;) r

In most practical examples of singly graded

spectral sequences, however, we normally have that r> r - 0

integers

all

~rO'

In this abstract definition, the integers can be arbitrary.

such that

is an isomorphism in the (ordinary, un-

graded) abelian category n,r

for each integer

(2)

we have a map

such that (3) r

r~rOi

all

(Le., all the

d*'s r

Cl.r=+l,

have degree +1).

all

A singly

graded spectral sequence with this property will be called conventional. Remark:

be a singly graded spectral se-

Let r~ro

quence in an abelian category integer

a

Eo

7

such that

Cl

r

A.

Suppose that we have a fixed

= deg d; = Cl.,

all integers

r

~

rD.

Then: dn n) ( En r' r,T r r>r - 0 is an (ordinary, ungraded) spectral sequence in the abelian

Case 1.

category

If

A.

sequences in

Cl. = 0,

then for each integer

n,

In this way the study of singly graded spectral A

such that all the

Cl.r'S

are

0,

reduces to

that of studying denumerably many (ordinary, ungraded) spectral

188

Section 4

A.

sequences in Case 2.

a f 0,

If

then for each integer

i,

0 ~ i ~ I a : - I,

define

all integers

o~ i

~

Ia I -

r

I,

~

r

o·

Then, for each of the

all

n E ;z ,

Ia I

integers

i,

we have a conventional singly graded spectral se-

quence: ( iEnr' idnr' iTn) r nE;z r~rO

In this way, we see that to give a singly graded spectral sequence in the abelian category the same non-zero integer

a,

A

such that all the ar's

are

is exactly equivalent to giving

lalconventional singly graded spectral sequences in the abelian category

A.

Combining Case 1 and Case 2, we see that, the notion of "singly graded spectral sequence in the abelian category ar's

~

all equal for different

r,"

A, really

reduces to those of "(ordinary, ungraded) spectral sequences in A"

and "conventional singly graded spectral sequences in

Example 4. group

0

The special case of Example 2 in which the additive is the group

(~x~,+).

Then a

tral sequence in an abelian category

A

spectral seguence in the abelian category if

A

A".

(~x~)-graded

spec-

is called a bigraded A.

Thus, explicitly,

is an (ordinary, ungraded) abelian category, then a bi-

graded spectral sequence in the abelian category with the integer

rO

is, an indexed family:

A

starting

189

Graded Abelian Category (EP,q dP,q TP,q) r ' r ' r p,qEz r.::.rO integers p,q,r with have a pair

(ar,Sr)

where (1)

(4)

TP,q r

0

A,

is an object in

of integers, and for every triple

dP,q = 0, r

all

A,

all

p,q,r E 71

A,

r

r

r

such that (3) . w~

th

> r _rOi

h an d were

[Ker (dP,q)/Im(dP-ar,q-Sr)]~ EP,q r r r+l

is an isomorphism:

in the abelian category

p,q,r

dP,q:EP,q ->-EP+ar, q+Sr

we have a map

in the abelian category dP+ar,q+Sr r

r

we

r:=:.r O'

of integers with

EP,q

all

p, q, r E 71

with

r'::' r

o.

Again, as in Example 3, in the above definition of bigraded spectral sequence in the abelian category

A,

no conditions imposed on the integers

all

r'::' r O.

r

and that

a , Sr E Z , r

However, in practice, it is usually true that Sr = -r + 1,

all integers

r.::. r O.

a

r

=

there are

Such a bigraded spectral se-

quence will be called conventional (the motivation for why such a condition should be so important will be clear when we study the spectral sequence of a bigraded exact couple in the next section.)

Thus, a conventional bigraded spectral sequence in

the abelian category

A starting with the integer

multi-indexed family:

(EP,q dP,q TP,q) r ' r ' r p,q,rE7i' r.:=:.rO

rO

is, a

where (1)

EP,q is an object in A, (2) all p,q,r E71 with r .:=:. rOi r EP'q .... EP+r,q-r+l dP,q: is a map in A, all p,q,r E71 with r r r such that (3 ) dP+r,q-r+1 odP,q=O, all p,q,r E71 r .:=:. rOi r r with

r >r i - 0

and where

(4)

TP,q r

[Ker(d~,q)/Im(d~-r,q+r-l) 1 '!E~~i all

p,q,rE71

is an isomorphism: in the abelian category

A,

with

Example 5.

Let

m

be any non-negative integer and let

D = (Zm, +) .

Then as a special case of Example 2, we have the

190

Section 4

~m-graded spectral sequence in any abelian cate-

notion of a gory

A.

This is also called an m-spectral diagram in the abel i-

an category

Thus, an m- spectral diagram for

A.

(ordinary, ungraded) spectral sequence;

m= 0

is an

a I-spectral diagram

is a singly graded spectral sequence; and a 2-spectral diagram is a bigraded spectral sequence. m> 3

An m-spectral diagram for

is a new kind of object, and is sometimes studied in cer-

tain cases. Remark 1.

Example 4, for the special case of conventional bi-

graded spectral sequences;

Example 3, for the special case of

conventional singly graded spectral sequences; and Example 2 are the most important and frequently used kinds of spectral sequences in mathematics. Remark 2.

Let

A be an abelian category and let

(E~,d~,T~)nE~ r~rO

be a conventional singly graded spectral sequence in if we define

A.

Then

then we ob-

tain a conventional bigraded spectral sequence (EP,q dP,q ,p,q) • r ' r ' r p,q,rE~

In this way, conventional singly graded

r~rO

spectral sequences can be regarded as special cases of conventional doubly graded spectral sequences (however, we shall not usually do this).

(The above procedure also allows us to turn

every singly graded spectral sequence in

A,

whether conven-

tiona I or not, into a doubly graded spectral sequence, such that the bidegree of r

~rO.

d** r

has first coordinate

r, each integer

If the procedure is so generalized, then a given singly

graded spectral sequence is conventional iff the associated doubly graded spectral sequence is conventional.)

Graded Abelian Category Remark 3.

191

A be an abelian category such that denumerable

Let

direct sums exist in an exact functor.

A and such that denumerable direct sum is

Then if

(EP,q dP,q TP,q) r 'r 'r

is an arbi-

p,q,rE~

r..::rO trary doubly graded spectral sequence in the abelian category

A,

then we can associate a singly graded spectral sequence by defining:

En = r

Ell

p+q=n

EP,q r'

dn = r

Ell

p+q=n

Tn = r

dP,q r'

Ell

p+q=n

TP,q r'

This procedure turns a conventional doubly graded spectral sequence into a conventional singly graded spectral sequence. Of course, the procedure of Remark 2 is perfectly general (requiring no hypotheses on the abelian category

A);

and a

singly graded spectral sequence can be recovered from its associated doubly graded spectral sequence. n T r

'=

T

n,O r

(Namely,

n,rE~,

all

(Both of these ob-

servations, of course, fail for the, more crude and more special, procedure for passing from doubly graded spectral sequences to singly graded spectral sequences, as described in Remark 3 above.) Remark 4.

Of course, analogous to Remark 2, there is a way of

passing from an ungraded spectral sequence: the abelian category n dn (E r' r,T n) r

n,rE~

r..::rO of degree +1,

A to

a singly graded spectral sequence:

-- n a me ly , d e f'lne Tn = T

r

r'

all

n,r

En = E , r

~ ~,

r

d

n =d r r

and

d* r

is

Again, this pro-

cedure enjoys all the nice properties of the procedure of Remark 2; and, the definition we have given is such that, the singly graded spectral sequence that we have just defined is

Section 4

192

d*

always conventional (since we have insisted that ree

+1,

all

r .:::.r ). O

abelian category

is of deg-

r

Also, analogous to Remark 3, if the

A has denumerable direct sums and if denumer-

able direct sum is an exact functor, then one can associate to (En d n n) r' r,T r n,rEl" an r.:::.rO where we define:

every singly graded spectral sequence: ungraded spectral sequence: E = Ell En, r nEl' r

(Of course,

these two procedures, when they both make sense, for passing from ungraded to singly graded, or from singly graded to ungraded spectral sequences, are not inverse to each other.

Simi-

larly, the procedures of Remarks 2 and 3 above, when they both make sense, are not inverse to each other.) Let us now return to study the general theory of spectral sequences.

That is, we have a fixed additive abelian group

A,

and a fixed D-graded abelian category

A.

spectral sequences in

D

and we are studying

A quick summary is, that essentially

all the results of section 1 go through to this more general situation, without any change in the arguments.

We keep the

same theorem numberings; statements only are given since the proofs are essentially the same.

We also keep the analogous

numbering of Definitions and Remarks. Definition 2'.

Let

D

be an additive abelian group, let

a D-graded abelian category, and let

(Er,dr,Tr)r>r

tral sequence in the D-graded abelian category the integer

rOo

Er ,

i.:::. 0,

two sequences

for each integer

r 2:.rO'

by

Zi (E ), Bi (E ) r r

r 2:. r 0'

quence of maps of degree zero in the category

be a spec-

A, starting with

Then we define, for each integer

induction on the integer of subobjects of

- a

A be

and also a se-

A,

called i-fold

Graded Abelian Category

193

for all integers

r

~

r O.

(The inductive construction is identical, word for word, to Definition 2 of section 2, so we don't repeat it.) Zi(E r )

the i-fold cycles in

daries in

Er'

Er

all integers

i

~

Bi (E ) r

and

the i-fold boun-

all integers

0,

We call

r

~rO'

(The

Remark following Definition 2 of section 1 is also applicable in this situation.) Proposition 1'.

Let

D

be an additive abelian group, let

A

be a D-graded abelian category and let

(Er,dr,Tr)r>r be a - 0 spectral sequence in the D-graded abelian category A, starting

wi th the integer

r O.

Then for every integer

r

Er .

Also, for every integer

~

rOwe have

that

as subobjects of integer 2)

j

~

0,

The map of degree zero "j-fold image":

(j-fold

= Zi+j (E r ) = B,+, (E ) 1 J r

(j-fold all integers integer

j

~

i,j,r 0

~rO

and every

we have that

epimorphism, and that 3)

r

with

i,j~O,

and every integer

I

,

r~rO'

r

~

r ' O

Z, (E ) -+ E

J

r

,

r+J

is an

as subobjects of

Moreover, for every there is induced a

specific isomorphism of degree zero:

(The proof is entirely similar to that in the ungraded case, see

194

Section 4

section 1.) Remark:

Let

D

be an additive abelian group and let

D-graded abelian category. degree zero of

A

Let

AO

all maps of degree zero of all objects of

gory.

AO

be a

be the category of maps of

(consisting of all objects of

seen in section 3,

A

A).

A,

and of

Then as we have

is an (ordinary, ungraded) abelian cate-

When we speak of a "subobject" (or a "quotient-object",

A

or a "subquotient object") of an object belian category

A,

in the D-graded a-

then we mean, as in Definition 5 of section

3, a subobject (or a quotient object, or a subquotient object) of

considered as object in the (ordinary, ungraded) abelian

A

category

AO.

(Therefore, for example, a subobject of

(B,i)

equivalence class of pairs: and

(B',i') iff .3 an isomorphism

i' = i

is an object in

B

is a monomorphism of degree zero, where

i: B .. A

tha t

where

0

is an

A

(B, i) '"

e:B'~B,

e of degree zero,

such

e) •

Remark 2'.

The hypotheses being as in Proposition 1', we have

that

is isomorphic (by means of

Er+l

degree zero)

to

Ker(dr)/Im(d r ).

natural way a subquotient of is a subquotient of

E . r

Er .

T

r'

an isomorphism of

Therefore

Er

itly

such that

A'

is

A" c: A'

Zi(E ) r

and

is in a

Er+l

i.::. 0,

Therefore, for

Er+l

Therefore, by the general theory of

subquotients, there must exist unique subobjects of

A

and such that

A"

is

Er+i

Bi(E ). r

A'

and

= A' /A".

A"

Explic-

(The analogue

of

Remark 3 of section 1 also holds good). Remark 3'.

Under the hypotheses of Proposition 1', if we look

at the corresponding spectral sequence in the dual D-graded category, the i-fold cycles in

Er

constructed in

O

A ,

call it

O

A

Z.

~

(E), r

is the quotient-object of

Er

that corresponds to the

Graded Abelian Category

195

o

subobject

Bi{E ), r

quotient-object of i~O,

Zi{E r ), Lemma 2'.

~

i E

0,

r

~

r O.

Similarly,

B~l (E r )

is the

that corresponds to the subobject

r

r~rO.

Let

D

be an additive abelian group and let

A

be

a D-graded abelian category such that denumerable direct sums (resp.:

denumerable direct products) exist in the D-graded

category

A.

Then

A

is closed under denumerable suprema (resp.:

infima) of subobjects. Proof.

Let

Ao

be the category of all objects, and all maps of

degree zero, of

A.

Then as we have seen in section 3,

closed under denumerable direct sums (resp.:

A

is

products) of ob-

jects as a D-graded category iff the (ordinary, ungraded) abelian category

AO is closed under denumerable direct sums (resp.:

products) of objects.

Lemma 2 (resp.:

Corollary 2.1) of sec-

tion 1 therefore completesthe proof. Definition 3'.

Let

D

be an additive abelian group and let

be a D-graded abelian category.

A

(Er,dr,Tr)r>r be a spec- 0 tral sequence in the D-graded abelian category A starting with the integer

r O.

Then for every integer

1)

Z (E ) = n z. (E 00 r i>O l

of

Er

2)

B

t ),

r.:: r 0'

we define

i f this denumerable infimum of subobjects

exists, and (E

00

Let

r

) =

J B.l

i>O

(E

r

if this denumerable supremum of subob-

),

is called the permanent cycles of is called the permanent boundaries of tegers

r

~

E , r

all in-

rOo

Proposition 3'.

Let

D

be an additive abelian group.

be a D-graded abelian category, let

rO

Let

A

be an integer and let

Section 4

196 (Er,dr,Tr)r>r

-

A starting with the integer

category teger

1)

r

be a spectral sequence in the D-graded abelian

0

~

r 0'

Z (E) 00 r

Z

2)

exists iff

(E)

r-r 0

r0

Zoo (E ) r -+

3)

Z

r-r

I f both

(E)

rO

O

Z (E 00

and

r

rO

E

)

exists, in which case,

under the epimorphism, "(r-rO)-fold

•

exists, in which case,

Boo(Er)

o

Boo (E ) r -+

rO

r

Boo(Er)

o

under the epimorphism,

" (r-r 0) -fold

•

exist as in conclusions (1)

) and

(2) above, then we have that Er

= Zo (E r

Boo (E ) r

~

)

~

...

Zl (E ) r

~

Z (E

rO

)

and

~

..

Bi+l (E r )

as subobjects of 00

E

exists iff

is the pre-image of image" :

Z (E 00

Boo (E ) r

Then for every in-

we have that:

is the pre-image of image" :

rOo

E , r

B (E 00

rO

,~Zi ~

(E ) r

~

Zi+l (E ) r

Bi (E ) r

~

...

~

for all integers

)

~

...

Bl (E ) r r

~

~rO'

~

Zoo (E ) r

BO (E ) r

~

= 0,

And, then

both exist, then for every integer

we have that the epimorphism

r':: r

' O

"(r-rO)-fold image" induces an

isomorphism:

4)

Z (E 00

rO

) /B (E 00

rO

) ":,. Z (E ) /B (E ). 00 r 00 r

(The proof is entirely similar to that of Proposition 3 of section 1). Definition 4'.

Let

D

be an additive abelian group, let

a D-graded abelian category, let (Er,dr,Tr)r>r

category E 00

A

rO

A

be

be an integer and let

be a spectral sequence in the D-graded abelian

0 starting with the integer

exists iff

Z

(E 00

r 0

)

and

B (E 00

rO

)

rOo

Then we say that

both exist, as defined in

Graded Abelian Category

197

Definition 3' above; in which case, we define

From Proposition 3', we immediately deduce Corollary 3.1'.

Let

D

be an additive abelian group, let

be a D-graded abelian category, let

rO

A

be an integer and let

be a spectral sequence in the D-graded abelian (Er,dr,Tr)r>r - 0 category A starting with the integer be any Let r fixed integer Boo(Er)

~.rO.

Then

Eoo

Zoo(E ) r

exists iff both

and

exist, and when this is the case there is induced a can-

onical isomorphism:

(The inverse of that isomorphism being induced by "(r-rO)-fold image". ) Remark 1'.

The hypotheses being as Proposition 1', notice that

(Er,dr'T~l)r>r

is a spectral sequence in the dual D-graded a-

- 0 belian category,

0

A.

1', it follows that:

From the Remark 3' following Proposition If

r

is any integer .:.. r

exists in the given spectral sequence in

A

O

iff

then

' B

00

(E ) r

Z

(E 00

r

)

exists

O

for the dual spectral sequence in A (call this latter o o "BA (E )"). BA (E ) in which case, is the quotient-object of 00 r ' 00 r o BA (E ) = E /Z (E ). Also, in A, 00 r roar given spectral sequence in

A

iff

Z

spectral sequence in the dual category, "ZAo(E )") r

00

Er

in

A,

'

in which case

o

ZAo(E) OJ

zA (E r ) = E r /B 00 (E r ). 00

Definition 4', imply that:

Eoo

r

exists for the dual

(E ) 00

r

O

A

(call this latter

is the quotient-object of

These two observations and exists for the given spectral

198

Section 4

A iff

sequence in the D-graded abelian category

Eoo

exists for

the dual spectral sequence in the dual D-graded abelian category

AO ;

in which case, they coincide.

Remark 21.

Let

D

A be a

be an additive abelian group, let

D-graded abelian category, let

rO

be an integer and let

be a spectral sequence in the D-graded abelian (Er,dr,Tr)r>r - 0 category A starting with the integer Then we have observed, in Remark 21 following Proposition 11, that sUbquotient of of

E

, rO all integers

Clearly

Er

is a

as subquotients

, Then notice that, by Introductior rO Chapter 1, section 5, Corollary 1.4, we have that: The deE

creasing sequence of subquotients an infimum iff the abutment tion 4 1

;

E00

Er ,

admit rO exists in the sense of Defini-

in which case, the abutment

r .:::.rO'

E00

of

E

is that infimum.

(Of

course, this gives another definition of the existence and value of

Eoo'

that is equivalent to Definition 4 1

,

entirely in terms

of infima of subquotients). Definition 51.

Let

D

be an additive abelian group, let A

a D-graded abelian category, let

rO

be

be an integer and let

(Er,dr,Tr)r>r and (IEr,ldr,ITr)r>r be spectral sequences in - 0 - 0 the D-graded abelian category A, starting with the same integer A map of spectral sequences from

into (Er,dr,Tr)r>r - 0 is a sequence (f ) indexed by the inter r.:::.rO f are maps, of arbitrary degrees (a priori r

perhaps differing for different A

from (Ml)

Er

into

0

in the D-graded category

I E r , for all integers

For every integer Id r

r),

fr = fr

0

dri

r

2:. r 0'

r

2:. r 0'

we have that

and such that

such that:

Graded Abelian Category (M2)

For every integer

r 2. r 0'

the diagram:

H(f ) H (Er ,dr ) _ _...;;r,--.;;» H ('E

'" l'r

'"

199

'd)

l'rr'

r

fr+l

-=-'--=---:> 'E

A is commutative.

in the D-graded category Remark 0.1. (fr)r>r

r+l

In practice, most of the maps of spectral sequences

that one encounters, are such that

-- 0

integers

deg(f r ) = 0,

all

r 2. r O.

Remark 0.2.

Let

(1) = (Er,d r " r) r>r and (2) = ('E r , 'dr'" r) r>r - 0 - 0 be spectral sequences in a D-graded abelian category starting with the same integer

r

a map from (1) into (2).

' O

and suppose that we have

(fr)r>r - 0 Then, from Axiom (Ml), we deduce that

deg('d ) +deg(f ) =deg(f ) +deg(d ), r r r r (3)

deg(d ) =deg('d ), r r

i.e.,

that

r~JO.

all integers

Thus, a necessary condition for there to exist a map from the spectral sequence (1) into the spectral sequence (2) is that all integers

r >r

-

sufficient, since if one takes, e.g., ree zero from

E

r

0

.

This condition is also

fr = (the zero map of deg-

all integers

into

then

(fr)r>r is a map from the spectral sequence (1) into the spec- 0 tral sequence (2). Given two spectral sequences in a given 0graded abelian category

A,

we call them of the same kind iff

they start with the same integer deg('d ), r

all integers

r 2.rO'

-- ---

r

' O

and if also

---- ---

deg(d ) = r

By what we have just observed,

two spectral sequences in the D-graded abelian category of the same kind iff there exists a map between them. Notice, also, from Axiom (M2), that if

A

are

Section 4

200

of spectral sequences in the D-graded abelian category deg(f r ) +deg("r) =deg(f + ) +degh )· r r l deg("r) = 0,

then

deg('r) =

this implies that

(4 )

deg (f ) = deg ( f

Let

D

r 0

r

),

all integers

r~rO'

be an additive abelian group and let

graded abelian category. rO

Since

A,

Consider a pair

is an integer and where

A

be a D-

(r, (ar)r>r)'

o

-

where

0

r '::.rO'

E D is a degree, all integers r We will call such a pair a sequence of degrees in the

abelian

~

D.

L'l.

Then we let

Spec.Seq. (r

(a) ) (A) de0' r r~rO note the category, having for objects all spectral sequences

(Er,dr"r)r>r starting with the integer rO in the D-graded - 0 abelian category A, such that deg (d ) = a for all integers r r r':' r ' O

and for maps, all maps of spectral sequences as defined

in Definition 5'. Proposition 0.3.

Let

D

be an additive abelian group, let

be an integer and, for all integers degree. gory

Let

A

r~rO'

let

be a D-graded abelian category.

Spec. Seq. (r O'

(a) ) (A) r r>rO

lYrE D

rO

be a

Then the cate-

has a natural structure as D-

graded additive category. Proof.

We have already observed the category structure.

For

(a) ) (A), by equa0' r r>rO tion (4) we have that deg(f) =deg(f ), all integers r~rO' r rO Define deg(f) = deg(f r ), all integers r ~rO' Then every map

f = (f r ) r>r - 0

in Spec. Seq. (r

becomes a D-graded category. (a) ) (A) r O' r r>r - 0 graded additive structure is obvious. (Namely, define Spec. Seq. (

The D-

Graded Abelian Category

201

are maps with the same domain and range and such that deg(f r ) =deg(gr ), o 0 deg (gr ) = deg (gr) ,

o

and therefore such that all

r 2.r ' O

deg (f ) = deg (f ) r rO fr + gr is defined).

so that

Q.E.D. Remark I'.

Let

(1) = (Er,dr,Tr)r>r

and (2) = ('Er,'dr,'Tr)r>r 0 - 0 be spectral sequences starting with the same integer rO in the

D-graded abelian category

A,

and let

arbitrary degrees), for all integers

fr:E r?. r O.

r

.... 'Er

be maps (of

Then, if Axiom

(Ml) holds, then Axiom (M2) is equivalent to the statement that, r ?.r ' the map fr+l O passing to the subquotients. Hence, if for every integer

spectral sequences, then for every integer that

is induced by

r?. r 0'

fr

by

it follows

f

is induced from f by passing to the subquotients. r rO Therefore a map of spectral sequences starting with the integer in the D-graded abelian category A is com(fr)r>r ' - 0 pletely determined by the initial map f rO Otherwise stated, an alternative, equivalent definition of a map of spectral sequences starting with the same integer in the D-graded abelian category

A,

rO

from (1) into (2), is:

"A map

f:E .... 'E , of arbitrary degree, such that there rO rO rO exist maps f r: Er .... 'Er for all integers r?. r 0 + 1 such that conditions (MI) and (M2) above hold". Example.

The construction of Remark 2 following Example 5 after

Definition 1 of this section can be seen to be related to that of Exercise 3 of section 3 ("change of grading groups, other direction" ) .

In fact, in the notations of Remark 0.2 above, we

have that the construction of Remark 2 is: the construction of a canonical additive imbedding of additive bigraded categories, from

Section 4

202

7lX7lSpec. Seq. (r

7l

(1)

r~rO

0' into

Spec. Seq. (r

(r -r+l) 0"

additive abelian groups p + q.

) (A

r>r - 0

and for all abelian categories

) (A )

7l X7l

A;

for each integer

),

where the homomorphism of is the sum map,

¢:7l x7l-+7l

More generally, for every abelian category

:7l x 7l -+7l

p + q,

¢(p,q) = A,

denotes the additive homomorphism such that

if again

(p,q) =

then the construction of Remark 2 following Exercise 5 is

A canonical additive imbedding of bigraded additive categories from 7l x7l Spe .

7l

() ) (A ) 0' O:r r-.::.rO 7lx7l

into for

Seq. (r

Spec. Seq. (r ' (r -r+o: ) ) (A O , r r-.::.rO every abelian category

A,

),

every integer

r 0'

and every

sequence

(O:r)r>r of integers. - 0 .Similarly, Remark 3 following Exercise 5 is related to the

construction in Exercise 2 of section 3 ("change of grading group, additive case".)

Namely, if

A

is any abelian category

such that denumerable direct sums of objects exist and such that denumerable direct sum is an exact functor, then the constructiol of Remark 3 following Example 5, is the construction, for each integer

r ' O

of a specific additive faithful functor of singly

graded additive categories I:

Spec. Seq. (

Spec. se q .(

r O'

r O'

( _ +1) r, r (1)

r>:t - 0

)

r2r O

) (A

7l

),

(A

7lx7l

)

7l

'\IV>

Graded Abelian Category

203

with respect to the additive homomorphism of additive abelian groups

¢:J' XJ' +J'

such that

¢(p,q) =p+q,

all

p,qEJ'.

Similarly, the constructions in Remark 4 following Example 5 after Definition 1, are the construction, for every abelian category

A

1)

and every integer

r

' O

of

A canonical additive imbedding of singly graded addi-

tive categories, J'spec. Seq.

(A)'VV> Spec. Seq. (r rO

(1)

0'

r~rO

)

J' (A ),

with respect to the (unique) homomorphism of additive abelian groups

¢:~+

{a};

and, if the abelian category

A

has denum-

erable direct sums and is such that the functor "denumerable direct sum" is exact, then also the construction of (2)

A canonical additive faithful functor of (ordinary,

ungraded) additive categories, I:

(Spec. Seq. (

Remark 2'.

(2)

Suppose that

= ('Er'~r,ITr)r>r

abelian category pose that

J' () )(A ) {O}'VV> Spec. seq.ro(A). r O' a r r>r - 0 (1) = (Er,dr,Tr)r>r

and - 0 are spectral sequences in the D-graded

- 0 A starting with the same integer

rOo

Sup-

:E + 'E is a map (of arbitrary degree) in the rO rO rO D-graded abelian category A, such that, for every integer r > r, -

0

f

degree as

f

induces a map

rO f

f

r

: Er + 'Er

(necessarily of the same

), by passing to the subquotients. (It is equivarO (Z. (E » C Z. (' E ) and lent to say that, "such that f rO ~ rO ~ rO f (B.(E »CB.('E ), for all integers i~O"). Then, is rO ~ rO ~ rO the sequence (f) necessarily a map of spectral sequences? r r~.r 0 As in the corresponding observation in section ~ the answer is

Section 4

204

in general "no".

Necessary and sufficient conditions for such

a constructed sequence (fr)r~ro

of maps (i.e., a sequence of

maps

r 2rO'

fr:E r "" 'E r , fr

duces

all integers

such that

fro

in-

by passing to the subquotients, all integers

to be a map of spectral sequences, is that Axiom (Mll above hold. Remark 2.1.

Let (1) and (2) be spectral sequences in the D-

graded abelian category

A

starting with the same integer

rOo

Let

(fr)r>r be a sequence of maps as in Remark 2' above- 0 that is, fr:E "" 'Er is a map (of arbitrary degree) in the r

D-graded abelian category fr

is induced from

A,

all integers

r .:.rO'

such that

f

by passing to the subquotients. Then, rO as we have noted, (fr)r>r need not be a map of spectral se- 0 quences, since Axiom (Ml) need not hold. Given such a sequence it is still true that r.::.rOl

gers for

deg (f ) = deg (f ), all inter rO however, it need not be true that deg(d ) =deg('d r r

r.:. r 0'

since Axiom (Ml) is not being assumed.

Therefore

such sequences

(fr)r>r can, and in fact do, exist between - 0 spectral sequences not of the same kind, as long as they start with the same integer

In fact, if

(1) = (Er,dr"r)r>r - 0 are spectral sequences in the D-grade(

and (2) = ('E r , 'dr' "r)r>r - 0 abelian category A starting with the same integer not necessarily of the same kind, then if we let the zero map of degree zero from

Er

into

'E , r

fr

but be, e.g.,

all integers

(fr)r>r is as in Remark 2' (but - 0 is not a map of spectral sequences in the sense of Definition

r .::.rO'

then the sequence

5' unless the spectral sequences (1) and (2) are of the same kind) .

Thus, if

A

is any D-graded abelian category, and

rO

Graded Abelian Category

205

is any integer, then one could speak of "the category of all spectral sequences starting with the fixed integer fixed D-graded abelian category (fr)r>r

A,

rO

in the

with maps all sequences

as defined in Remark 2' above."

That category is an

- 0 additive, in general non-abelian, D-graded category, but it is not very interesting. (r

o'

(ar)r>r)

-

(For every sequence of degrees

in the additive group

Spec. Seq. (

()

ro, a r

) (A)

r>r

cons tructed above is an addi ti ve D-

be an additive abelian group and let

D

Let

graded abelian category.

Let

i

~

and

0,

B.

1.

a map in

Spec. seq'r

(resp.: ~a)

Spec. Seq. rO

in

F',F")

r':: r O.

Then for each

Spec. Seq.

(Therefore i f of degree

(A)

() (A) r O' a r r~rO is f = (f ) r>r r - 0 d, then Zi(f)

r r~rO

A

are maps in F

A.

(a)

0' and

be a D-

are covariant additive functors

from the D-graded additive category into the D-graded category

A

be a fixed integer and let

be fixed degrees, all integers

integer

If

the D-graded category

subcategory-o~ that category).

graded

ar E D

D,

0

d,

of degree

all integers

i

~

0).

are the full D-graded subcategories of

(A)

generated by those spectral sequences

r r.::rO

(a)

Spec. seq'r 0'

(A)

such that

(resp. :

r r.::rO

)) exist, which are D-graded ) and B (E 00 rO rO ) additive categories, then Z (E (resp.: B (E ); E) are 00 rO 00 00 rO covariant, additive functors from the D-graded additive cate-

Boo(Er ); both

o

gories

Z (E 00

F

(resp.:

Proposition 4'.

F',F")

Let

D

into

A.

be an additive abelian group, let and

be an integer and let (1) = (Er,dr,Tr)r>r

( 'E 'd

r

r'

-

(2) =

0

be spectral sequences in the D-graded abelian

'T)

r r>r

rO

0

206

Section 4

A starting with the same integer

category

rOo

Let

f = (f r ) r>r be a map of spectral sequences in the sense of - 0 Definition 5' from (1) into (2). Then the following two conditions are equivalent. is an isomorphism (of arbitrary degree)

(1) in the category f= (fr)r>r - 0

(2)

Remark: f

A.

is an isomorphism of spectral sequences.

Of course, condition (2) of Proposition 4' means that

is an isomorphism in

Spec. Seq. (r 0'

ar=deg(dr)=deg('d r ), Corollary 4.1'.

all integers

(a) ) (A), r r>rO

where

r~rO.

Under the hypotheses of Proposition 4', sup-

pose that the two equivalent conditions (1) and (2) of Proposition 4' hold.

Then

Z (E

If

00

if

Z

(E 00

rO

)

and

rO

)

f

r

: Er -+ 'Er

exists (resp.:

B (E 00

rO

)

exist),

Boo(f): Boo(E r ) -+Boo('E r

(resp.:

o

an isomorphism in the category deg (f r

is an isomorphism, all integers

);

If

Boo(E r

then resp.:

)

o

exists;

Zoo(f): Zoo(E r

resp.:

) -+ Zoo('E r

o

E",(f): Eoo-+ 'E,,)

.

0

is

0 A, of degree equal to deg(f)

).

o

(The proofs of Proposition 4' and of Corollary 4.1' are entirely similar to those of Proposition 4 and of Corollary 4.1 of section 1). Remark 1'.

In the statement of Proposition 4', suppose we weaken

the hypothesis that

"f = (f ) be a map of spectral sequences r r~rO in the sense of Definition 5'" to the weaker assertion, that "(fr)r>r be a3 in the hypotheses of Remark 2' following Defi- 0 nition 5'." Then Proposition 4' and Corollary 4.1' remain valid (where, in condition (2) of Proposition 4', by "isomorphism of

Graded Abelian Category

207

spectral sequences" one means "isomorphism in the D-graded category of all spectral sequences starting with the fixed integer ro

in the fixed D-graded abelian category

A,

with maps all

sequences

(fr)r>r as defined in Remark 2' above.") Of course, - 0 then condition (2) is equivalent to the statement: (2')

fr

is an isomorphism (of arbitrary degree) in the

D-graded category

A, all integers

r

~

rOo

(Notice, that an "(f ) > " as in Remark 2' following Definir r_rO tion 5' induces a map from Zoo(Er ) into Zoo (' Er ) whenever 0

0

both are defined; similarly for

B 00

and

EcO> .

Section 5 The Spectral Sequence of an Exact Couple, Graded Case

This section closely parallels section 2.

We therefore

will use a similar theorem numbering, and we will omit proofs when they are identical to the corresponding proofs in section 2. We let

D

be a fixed additive abelian group, throughout

this section. Definition 1'. V

Let

be an object in

A A

be a D-graded abelian category, let and let

essarily of degree zero) quence of subobjects,

from

(Ker t

sequence of subobjects of

t:V-*V

r

V.

lection of all subobjects of

V

be a map in

into itself. r ~ 0,

),

of

A

(not nec-

Then the se-

V,

is an increasing

If a supremum exists in the colV,

(t-torsion part of

then it is called the t-torsion

L Ker(t r ).

V)

r>O Similarly, the sequence of subobjects,

is a decreasing sequence of subobjects of

V.

If an infimum ex-

ists, then it is called the t-divisible part of (t-divisible part of

V) =

n

V,

r Im(t ).

r>O Definition 2'.

On the other hand, we can consider the inverse

system, indexed by the positive integers, such that i':l,

and such that

t(i+l)~v(i+l) 208

-*v(i)

is the map

V(i) =V,

t,

all

Exact Couple, Graded Case integers

i > 1.

lim(v(i) t(i» i>l

If an inverse limit

the D-graded category

A,

209

exists in

I

"",0.

then we have the natural map:

If the inverse limit exists, then the image of this map, a subobject of

V,

is called the infinitely t-divisible part of

V.

Finally, consider the direct system indexed by the positive integers, such that

V (i) = V,

t (i):V (i)

is the map

->

V (i+l)

this direct

i

~ 1,

t,

and such that i > 1.

If a direct limit of

system exists, then we have the natural map:

If the direct limit exists, then we call the kernel of this map the infinite t-torsion part of Example 1'.

V.

If in the D-graded abelian category

A,

denumer-

able direct products of objects exist, then by Proposition 6 of section 3,inverse limits of inverse systems in the category

A

indexed by denumerable directed sets exist, and also by Corollary 2.1 of section 1 denumerable infima of subobjects of objects of

A

exist.

Therefore, in this case,

t:V

->

V

is any map (of

arbitrary degree), then both the t-divisible part of the infinitely t-divisible part of D-graded abelian category

A

V,

exist.

V,

and

Similarly, if the

is such that denumerable direct

sums of objects exist, then direct limits of direct systems indexed by denumerable directed sets in the category

A

exist,

and also denumerable suprema of subobjects of objects of exist.

Therefore, in this case, whenever

the D-graded category

A

t:V+V

A

is a map in

(of any degree), then the t-torsion

210

Section 5

part of

V

and the infinite t-torsion part of

Example 2'.

Let

A

t:V .... V

both exist.

be a D-graded abelian category, let

be an object and let Then

V

t:V .... V

V

be a map (of arbitrary degree).

is also a map in the dual D-graded abelian cate-

It is easy to see that the t-divisible part, resp.:

gory

t-torsion part; infinitely t-divisible part; infinitely t-torsion part of the map in the dual D-graded abelian category t:V"" V,

exists iff the t-torsion part, resp.:

t-divisible

part; infinite t-torsion part; infinitely t-divisible part of t:V"" V

A·,

exists in the given D-graded abelian category

which case, the t-divisible part (resp.:

in

t-torsion part; in-

finitely t-divisible part; infinite t-torsion part) of

t:V .... V

considered as a map in the dual D-graded abelian category, is the quotient-object in t-torsion part (resp.:

A

of

V,

V/Q,

where

Q

is the

the t-divisible part; the infinite t-tor-

sion part; the infinitely t-divisible part) of

t:V .... V

as a map in the given D-graded abelian category

considere(

A.

That is, under the operation of passing to the dual Dgraded abelian category and then replacing quotient objects by the corresponding subobjects, the notions of nt-divisible part" and nt-torsion part" interchange, and similarly the notions of "infinitely t-divisible part" and "infinite t-torsion part" interchange. Example 3'.

If

t:V""V

D-graded abelian category of

V

is a map (of arbitrary degree) in the A,

and if both the t-divisible part

and the infinitely t-divisible part of

V

exist in the

sense of Definition I', then we always have (infinitely t-divisible part of

VIc (t-divisible part of

V).

Exact Couple, Graded Case

211

(The proof is similar to that of the corresponding assertion in section 2). Example 4'. object

V

If

t:V-+V

is a map (of arbitrary degree) from the

if the t-torsion part of V

A,

into itself in the D-graded abelian category V

and

and the infinite t-torsion part of

both exist as defined in Definition 2', then we always have (t-torsion part of

V)

C

(infinite t-torsion part of

V).

(The proof is similar to that of Example 4 of section 2). Example 5'.

Proposition I'.

Let

A be a D-graded abelian cate-

gory such that denumerable direct sums of objects exist and such that the denumerable direct limit over the directed set of positive integers is an exact functor.

Then if

is a map (of arbitrary degree) from an object

V

then the of

V)

(t-torsion part of

V)

t:V

-+

V

into itself,

and the (infinite t-torsion part

both exist, and

(t-torsion part of

(infinite t-torsion part of

V)

V).

(The proof of this assertion is similar to that of Proposition 1 of section 2).

Example 6'.

Consider the special case, in which the D-graded

abelian category

A is the category of D-graded abelian groups,

D A = {abelian groups} . d

V = (V ) dED' t:V->V

where

be a map in

d td:V -> vd+dO

Let

d V A

V

be an object in

is an abelian group, all of degree

dO'

Then

Then

A.

dE: D.

Let

d

t= (t ) dED'

is a homomorphism of abelian groups, all

where

dE D.

Then explicitly, (1)

(t-torsion part of

V)

d

(T ) dED'

where

T

d

= {v E V

d

such

212

Section 5

that there exists an integer td+(i-l)dO

0

.t d + 2dO

.,

~

i

every d-dO

td-2dO

=

(5

d

)

t d-idO ( v. )

= v },

~

all

where

S

d

d-(i+I)d O (

i

v i +l

~

dE

)

0, such that

(infinite t-torsi6n part of

dE d

= {v E V:

V) = (H

v = va'

For

such that

d

)dED'

where v. E vd-idO ~

and such that i > O.

(t-torsion part of

V)

o.

o.

all integers

= vi'

all

vi E vd-ido,

There exists a sequence of elements

for every integer t

v d + idO },

in

dEO'

(infinitely t-divisible part of

Hd = {v E vd:

( 4)

=0

td(V)

0

V)

such that

there exists an element

0,

t o o

(3)

t d + dO

0

(t-divisible part of

(2)

i >0

V),

this latter by Proposition 1'. Also, in the category of D-graded abelian groups, if d

B

A = (A )dEO' f:A .... B

= (B d ) dEO

is a map with

fd:Ad .... B d + e

are D-graded abelian groups, and if e = deg (f) ,

so that

d

where

f= (f )dED'

is a homomorphism of abelian groups, all

dE D,

then (5)

d

Ker f = (Ker (f )) dEo'

(where

"Ker fd",

and

"1m fd-e"

1m f = (Im(f

d B = {B )dEO

) )dEO

denote the usual kernels and images

of the homomorphisms of abelian groups). and

d-e

Also, if

are D-graded abelian groups, and if

d

A = (A ) dEO is

eEO

a degree, then (6)

An additive relation

simply an indexed family Ad

lation from (i.e., where Ad x Bd +e ,

all

Rd

into

R

from

d

(R )dEO'

d+e B ,

all

A

into

where

B Rd

of degree

e

is

is an additive re-

dEO

is an additive subgroup of the abelian group dEO).

By Corollary 5.1 of section 3, the Exact Imbedding Theorem

Exact Couple, Graded Case

213

for D-Graded Abelian Categories, every D-graded abelian category that is a set admits an exact

imbeddin~

of D-graded abelian

categories, into the category of D-graded abelian groups.

This

result can be used, in the same fashion as the usual Exact Imbedding Theorem for (ordinary, ungraded) abelian categories, to prove certain theorems that are

"finite"

(i.e., that do not

involve infinite constructions) in arbitrary D-graded abelian categories, by simply verifying them for the case of, the specific D-graded abelian category, the category of D-graded abelian groups.

For example, most of the theorems in section 4 can be

so handled (e.g., the construction by induction of the "r-fold cycles", the "r-fold boundaries", and of the epimorphism, "rfold image", all integers

r

~

0,

given in Definition 2' of sec-

tion 4; or Proposition I' of section 4, etc.) in this section, in the proofs of:

The same is true,

Proposition 2'; in Defini-

tion 4'; Corollary 2.1'; Corollary 2.2'; and e.g., in the proofs of conclusions (lr) and (2r) of both Corollary 3.1' and Corollary 3.2'; - but not when such concepts as "infinite direct sums", "infinite direct products", "infinite suprema of subobjects", "infinite infima of subobjects", "t-divisible part", "t-torsion part", "infinitely t-divisible part", "infinite t-torsion part", or any other construction involving infinite direct or inverse limits is involved.

The reason why the Exact Imbedding Theorem

doesn't apply in that case, is that exact imbeddings do not, in general, preserve such infinite constructions (e.g., in general do not preserve "infinite direct sums", "t-divisible part," etc.). In the theorems noted below, which are parallel to theorems in section 2, the proofs that are left out are always virtually

Section 5

214

the same as the corresponding assertions in section 2.

When a

proof of a corresponding assertion in section 2 uses the (ordinary, ungraded) Exact Imhedding Theorem to reduce the proof to the case of abelian groups, the proof of the assertion in this section uses the Exact Imbedding Theorem for D-Graded Abelian Categories (i.e., Corollary 5.1 of section 3) to reduce the proof to the case of D-graded abelian groups.

(Similarly for

the theorems in section 4 that are parallel to theorems in section 1). Definition 3'.

Let

A

be a D-graded abelian category.

exact couple in the D-graded abelian category

A

Then an

is a diagram:

t

\)V

(1)

E

in the D-graded category

A

(in which the maps

t,h,k

can have

arbitrary degrees), such that we have exactness at each of the three corners.

That is, it is a pair of objects

D-graded abelian category h:V-+-E

and

Ker h,

Im h

k:E-+-V,

= Ker

Proposition 2'.

k, Let

A,

A,

in the

t:V -+- V,

of arbitrary degrees, such that

1m t=

Im k = Ker t. A

be a D-graded abelian category, and

let (1) be an exact couple in exact couple in

together with maps

V,E

A.

Then there is induced another

215

Exact Couple, Graded Case

called the derived couple of the exact couple (I), such that VI = 1m t, maps t,

tl,h

El = (Ker d) / (1m d), l

and

kl

where

d =h

0

k,

and where the

are induced, respectively, from the map and the map

the additive relation

k,

by passing

to the 3ubquotients (as defined on the next to the last page of section 3).

In particular,

deg(t ) =deg(t), l

deg(h ) =deg(h)l

deg(t),

deg(k ) =deg(k). l (The proof is entirely similar to that of Proposition 2 of

section 2). Definition 4'.

Let

A

be a D-graded abelian category, and let t

(1)

be an exact couple in the D-graded abelian category for each integer

r > 0

abelian category

A,

A.

Then

we define an exact couple in the D-graded which we call the r'th derived couple

of the exact couple (1), and denote:

each integer

r > O.

The construction is by induction on

r.

The zero'th derived couple of (1) is defined to be the derived couple (1). teger

r

~

0,

Having defined the r'th derived couple for any indefine the

(r + 1) 'st derived couple of (1) to be

the derived couple, as defined in Proposition 2', of the r'th derived couple of (1).

216

Section 5

Notation: r

.:.0,

We will write

whenever

object

V

trV

t:V->-V

for

1m (tr:v ->- V),

all integers

is a map (of arbitrary degree) from an

A.

into itself in a D-graded abelian category

By induction on

r,

Corollary 2.1'.

Let (1) be an exact couple in the D-graded

abelian category

we deduce,

A.

Then for each integer

derived couple (lr) of (1) r V =trV=Im(t ), Er

~

0,

the r' th

and

k :E ->- V r r r

is such that

a subobject of

r

r

is a subquotient of

V,

E,

and such that the maps

h :V ->- E r r r

are induced, by passing to the subquotients, respectively by the t:V->- V,

map

the additive relation

k: E ->- V, all integers

r> 0.

deg (h ) = deg (h) - r • deg (t) r r> 0.

and the map

In particular, and

deg (k ) = deg (k) , r

Also,

(therefore

deg (t ) = deg (t) , r

where d

r

0

d

r

= 0),

all integers

all integers d

r

=h

r

0

k

r

r> 0.

The proof of the Corollary follows immediately from the Proposition. Remark 1': (1)

We might also picture the r'th derived couple of

as:

where

It

til,

"h

0

t-r",

and

D-graded abelian category quotients by the map the map

k,

t,

respectively.

It

k"

denote the unique maps in the

A deduced by passing to the subthe additive relation

and

Exact Couple, Graded Case Corollary 2.2'.

Let (1) be an exact couple in the D-graded

abelian category all integers

217

r >

A,

o.

and let (1 ) r

be the r'th derived couple,

Then for each integer

r >0

we have the

short exact sequence

o +-

[(Ker t)

n

in which the maps by

hand

k,

(trV) 1

"h"

<~ Er ~

and

"k"

[

tr~

(V/Ker t· (V/Ker t

)

]"'-0,

are the unique maps induced

respectively, by passing to the subquotients

(and therefore have the same degrees as

hand

k,

respec-

tively) . (The proof is entirely similar to that of Corollary 2.2 of section 2.) Remark'

The short exact sequence (*r) of Corollary 2.2' is

such that, the objects at the right and left depend only on and the endomorphism

t

object in the middle is

(of arbitrary degree) of E . r

"a sort of computation" of

Vi

Th(refore the sequence E

in terms of

r

V

and

V

yet the (* r) gives t.

(Of

course, this computation is not perfect, since it is only a short exact sequence). Definition 5'.

Let

A

be a D-graded abelian category and let t

(1)

be an exact couple in the D-graded abelian category

A.

Then

we associate with the exact couple (1) a spectral sequence (Er,dr,Lr)r>r' starting with the integer - 0 abelian category A, as follows. Let Er

0,

in the D-graded

be the object in the

Section 5

218

r'th derived couple of

(1)

as defined in Definition 4',

(as in Definition 4'). in Corollary 2.1', we have that Let Ker (d ) lIm (d ) r r

onto

(Er,dr,Tr)r>O

A,

category

E + , r l

d

0

r

d

r

= 0,

be the identity mapping from

Tr

all integers

r> O.

Then

is a spectral sequence in the D-graded abelian starting with the integer

abelian category ':

Then as noted

and that

O.

the spectral sequence of the exact couple (1)

Remark

r:"O,

in the D-graded

A.

As we have noted in section 4, a spectral sequence

in the D-graded abelian category

A,

starting with a given in-

teger, by re-indexing, can be re-interpreted as a spectral sequence in the D-graded abelian category other integer.

A starting with any

In particular, given the exact couple (1) in an

arbitrary D-graded abelian category, and given any fixed integer r

' O

we can regard the spectral sequence of the exact couple (1)

as starting with the integer

r

' O

by re-indexing the spectral

sequence of (1), as defined ill Definition 5'; we obtain Definition 5.1'.

Let (1) be an exact couple in the D-graded

abelian category

A.

Then the spectral sequence of the exact

couple (1) re-indexed to start with the integer

rO

spectral sequence in the D-graded abelian category with the integer

where

Er,dr,T r

r

is the

A starting

' O

are as in Definition 5' above, all integers

r > O.

(The reason for our inclusion of Definition 5.1' is that we shall actually have occasion, in studying bigraded exact couples

Exact Couple, Graded Case

219

and their spectral sequences, occasionally to regard spectral sequences as being so re-indexed. to start usually with or

2.

rO = 1

The motivation is, in order to make the resulting bi-

graded spectral sequences conventional in the sense of Example 4 of section 4). The spectral sequence of an exact couple, graded case, was first introduced in [ECJ, for certain D-graded abelian categories and certain

D.

The next theorem describes explicitly the r-fold cycles, r-fold boundaries, permanent cycles, and permanent boundaries in the spectral sequence of an exact couple in a D-graded abelian category. Corollary 3.1'.

Let

D

be an additive abelian group and let

A be a D-graded abelian category.

Let

t

(1)

be an exact couple in the D-graded abelian category (Er,dr,Tr)r>O so that (1)

EO = E.

Then

Zoo(EO)

V)

exists, then the per-

exists, and we have that

Zoo (EO) = k -1 (t-divisible part of (lr)

Let

be the spectral sequence of the exact couple (1),

If the (t-divisible part of

manent cycles

A.

Always, for every integer

r

~

V). 0,

we have that

Section 5

220 (2)

If the

nent boundaries

(t-torsion part of Boo (EO)

V)

exists, and we have that

Boo (EO) = h (t-torsion part of (2r)

Always, for each integer

Corollary 3.2'.

exists, then the perma-

V). r

~

0,

we have that

Under the hypotheses of Corollary 3.1', we have

that, always, r > O.

all integers If the (t-divisible part of

V)

exists, then also

(t-divisible part of

V)

n Ker

t.

Always, we have [Ker(tr:V+V)] +tV. If the (t-torsion part of

V)

exists, then also

(t-torsion part of Corollary 3.3'. both E

Boo(EO)

Under the hypotheses of Corollary 3.1', if

and

= EO = Zo (EO)::::;)

V) + tV.

Zoo(EO)

exist, then we have that

Zl (EO)::::;)···::::;) Zr (EO)::::;) ... ::::;) Zoo(EO)::::;) Ker k

=

Im(h) ::::;) Boo (EO) ::::;) ... ::::;) Br (EO) ::::;) ... ::::;) Bl (EO) ::::;) BO (EO) = O. (The proofs of Corollaries 3.1', 3.2', and 3.3' are analogous to the proofs of the corresponding assertions in section 2). Remark 1'.

In Corollary 3.3', if either

Boo(EO)

or

Zoo(EO)

or both do not exist, then the conclusion of Corollary 3.3' re-

221

Exact Couple, Graded Case mains valid, if we simply delete the single occurrence of liB (E ) II 0

00

or of

'

or both in the conclusion, which-

ever or both do not exist. Remark 2'.

In Corollary 3.1', conclusion (1), a sufficient con-

dition for

II

Z

(E 00

divisible part of

0

)

II

to exist is given, namely that the

V)

exists.

This result can be improved.

Namely, necessary and sufficient conditions for

Zoo(EO)

exist is that the decreasing sequence of subobjects of Ker (t)

n

r ~ 0,

(trV),

(t-

have an infimum,

And, when that infimum exists,

Zoo(EO)

to V,

n [(Ker t) n (trV»). r>O is-always the pre-image

n [(Ker t) n (trV»). A similar improvement in r>O conclusion (2) of Corollary 3.1' holds. Namely, necessary and under

k

of

sufficient conditions for

Boo(EO)

to exist is that the inV, (tV) + [Ker (tr:v ->- V) ) ,

creasing sequence of subobjects of r > 0,

L [(tV) + Ker (tr:V -+ V»). And, when r>O Boo(EO) -is always the image under h of

have a supremum,

this is the case,

I [(tV) +Ker(tr:v->-V»). r>O We take the next theorem seriously.

this supremum:

Theorem 4'.

Boo(EO) =h(

Let t

(1)

be an exact couple in the D-graded abelian category that the V)

(t-divisible part of

both exist.

V)

and the

A.

Suppose

(t-torsion part of

of the exact couple (1)

(E ,d ,1 )r>r r r r - 0 in the D-graded abelian category A, we

have that

Also, we have the short exact sequence

Eoo

Then, for the spectral sequence

exists.

in the D-graded abelian category

A:

222 (*)

Section 5

V)l~Eoo ~

O-<-[(Ker t)f"'i (t-divisible part of V/ (t-tor sion part of V)] f -V:':+-),) -<- 0 , [-:-t-';(-':-v=-';(;-:=t'---:t:':o:':r:":s:'::i;:"=o:":n.:........lp;:;';a=:r=-t;="-o=7

in which the maps

"h"

and

"k"

passing to the subquotients.

are induced by

In particular,

hand

k

by

deg("h") = deg h,

deg("k") =deg(k). (The proof is entirely similar to that of Theorem 4 of section 2). Remark I'.

We will call the rightmost group in the short exact

sequence (*), E"" (*),

which is a subobject of

or sometimes the direct part of

Eoo'

The leftmost group in will be called the

or sometimes the inverse part of

that the right part of that the left part of Remark 2'.

the right part of

E",.

which is a quotient-object of

left part of

E""

Eoo Eoo

is a quotient-object of is a subobject of

E . 00

V,

Notice and

V.

As in the corresponding Remark in section 2, we ob-

serve that, if (1) is an exact couple in the D-graded abelian category

A,

then we have a dual couple,

(1°),

which is an

exact couple in the dual D-graded abelian category couple (1) in the D-graded abelian category theses of Theorem 4' iff dual couple (1°) abelian category

O

A

A

AO.

The

obeys the hypo-

in the dual D-graded

obeys the hypotheses of Theorem 4'.

When

this is the case, then by Theorem 4' we have the short exact sequence (*) - and by Theorem 4' applied to the dual couple (1°) in the dual D-graded abelian category short exact sequence,

(*0),

"written backwards".

That is,

with

E",

for

°

(1 )

in

AO. Eoo

AO (*0)

we have a similar is simply (*)

for (1) in

A

coincides

(so the middle terms of (*) and (*0)

Exact Couple, Graded Case coincide), and the left (resp. :

223

right) part of

coincides with the right (resp. : left) part of

Eoo E

for (1) for

In particular, the dual concept of the "right part" of the "left part" of

E00'

and converselYi while

E

00

(1°) •

E

00

is

itself is

"self-dual" . Remark 3 of section 2 does not generalize very readily to D-graded abelian categories (unless the map

t:v

-+

V

is of deg-

ree zero, which is certainly not always the case in the most important applications). Remark 4'.

Remark 4, through the end of Corollary 4.1, gener-

alizes word for word to gories.

exact couples in D-graded abelian cate-

We won't write down the generalization (but we will re-

fer, e.g., to "Corollary 4.1''', which is Corollary 4.1 of section 2, with the single change in statement: "Theorem 4'. ")

"Theorem 4" to

(The Adams spectral sequence, noted at the end

of Remark 4 of section 4 occurs really in a graded category, so that portion of Remark 4 of section 2, technically speaking, actually belongs here). The analogues of Remarks 5 and 6 after Theorem 4 of section 2 go through to

exact couples in D-graded abelian categories,

essentially without change.

(The only changes necessary are:

"Corollary 3. i" by "Corollary 3. i ' " ,

i = 1,2,3,

and delete the

one occurrence of "Theorem 3" in Remark 6). Definition 5'.

Let

D

be an additive abelian group and let

be a D-graded abelian category. t (1)

Let

A

224

Section 5

and

t' V'

~V'

~}I

(2)

E¥,

be exact couples in the D-graded abelian category map of exact couples from (1) into (2) is a pair g:V-+V'

and

f:E+E'

D-graded category t'og=got, Proposition 5.0.

A,

Then a

(g,f)

where

are maps (of arbitrary degrees) in the such that

h' og=foh, Let

A.

D

k'of=gok.

be an additive abelian group.

Let

A

be a D-graded abelian category and let (1) and (2) be exact couples in the D-graded abelian category

A.

Then the following

two conditions are equivalent: (1)

There exists a map of exact couples from the exact

couple (1) into the exact couple (2). (2)

(a)

deg(t)=deg(t ' ),

and

(b)

deg (h) + deg (k)

Proof:

=

Suppose (1), and let

deg (h I (g,f)

)

+ deg (k I

)

•

be a map of exact couples

from the exact couple (1) into the exact couple (2). t'og=got,

h' og=foh

and

k'of=gok.

(3)

deg (t ') + deg (g) = deg (g) + deg (t) ,

(4)

deg(h') +deg(g) =deg(f) +deg(h),

(5)

deg(k') +deg(f) =deg(g) +deg(k).

Equation (3) implies (2) (a). additive abelian group (6)

D

Then

Therefore,

and

Adding equations (4) and (5) in the

yields

deg(h') +deg(k') +deg(g) +deg(f) deg (h) + deg (k) + deg (g) + deg (f) ,

Exact Couple, Graded Case which implies (2) (b). holds. to

Conversely, suppose that condition (2)

Then let

9

be the zero map of degree zero from

and let

f

be the zero map of degree

V',

from

225

E

into

E'.

V

in-

deg(h')-deg(h)

Then by construction, equation (4) holds.

By (2) (a), equation (3) holds, and by (2) (b), equation (6) holds. Subtracting equation (4) from equation (6) we deduce that equation (5) holds. f

Therefore equations (3), (4) and (5) hold.

are zero maps (of various degrees), t 'og,

Since

and

9

h' ° g,

f

rees).

Therefore, equations (3), (4) and (5) imply, respectively,

0

k' ° f

h,

that

t' og=got,

fore

(g, f)

and

go k

h' og=foh

got,

are zero maps (of various deg-

and

k' of=gok.

And there-

is a map of exact couples.

The last proposition motivates the following definitions. Definition 5.1.

Let

D

A

be an additive abelian group and let

be a D-graded abelian category.

If (1) and (2) are exact couples

in the D-graded abelian category

A,

and (2) are of the same kind iff

deg (t) = deg (t' )

then the exact couples (1) and

deg (h) + deg (k) = deg (h' ) + deg (k' ) . Definition 5.2.

Let

D

be a D-graded abelian category. T,pED).

let in

EC p,T

(A)

and such that

A

be an additive abelian group and let Let

T

Then we define a category

and

p

EC p,T

be degrees (i.e., (A) •

are the exact couples (1) such that deg (h) + deg (k) = p.

The maps in

The objects

deg(t) = T EC

(A) are p,T the maps of exact couples, as defined in Definition 5' above.

One way of interpreting Proposition 5.0 is that, if an additive abelian group and if gory,

D

is

A is a D-graded abelian cate-

then the category having for objects all exact couples in

the D-graded abelian category

A, and for maps all maps of exact

226

Section 5

couples in the sense of Definition 5', splits up into the disjoint union pairs

(or direct sum) of the categories

EC

p,T

for all

(A)

(P,T) ED x D. Let

Remark

(g,f)

be a map from the exact couple (1)

the exact couple (2) in the D-graded abelian category

A.

into Then

from equation (4) in the proof of Proposition 5.0, we have that deg (g) = deg (f) + deg (h) - deg (h') .

(4')

Therefore, the degree of either of

f

or

9

determines the

degree of the other. Corollary 5.0.1. A

Let

0

be an additive abelian group and let

be a D-graded abelian cabegory. the category

P, TED,

rees

EC

p,T

Then for every pair of deghas a natural structure

(A)

as D-graded additive category. Proof:

If

(g, f)

is a map in

deg(g,f) =deg f. ECp,T(A)

Then if

EC

(g,f)

p,T

then define

(A),

and

(g',f')

are both maps in

with the same domain, range and degree, say from the

exact couple (1) into the exact couple (2), then by definition deg f=deg(g,f) =deg(g',f') =deg f', the last Remark,

deg 9 = deg f + deg h - deg h' = deg f'+ deg h-

deg h' = deg g'. and

Therefore

(g+g',f+f')

Remark:

EC

p,T

9 + g'

is a map in

domain, range and degree as readily that

and by equation (4') of

(A)

and EC

(g,f)

P,T

f + f' (A),

and

both make sense,

again with the same

(g',f').

It follows

is a D-graded additive category.

Of course, the D-graded additive category

EC

P,T

(A)

as defined in Definition 5.2, is almost never aD-graded abelian category. If (1) and (2) are exact couples in the abelian category

A,

and if

(g,f)

is a map of exact couples in the sense of

Exact Couple, Graded Case Definition 5' P

and

T

t:;

227

(by Proposition 5.0, it is equivalent to say: EC

if (1) and (2) are objects in

D,

(g,f) is a map in

EC

then

(A) "),

p,T

p,T

of (1) and (2) are in

EC

P,T

induces a map, by

(g, f)

(A),

ECp_T,T(A).

and if

(A)

passing to the subquotients, on the derived couples. that, if (1) and (2) are in

"If

Notice

then the derived couples

(This is because, in the

derived couple:

of the exact couple (I), we have that

kl

is induced by

k,

so

deg (k l ) = deg (k), while h is induced by the additive relation: r 0 r- l by passing to the subquotients, so that

that

h

t

deg (hI) = deg (h) - deg (t) ).

The assignment, "derived couple",

therefore is a covariant, additive functor of additive, D-graded categories from

EC

p,T

is of degree

(g,f)

d

in

(gl,f ) l

then the map

into

(A)

EC

(resp.:

deg

(f) ,

g)

p,T

P-T,T

fl

(g,f)

and therefore

gr'

d = deg f),

on the derived couples

(resp.:

gl)

deg (g, f) r

~

0,

= deg

a map

f

are induced by

resp.:

fr

(g, f)

of degree

is induced by

(gr,f ) r

g,

resp.:

d

r

~

0,

and every

p, TED,

of

of degree f,

passing to the subquotients) on the r'th derived couples. fore, for each integer

deg(f ) 1

= deg (f 1 ) = deg (gl' f l ) ) .

exact couples therefore also induces a map (and

if

by passing to the subquotients, so that

By induction on

d

(Notice that if

(A).

(i. e.,

(A)

induced by

has the same degree d- since f

EC

by There-

we have a co-

variant additive functor of D-graded additive categories, r'th

228

Section 5

derived couple, from if

(g,f)

into

EC p ,1(A)

is a map of degree

d

EC

(A).

Therefore,

P- r 1,1

from the exact couple (1)

into the exact couple (2) in the D-graded abelian category then there is induced a map, of the same degree

d,

A,

in the

sense of Definition 5' of section 4, of the associated spectral sequences (which start with the integer zero) in the abelian category

A.

Explicitly, the induced map of spectral sequences

is the sequence on the r'th derived couples,

r> O.

We have proved Proposition 6. A

Let

D

be an additive abelian group, and let

be a D-graded abelian category.

Let

p, 1

i;:

D.

Then the as-

signment which to every exact couple t (1)

in

(E ,d ,1 r )r>0' r r is a covariant, additive functor of D-graded additive categories, from

EC p ,1(A)

EC

p,1

Example 1.

associates its spectral sequence

(A) Let

into D

Spec. Seq. (0 ( _ ) ) (A). , P rt r> 0 be an additive abelian group, let

A

be a

D-graded abelian category and let t

(1)

be an exact couple in the abelian category derived couple

A.

Then we have the

229

Exact Couple, Graded Case "t" tV--·-·-·-~tv

It

/

l "k'\ / : ' h t - "

(1' )

El where, respectively, k and

"ht- l "

"t"

is induced by

is induced by

fh

t,

(f )-1 t '

0

"k"

is induced by

by passing to the sub-

quotients. The exact couple (1) defines an exact couple, call it (1)°, in the dual D-graded abelian category

where

VO = V,

EO = E,

to = t,

hO = k

couple of the exact couple (1)°

in

and O

A

k

O

= h.

The derived

is of the form

litO"

(V/Ker t)

) (V/Ker t)

"k~/hO(tO)-l" El

where, respectively, induced by

f

0

hO abelian category

(f

"to"

1:9

) -1

O

A

is induced by and

"ko"

to,

"ho(to)-l"

is induced by

k

by passing to the subquotients.

the exact couple (1°')

as an exact couple in

A,

exact couple II

ttl

(V/Ker t)

>(V/Ker t)

"t-l~/h" El

O

is

in the Rewriting

we obtain the

230

in

Section 5 the O-graded abelian category

"h" and

(r t)

-1 0

"t-lk"

r k'

A,

in which the maps

are induced, respectively,

by

by passing to the subquotients.

exact couple

is in

(1)

EC

p,

T

(A). P-T,T morphism of

g: (V/Ker t) ~ tV

Let t,

exact couples

and (1

0

,0)

D-graded additive category degree zero:

If the original

(1 0

,0)

are both in

be the factorization iso-

an isomorphism of degree

(l)

t, hand

(A) - i.e., if P = deg h + deg k,

T = deg t - then the exact couples (1') and EC

"tn,

T = deg (t).

Then the

are canonic,ally isomorphic in the ECp_T,T(A),

through the map of

(g, idE ). 1

Remark: quence

By Proposition 6, it follows that every spectral se(Er,dr,Tr)r>O

in the O-graded abelian category

A,

starting with the integer zero, that comes from an exact couple, has the property that there exist degrees

P

and

,

in

0

such that (1)

Of

deg(d ) = P - n, r

cou~se,

P

and

T

all integers

r> O.

are uniquely determined by these proper-

ties. In general, given a spectral sequence

(Er,dr,Tr)r>O

starting with the integer zero in a O-graded abelian category A,

and if

P

and

TEO,

then we will say that the spectral

sequence is of type

(p,,)

all integers

Then clearly, the spectral sequences of

type

(p,T)

r > O.

iff condition (1) above holds for

in the O-graded abelian category

the objects in

Spec. Seq.

(0

(_)

, P r, r>O

)

(A).

A

are precisely

Therefore, we will

sometimes denote this latter O-graded additive category by Spec. seq.P"(A). Then Proposition 6, in this notation, is equivalent to

Exact Couple, Graded Case saying that, for every pair every exact couple in quence, is a

which T

(A)

associates its spectral se-

EC

p,T

into

(A)

A special case of the last Remark is the case in is the additive group of integers

0

= O.

p,T

the assignment which to

covariant, additive functor of D-graded additive

categories from Example 2.

EC

p,T ED,

231

Then if

A

is a

(;r ,+),

p

=1

;r-graded abelian category, the objects

1 0 of Spec. Seq. ' (A)

are precisely all spectral sequences

all integers

Also, the objects of

r> O.

EC1,0(A)

of this Example are all exact couples (1) in the abelian category Example 2.1.

Let

gory, and let

A

such

S

and

that

deg(t) =0,

in the case

;r-graded

deg(h)+deg(k)=+l.

be an (ordinary, ungraded) abelian cate-

A = S;r,

the

;r-graded abelian category of all

singly graded objects in the abelian category

S.

Then Example

2 above applies. Notice that in this case the spectral sequences

in

A = S;r

are what we have called in section 4 the "singly

graded spectral sequences in the abelian category objects of

Spec. Seq.

I 0 ;r ' (S )

S",

and the

are what we have called the"con-

ventional singly graded spectral sequences in the abelian category

S

starting with the integer zero".

couples in the

;r-graded abelian category

We will call the exact S;r,

graded exact couples in the abelian category jects of

the singly S,

and the ob-

the conventional singly graded exact couples

in the abelian category

S.

Thus, a very special case of Propo-

sition 6, is that, for every abelian category

S,

we have the

functor of singly graded additive categories, "associated spectral sequence", from the category of conventional singly graded

Section 5

232

S

exact couples in the abelian category

into the category of

conventional singly graded spectral sequences starting with the

S.

integer zero in the abelian category Example 3. which

Consider the special case of the last Remark in is the additive group

D

any integer.

Then let

(:l x :l , +).

p = (rO,-r

O

+ 1)

and

Suppose that

r0

l = (-1,+1).

In

is

this case, it is the custom to re-index spectral sequences in Spec. Seq.P,l

so as to start with the integer

(A)

A

More precisely, let

be a

rOo

(:l x :l)-graded abelian cate-

gory (that is, a bigraded abelian category), and let integer.

be an

Then we have a natural isomorphism of bigraded addi-

P

Spec. Seq. '

tive categories between Spec. Seq. (r

(r -r+l)

= (-1,+1).

) (A),

l

(A)

where

and

P = (rO,-rO+l)

and

r>r

0"

1

rO

- 0

(Namely,

Spec. Seq.P,1 (A)

Spec. Seq. (0 ( _ ) ) (A). If , P r1 r> 0 then define the corresponding object of Spec. Seq. ( (E r r

o

,d _ r r

( _ +1) ) (A) r O' r, r > r_rO

0

,1 _ r r

)r>r')

0

to be the spectral sequence:

Composing this isomorphism of

- 0

(:l x &?)-graded additive categories with the functor of Proposi-

tion 6, we see that: For every teger

r

' O

(:l x 7l) -graded abelian category

we have an additive functor of

A

and every in-

(:l x7l)-graded addi-

tive categories, "associated spectral sequence", from the category

EC

(rO,-rO+l), (-1,+1)

Spec. Seq. (

r O'

Spec. Seq. (r

( _ +1) r, r (r -r+l)

0" (Er,dr"r)r>O

> r_rO

(A) ) (A).

into the category Notice also that the objects of

) (A) consist of all spectral sequences r>r - 0 starting with the integer rO in the bigraded

Exact Couple, Graded Case abelian category gers

A,

(r,-r+l),

deg(d r )

such that

233

r::..r o.

Example 3.1.

Let

13

be an (ordinary, ungraded) abelian cate-

A = S&'x&',

gory, and let

the

(&' x &') -graded abelian category of

s.

all bigraded objects in the abelian category 3 above applies. in

all inte-

A =SlX&,

Then Example

Notice that in this case the spectral sequences

are what we have called in section 4 the "bigraded

spectral sequences in the abelian category is any integer, Spec. Seq. (r

13",

and that if

then the objects of

(r -r+l) 0"

) (13 r> r

-

&,x&'

)

are what we have called the

0

"conventional bigraded spectral sequences in the abelian category

13

starting with the integer

r

O· "

We will call the exact

couples in the (&' x &') -graded abelian category

S&'x&' ,

the bi-

graded exact couples in the (ordinary, ungraded) abelian cate~

S.

A bigraded exact couple in the abelian category

will be called conventional if there exists an integer that the exact couple is an object in

EC

13

rO

such

(rO,-rO+l), (-1,+1)

(SlX&').

(If the exact couple is the exact couple (1), then it is equivalent to say that: where

(Y, 6),

"if

deg(t) = (-l,+l),

y + 6 = + 1" . )

and

deg(h)+deg(k)=

More specifically, a bigraded exact

couple in an (ordinary, ungraded) abelian category

13

will be

called conventional starting with the integer

rO if the exact lx&' couple is an object of EC ). Then, by (ro,-rO,+l) , (-1,+1) (13 Example 3 above, in the case in which the bigraded abelian category

A

is the category of all bigraded objects in the fixed

(ordinary, ungraded) abelian category

S'

A = Slx&',

we have

that: For every (ordinary, ungraded) abelian category

13,

and

Section 5

234

every integer

r ' O

we have the canonical functor of bigraded

additive categories, associated spectral sequence, from the

(B~x~) of all (rO,-rO+l), (-1,+1) conventional bigraded exact couples in the abelian category B

bigraded additive category

starting with the integer category

Spec. Seq.(

EC

r ' O

into the bigraded additive

( _ +1) ) (A) of all conventional r O' r, r r>r bigraded spectral sequences in the ~belian category B starting with the integer

rD.

Section 6 Filtered Objects

Definition 1.

A is a pair

object in in

A,

F p (A)

::::>

F p+l (A) ,

A,

If

all

(A,F*A)

all

p

(A,F*A)

into

(B,F*B)

(B,F*B)

(A,F*A)

FpA,

is an object

is a subobject of

A,

We sometimes say that the

~

filtered

are filtered objects in the

A, then a map of filtered objects from is a map

A,

the abelian category If

f:.~.

A

p E~ . and

abelian category

where

F A p

p<=;Y

is a filtered object" and call

piece of

A,

(A, (F A) E) p p ~

and where for every

such that "A

A be an abelian category. Then a filtered

Let

f:A->-B

such that

from

A

f(FpA)CFpB,

into all

B

in

pE;y.

is a filtered object in the abelian category

then define GA=FA/F+IA,

P

all integers object

A,

P

p.

P

GpA

is the

all integers

p.

~

graded piece of the filtered

The sequence:

the associated graded of the filtered object abelian category

A.

If

A

G*A=

(GpA)pE~

(A,F*A)

is

in the

is an abelian category, then the

class of all filtered objects in

A,

together with all maps of

filtered objects, forms an additive (but, except in trivial cases, not abelian) category, and the assignment: is a functor from the additive category of all filtered objects 235

236

Section 6

in A

A

into the abelian category

and maps of degree

Definition 2. gory

Let

A

A?

of

?-graded objects in

O. be a filtered object in the abelian cate-

A. i)

The filtered object

only if there is an integer ii)

only if iv)

A If

F (A) = O. P is said to be codiscrete if and

A a

The filtered object

is said to be discrete if and such that

b

The filtered object

only i f there is an integer iii)

A

such that A

Fa (A) = A.

is said to be finite if and

is both discrete and codiscrete. A

is a filtered object, then for each integer

p,

we have the natural mapping:

Therefore, the sequence: dexed by the integers.

(A/FpA)pE? is an inverse system inIf the object

A,

together with the

natural mappings:

A-+A/F A, is an inverse limit of Pc? , P this inverse system, then we say that the filtered object (A,F*A) v) A,

is complete. If

A

is a filtered object in the abelian category

then for each integer

p

we have the natural inclusion:

Therefore the sequence: system indexed by the integers. with the natural inclusions:

F

is a direct

If the object

-p

A

-+

A,

A,

all integers

together p,

is a

direct limit of this direct system, then we say that the filtered object vi)

(A,F*A)

is co-complete.

The filtered object

A

is Hausdorff if and only if

the infimum of the decreasing sequence of subobjects

FpA,

Filtered Objects p E 1',

of

vii)

is zero:

A,

n F A:o pE1' P

The filtered objects

237

o.

A

is cO-Hausdorff if and only

if the supremum of the increasing sequence of subobjects of F

-p

A,

pG1',

is

A,

A:

I F A:o A. pE;r -p Remark

1.

A be an abelian category and let

Let

A:o (A, (F pAl pE1')

A.

be a filtered object in the abelian category

Then we have the induced filtered object

AO,

in the dual category ject.

o

A

:0

-p

A) E ), P l'

which we call the dual filtered ob-

Then the dual of the dual of

A,

Oo A ,

is of course

The reader will verify that, the filtered object

A is discrete (resp.:

abelian category

(A, (A/F

A

A.

in the

co-discrete, finite,

complete, co-complete, Hausdorff, cO-Hausdorff) iff the dual O A

filtered object

in

AO

is co-discrete (resp.:

discrete,

finite, co-complete, complete, co-Hausdorff, Hausdorff).

Thus,

the concepts of "discrete" and "co-discrete" are dual to each other, and similarly for "complete" and "co-complete", and for "Hausdorff" and "co-Hausdorff"; while the property of being "finite" is self-dual. Remark

2.

Let

A be an abelian category.

Then the filtered

objects that we have defined are sometimes called filtered objects with decreasing filtration in the abelian category

A.

One can define a filtered object with increasing filtration in the abelian category A

isan object in

p E 1',

such that

tiona I convention:

A,

A to be: and

FPA

FPAC FP+lA,

a pair

P (A, (F A)PE1')'

is a subobject of all

all

p E 1'. p E 1',

A,

where for all

If we make the notathen every filtered

Section 6

238

A

object with increasing filtration in the abelian category

becomes a filtered object with decreasing filtration, and vice versa.

We define a map of filtered objects with increasing

filtration in the abelian category

A to be a map of the cor-

responding filtered objects with decreasing filtration in the abelian category

A;

and we call a filtered object with in-

creasing filtration discrete, co-discrete, finite, complete, cocOHlplete, Hausdorff, or, respectively, co-Hausdorff, iff the corresponding filtered object with decreasing filtration is discrete, co-discrete, finite, complete, co-complete, Hausdorff, or, respectively, co-Hausdorff. Remark 3.

If the abelian category

A is such that denumerable

direct sums of objects exist, and is also such that denumerable direct limit is an exact functor, then it is easy to see that a filtered object

(A,F*A)

in the abelian category

co-Hausdorff iff it is co-complete.

A is

(E.g., this is the case if

A is the category of abelian groups). Remark 4.

A is

In the special case that the abelian category

the category of abelian groups, we can put a natural topology on

A

such that

Namely, we give

A A

becomes a topological abelian group. the topology, such that a complete system

of neighborhoods of zero is

{F pA: p E 'l! } •

see that, the filtered abelian group spectively:

Hausdorff; respectively:

Then, it is easy to

(A,F*A)

is complete (re-

discrete) in the sense

of Definition 2 above iff the topological abelian group complete as uniform space (respectively: cal space; respectively: Definition 3.

Let

A

is

Hausdorff as topologi-

discrete as topological space).

(A,F*A)

be a filtered object in the abelian

Filtered Objects category f:B -+ A

A.

B

If

integers

tration on the object

B

of filtered objects"). of the filtration of A,

then there is induced a F B = f- l (F A), p p

such that the map

P

=

(F A)

P

B

As a special case, if

n B,

all

duced filtration from

p

all integers

I

B,

f

is a subA,

which we

such that

is a filtered object

(A,F*A)

A, then for every integer

is a subobject of

F

is a map

p (: J'.

in the abelian category FpA

B

inherits a filtration from

As further special case, if

that

all

We will call this the pre-image under A.

then

f:B-+A

call the induced filtration on the subobject F B

and i f

(This can be described as, "the coarsest fil-

p E J' .

object of

A,

namely the one such that

B,

A,

is any object in the category

is any map in the category

filtration on

239

A.

A,

p

we have

and therefore we have the in-

Explicitly, the p' 'th filtered piece

if

p' ':'p,

if

p' ':::'p,

of

(F A)

p

p'.

We have also another similar construction. Namely, if is a filtered object in the abelian category B

is an object in

A

and if

is induced a filtration on tegers B

p.

f:A+B

B,

is a map in

such that

A,

if

A, then there

FpB=f(FpA),

all in-

(This can be described as, "the finest filtration on

such that the map

f:A-+B

We call this the image under special case, if

B

is a map of filtered objects"). f

of -the -

filtration of

is a quotient-object of

herits a filtration from

A,

tion on the guotient-object

A,

then

A. B

As a in-

which we call the induced filtraB.

Section 6

240

As a further special case, if

A,

ject in the abelian category have the quotient-object

(A,F*A)

then for every integer

A/FpA

of the object

fore we have the induced filtration from p' 'th filtered piece

all integers

p'.

Remark:

(A,F*A)

Let

A,

gory

and let

of

Fpl (A/FpAl if

p' ':'p,

if

p'

is a filtered ob-

A.

A,

p

we

and there-

Explicitly, the

A/F p A

is

~p,

be a filtered object in the abelian cate-

{(B,R)}

be a subquotient of the object

A.

Then there are two obvious ways to induce a filtration on the object

B.

First, if we write

B=A'/A",

are uniquely determined subobjects of A'

has the induced filtration from

subobject of tion from

A,

A'

and then

A A

B=A'/A"

A/A"

A/A".

two filtrations on

then

by virtue of being a

A'.

A;

A/A"

and therefore

But A

B = A I /A"

by virtue of being a sub-

B

coincide.

Therefore, whenever {(B,R)}

subquotient of the filtered object A,

A,

then

B

inherits

which we call the induced filtra-

(Explicitly, if one fixes an exact imbedding from some

full exact abelian subcategory

A'

of

A that is a set into

the category of abelian groups, such that all

pE&"

by

We leave it as an exercise to prove that these

a natural filtration from

FpA,

A" c A',

A"

has the induced filtration from

has the induced filtration from

tion.

and

has the induced filtra-

virtue of being a quotient-object of

is a

with

A'

by virtue of being a quotient-object of

on the other hand,

object of

where

B, and

A'

and

A",

A'

where

contains

A,

{(B,R)}=A'/A",

Filtered Objects then

F B= {a' +A":a' EA', p

Lemma 1.

a' EF A}, p

filtered object in the abelian category of

A.

Regard

duced filtration from integer (B,F*B)

all integers

A be an abelian category.

Let

subobject

241

B

A,

(A,F*A)

and let

be a

B

be a

as a filtered object with the in-

(A,F*A).

Suppose that there exists an

AcB. Then the filtered object Po is complete iff the filtered object (A,F*A) is com-

Po

such that

Let

pE;:r).

F

plete. Proof:

Case 1.

A is closed

Suppose that the abelian category

under denumerable direct products of objects.

Then, notice that,

since denumerable direct products of objects exist in the abelian category

A,

we have that inverse limits over denumer-

able directed sets exist in we have that F pB = (F pAl

n B = F pA.

A.

For each integer

p

such that

F Ac F AC B, and therefore that P Po Therefore, we have the short exact se-

quence:

Passing to the inverse limit for

P'::' Po

(as

p .... + 00),

we have

the exact sequence:

.... -A

(1)

B

Therefore, we have the commutative diagram with exact rows: (2)

0---':1 o

--.;>

B

0 :1 :r--:> A

ljm [F 13]--:> lim [F A]--:> AlB P'::'PO P P'::'PO P

242

Section 6

Diagram chasing in this commutative diagram, we see that the map

P

is an epimorphism.

Therefore the sequence (1) is a

short exact sequence.

phism, and

(B,F*B)

B

is complete iff

By definition (A,F*A)

is complete iff

a

is an isomor-

is an isomorphism.

from the commutative diagram (2) and the Five Lemma, B

isomorphism iff

a

But

is an

is an isomorphism, completing the proof of

Case 1. Case 2.

Then notice that, for every fixed object tor

A is arbitrary.

Suppose that the abelian category

G = Hom (C, ) A C

C

A,

in

the func-

is a left-exact functor, and is also a func-

tor that preserves arbitrary inverse limits indexed by directed sets, whenever they exist.

(In fact, this functor even preserves

arbitrary inverse limits indexed by set-theoretically legitimate categories, whenever they exist.) (GC(A),GC(Fp(A))pE~)

we have that

Since

G C

is left exact,

is a filtered object in the

category of abelian groups; denote this filtered abelian group by

G (A).

c

And

G (B) C

that the functors (1)

If

0

G

c

is a

subgroup of

GC(A).

have the following property:

isadirectedset,if

(X.,a .. ). 1.

the object

X

ai:X->-X

i

A

X

is an

are maps in the category

A,

then

together with the maps

iff for every object

together with verse system groups.

G (ai)' C

is an inverse if

verse limit of the inverse system gory

'ED

1.J 1., J

system of objects and maps in the category object, and if

Notice also,

i E 0,

C

A,

ai'

(X. ,a .. ). 1.

in

i ED,

are an in-

'ED

in the cate-

1.J 1.,J

A,

we have that

GC(X)

are an inverse limit of the in-

(GC(Xi),GC(aij))i,jED

in the category of abelian

(Of course, this property generalizes, beyond existence

Filtered Objects

243

of inverse limits of inverse systems indexed by directed sets, to existence of inverse limits of functors defined on categories that are sets).

From property (1) above, it follows that, the

filtered object

A

(resp.:

B)

complete, iff for every object group

(resp.:

GC(A)

abelian groups.

GC(B»

in the abelian category C

in

A,

A

is

the filtered abelian

is complete in the category of

Therefore, to prove the theorem in general, it

suffices to prove it in the case that the abelian category is the category of abelian groups.

A

And Case 1 covers that

case.

Q.E.D.

Remark:

Let

A

be the category of abelian groups and let

be a filtered object in

A.

If

B

A

is any subobject (i.e., sub-

group) of

A,

then in Lemma 1 above we have considered the con-

dition on

B,

that "there exists an integer

B::J F A". p

In terms of the topology on

A,

p

such that

introduced in Remark

4 following Definition 2, this condition is equivalent to, is open in

"B

A."

Definition 4.

Let

A

be an abelian category, and let

a filtered object in the abelian category exists in the abelian category the completion of

A,

A,

A.

If

A

be

lim A/F A

p",++oo

p

then this object is called

and is denoted

1\

A.

If

lim F p->-+oo

A -p

exists, then this object is called the co-completion of

A,

and

is denoted co-comp(A). Of course, from Definition 4 above, it follows immediately that, if denumerable direct products (resp.: ect sums) exist in the abelian category filtered object A

(resp.:

A

in

A,

A,

denumerable dirthen for every

we have that the completion

the co-completion co-comp(A)

of

A)

exists.

1\

A

of

Section 6

244

Also, from Definition 2, course follows that, if

A

(iv), and Definition 4, it of

is an abelian category, and if

is a filtered object in the abelian category complete iff both (i) A

A -+ A

Clearly, if

category

A

A

is

A.

is a filtered object in the abelian

then a completion of

A,

then

exists, and (ii) the natural mapping:

is an isomorphism in the abelian category

Remark 1.

A

AA

A,

A

in the abelian category O

exists iff a co-completion of the dual filtered object

in the dual abelian category incide.

A

O

A

A

exists, in which case they co-

Passing to the dual categories, it follows that a co-

completion of

A

in

A

O

exists iff a completion of

A

in

O

A

exists, in which case they are the same. Corollary 1.1.

Let

A

be an abelian category such that de-

numerable direct products of objects exist. tered object in B

A,

and let

B

Let

be a subobject of

A

be a fi1A.

Regard

as being a filtered object with the filtration induced from

A.

Suppose that there exists an integer

p

such that

F AC B. P

Then

and

1)

The natural map:

BI\ -+ AI\

2)

The natural map:

A-+AA

is a monomorphism, induces an isomorphism:

AI\ IBI\ "" AlB. Proof:

By the proof of Case 1 of Lemma 1, we have the short

exact sequence 1)

0

-+

BI\ -+ AI\

-+ AlB -+

0,

(equation (1) in the proof of Case 1 of Lemma 1). lent to the conclusions of the Corollary. Remark 2.

A slight refinement of Corollary 1.1 is

This is equiva-

Filtered Objects Corollary 1.1.1.

245

A be an abelian category such that de-

Let

numerable direct products of objects exist. tered object in

A,

let the subobject A,

and let B

of

A,

B

Let

A

be any subobject of

and the quotient object

be a filA.

Then

A/B

of

both be regarded as filtered objects with the filtrations

induced from

A.

Then the induced map:

B/\

-+-

A/\

is a monomor-

phism, and there is induced a natural monomorphism:

in the category

A.

If the abelian category

A

is such that

denumerable direct product is an exact functor, then the natural monomorphism (l) is an isomorphism. Since

Proof:

F B = (F A)

P

p

n B,

we have the short exact sequence

0'" (B/F B) ... (A/F A) ... {A/ (B + F A)) ... 0, p p p But

F (A/B) = (B + F A) /B.

P

Therefore

P

all integers

p.

A/ (B + F A) "" (A/B) / (F (A/B)).

P

P

Therefore, the above short exact sequence can be rewritten as the short exact sequence: 2)

0'" (B/F B) ... (A/F A) ... (A/B) / (F (A/B)) ... 0, p p p

all integers

p.

Passing to the inverse limit as

p ... + 00,

we

obtain the exact sequence 3)

/\

O"'B"'A

/\p

/\

-+(A/B) ,

which proves the first part of the Corollary.

If the functor,

denumerable direct product, is exact in the abelian category A,

then throwing the sequences (2) through the exact connected

sequence of functors (in fact, sequence of derived functors)

246

Section 6 l [lim (B/F B) 1 = 0,

and noting that

p:-+oo

B

constant inv. system we deduce that the map

by an epimorphism,

maps into (B/F B) E p

p

since the

p p 'Z'

in the sequence (3) is an epimorphism.

This proves the Corollary. corollary 1.2.

Let

A

be an abelian category such that denum-

erable direct products of objects exist.

A.

object in the abelian category The natural mapping:

1)

for all integers 2)

Let

A

be a filtered

Then

A

is an epimorphism,

A ->- (A/FpA)

p.

The kernel of the mapping in (1) is

pletion of

where the subobject

F A

P

of

A

is regarded

as a filtered object with the filtration induced from 3)

For all integers

p,

the natural mapping:

A. A->-AA

induces an isomorphism,

Proof: 1.1.

Let

Then

B

obeys the hypotheses of Corollary

The conclusions of Corollary 1.1 then imply the conclu-

sions of Corollary 1.2. Remark 4.

Let

A be any abelian category such that denumerable

direct products of objects exist, and such that the direct limit over the directed set of non-negative integers is an exact functor.

Then a filtered object in the abelian category

complete iff it is cO-Hausdorff. object in

A.

Then

A

(Proof:

is co-Hausdorff iff

Let

A

A

is co-

be a filtered

l F A = A, pEiI' P

and

A

is co-complete iff A,

lim F A = A. Since, in our abelian category pciI' -p we have that denumerable direct limit is an exact functor,

it follows that for any increasing sequence of sub-objects, the

Filtered Objects

247

direct limit is the supremum as subobjects.

I F A = lim F A, pEil -p pE;p: -p F -p A

of

A,

In particular,

for the increasing sequence of subobjects

p Eil).

It follows readily that, if

a filtered object in such an abelian category completion of the filtered object

I F A pEil -p

subobject

of

A

A

A,

(A,F* (A)

is

then the co-

exists, and is simply the

with the filtration induced from

A.

Of course, the category of abelian groups obeys the hypotheses of this Remark. Lemma 2. A.

Let

teger A/\

Let B

A

be a filtered object in the abelian category

be a subobject of such that

A,

such that there exists an in-

A) C B. Suppose that the completion Po exists in the abelian category A. Regard B as

PO

of

Let us now return to the general situation.

A

(F

being filtered object with the induced filtration from Then the completion A,

where

Note:

B

B/\

of

B

A.

exists in the abelian category

is given its induced filtration from

A.

Actually, under the hypoth2ses of Lemma 2, a bit more can

be said.

Namely, one

exists in

A

CiiTI

show that the completion

iff the completion

B/\

of

B

A/\

of

A

exists in

A.

We

will not make any use of this stronger observation, and we do not include a proof of this stronger observation. Proof: l~m

We know that

B/FpB

lim A/F A pEil p

exists; we must show that

exists.

pEil For

p

~

PO'

we have

short exact sequence: (1)

F B = FA. P p

Therefore we have the

248 Let

Section 6

8: [l)m

A

A B

+-

F A

F':'PO

P

be the map induced by the Then for every

Pl':'PO'

and let

8 , P

K = Ker 8.

we have the commutative digram with

exact rO'ils: --> A

B

[I

8

Pl

--.....:::..>

from which we deduce a mapping

A B

K+ (B/F

B),

all integers

Pl I claim that (2)

The object

K,

together with the maps

for

are an inverse limit of the inverse system in the category

(B/F

Pl ':'P O' Pl

B)

>P

Pl- 0

A.

In fact, as in the proof of Case 2 of Lemma 1, to prove this, it is necessary and sufficient to show that, for every object in

A,

that the abelian group

Hom (C,K) A

C

together with the

is an inverse for all ), Pl limit of the inverse system of abelian groups:

homomorphisms

HomA(C,a

Since the functor

Hom (C,) A

preserves

inverse limits over directed sets whenever they exist, and preserves kernels of maps, we are therefore reduced to proving assertion (2) in the case that groups.

A

is the category of abelian

But then, since inverse limit is a left exact functor,

passing to the inverse limit for tain the exact sequence

P':'PO

in equation (1), we ob-

Filtered Objects

249

(3)

B

Exactness of the sequence (3) implies that

K = lim F B'

therep Therefore assertion (2) is true for p~'po

by proving assertion (2).

the category of abelian groups, and therefore, for an arbitrary

A.

abelian category in

And in particular,

lim (B/FpB) p~'po

A.

Corollary 2.1.

Q.E.D.

Under the hypotheses of Lemma 2, we have that

(1)

The natural map:

B/\->-A/\

(2)

The natural map:

A->-A/\

Proof:

exists

is a monomorphism, and induces an isomorphism:

In the course of proving Lemma 2, we have established

an exact sequence:

The composi te:

A ->- A/\

~

A/B

therefore an epimorphism.

is the natural mapping, and is Therefore

is an epimorphism.

p

Therefore we have the short exact sequence (1)

/\

/\

o ->- B ->- A +(A/B) ->- O.

This short exact sequence is equivalent to the conclusions of the corollary. Corollary 2.2.

Let

a filtered object in

A A

be an abelian category, and let such that the completion

A/\

A of

be A

exists. Then (1)

The natural mapping:

for all integers

p.

/\

A ->- (A/F A) p

is an epimorphism,

Section 6

250

regarded

F A,

the subobject

p,

For every integer

(2)

P

as filtered object with the induced filtration from such that the completion, tion, the completion

(FpA)A

(F A)A

of

P

P

is

In addi-

exists.

of F A

A,

is naturally isomorphic

to the kernel of the mapping (1). For all integers

(3)

p,

the natural mapping:

A -+ AA

in-

duces an isomorphism

AI FA"':. AAI p

Proof: 2.1.

Let

B = F A. p

(F A) A .

P

Then

B

obeys the hypotheses of Corollary

The conclusions of Corollary 2.1 then

imply the conclu-

sions of Corollary 2.2.

Q.E.D.

Notice that Corollaries 2.1 and 2.2 above are direct generalizations of Corollaries 1.1 and 1.2 respectively. Lemma 3.

A.

Let

A

be a filtered object in the abelian category

If the filtered object

Corollary 3.1. category

A.

Let

A

A

is complete then

A

be a filtered object in the abelian

If the completion

AA

of

A

exists, then the in-

fimum of the decreasing sequence of subobjects of p E ~,

exists.

n pEJ"

is Hausdorff.

A:

FpA,

And then

(F A) = Ker(A -+AA). p

Proofs of Lemma 3 and of Corollary 3.1:

If

C

is a subobject

A AA=lim(A/F A), the composite: C-+A+A pE~ p is zero iff the composites: C-+A+(A/FA) are zero, all inte-p gers p, iff C C F A, all integers p. It follows that

of

A,

then, since

p

is an infimum of the subobjects

F A

P

of

A, P E Z.

Filtered Objects

251

n F A exists, and Ker (A -+ AI\) = n F A, proving pE~ p pE~ P Corollary 3.1. By Definition 2, part (iv), and Definition 4, Therefore

we have that

A

is complete iff

is an isomorphism.

Therefore, if

AI\ A

exists and the map

is complete, by Corollary

n F A =Ker(A -+AI\) = 0, pEZ' p

3.1 we have that

A -+ AI\

and

A

is Hausdorff. Q.E.D.

Corollary 3.2.

Let

Then

A

be an abelian category, and let

A such that the completion

filtered object in ists.

A

AI\

of

is Hausdorff iff the natural mapping:

A

be a

A

ex-

A -+ AI\

is a monomorphism. Proof:

By Corollary 3.1,

n

Hausdorff iff

F A=

pE;r. p

°

n F A = Ker (A -+ AI\). There fore pEZ' p iff A -+ AI\ is a monomorphism.

is

A

Q.E.D. Definition 5.

Let

A be an abelian category, and let

filtered object in the abelian category AI\

completion

of

A

A.

of

A

duced filtration from

/\

A,

completion,

(FpA)I\, A/\,

if we regard the

(F A)I\ of F A P P We define a filtration on

then the completion AI\ .

by defining the p'th filtered piece

the subobject

p,

Then,

as being a filtered object with the in-

exists, and is a subobject of A ,

be a

Suppose that the

as defined in Definition 4 exists.

by Corollary 2.2 above, for every integer subobject

A

all integers

p.

of any filtered object

ists, in the arbitrary abelian category

Fp(A/\)

of

AI\

to be

We thereby regard the A,

A,

whenever

AI\

ex-

as being a filtered

object. Thus, if category

A,

A

is an arbitrary filtered object in any abelian

and is such that the completion

AI\

of

A

exists,

Section 6

252 then

AA,

by Definition 5 above, is itself naturally regarded

as being a filtered object, and explicitly the p'th filtered piece

is

of

p.

for all integers Proposition 4.

A.

gory

Let

A

be a filtered object in an abelian cate-

Suppose that the completion

(1)

AA

(2)

We have a natural mapping

AA

of

A

exists.

Then

is complete as filtered object. l:A ... AA

of filtered

objects.

(3)

A (A , l)

The pair

is universal with these properties.

Proof:

By Corollary 2.2, we have that

A/FpA.

Therefore

Definition 1,

P

P

lim AA/F (AA) '" lim A/F A=AA, so that by PE~ p pE~ P (iv), the filtered object AA is complete. (F A)A, F A into P P and is therefore a map of filtered objects.

l:A .... AA

The natural map all integers

AA/F (AA) =AA/(F A)A",

p,

clearly maps

It remains to prove universality. In fact, gory

A

let

B

be a filtered object in the abelian cate-

that is complete, and let

objects.

we deduce a map,

quotients.

Since

B

such that

B00

0

B : A/F A .... B/F B p p p

is a map

B :AA .... B. 00

l = B.

For every integer

by pass ing to the

is complete, we have that

Therefore the inverse limit of the maps

p,

be a map of filtered

We must show that there exists a unique map of fil-

tered objects B00 :AA .... B p,

B:A .... B

Bp'

B = lim B/F B. pE~ p for all integers

From the commutative diagrams:

253

Filtered Objects B A

~B

1

A/F A P for all integers

p,

Bp

1

) B/F B, P

if we pass to the inverse limit for

p E 'J' ,

we deduce the commutative diagram: A

B

>B ------------~

lL

lid B

Boo

) A ------

which proves existence of Since

B

Boo.

It remains to prove uniqueness.

is complete, we have that

fore,to give a map in the category to giving maps:

A .... B/F B

whenever

the digram:

q.:::. p,

p

B,

A,

B = lim B/F pB.

ph

8:A ->- B,

for all integers

p,

There-

is equivalent such that,

A~l p

is

commutative:

The map

will be a map of filtered integers to

0,

8: A ->- B

in the abelian category

objects iff

p; equivalently, iff the maps: all integers

p.

whenever

A) cF

p

A ->- B/F pB

for all

(B),

F A inP 8:A->-B of

map

A is equivalent to such that,

8 p.•

q.:::. p,

p

Therefore, to give a map

filtered objects in the abelian category giving maps

8(F

A

the diagram:

254

Section 6 8

AT

0

A/FpA

is

commutative.

replacing

'BT

q

p

;>B/F B P

Applying these same considerations, with

A, we see similarly that, to give a map

filtered objects in the abelian category giving maps

P : A/\ /F (A/\) -+ B/F B p p p

that, whenever

q

~

P9,

1

is equivalent to

for all integers

p,

of filtered objects

f

) B/F pB

~

8

/\

A

A/F A'::;. A /F (A ). p

from

A

into

B

A-+AA

Therefore, the maps

p

are in one-to-one

correspondence with the maps of filtered objects the correspondence being given by

8=

p

from

pOl.

proves the uniqueness part of universality. Corollary 4.1.

such

But, by Corollary 2.2, the natural map:

induces an isomorphism:

B,

of

»B/F B

Pp

A/\/F (A/\) P

into

-+ B

p, the diagram:

A/\/F (A/\) q

is commutative.

A,

p :A/\

AA

A/\

This Q.E.D.

The hypotheses being as in Proposition 4, we

have that the map induced by the natural map,

l:A -+A/\,

on the

p'th graded pieces is an isomorphism

for all integers Proof:

p.

For every integer

with exact rows:

p,

we have the commutative diagram

Filtered Objects

By Corollary 2.2, the maps fore, by the Five Lemma, Theorem 5.

Let

A

G (l)

p

yare isomorphisms.

is an isomorphism.

be an abelian category, let

A

A,

fil tered obj ects in pose that

Sand

255

and let

f: A .... B

Q.E.D.

A

and

B

A.

be a map in

is complete and co-Hausdorff, and that

complete and Hausdorff.

There-

B

be Sup-

is co-

Then the following two conditions are

equivalent. (1)

f

is an isomorphism of filtered objects.

is an isomorphism from restriction of

A from

F A p

(2)

p

(f):

Proof:

p

(A) ....

in the category

A,

for all integers

G

p

p, (B)

d~l,

and the

p) .

is an isomorphism of graded objects.

Clearly,

the integer

B

f

is an isomorphism in the category

F B, p

onto

G* (f)

G

onto

to

f

for each integer G

A

(I.e.,

the map in the category

(I.e.

I

A,

is an isomorphism.)

(1)==> (2).

Assume (2).

Then by induction on

I claim that, for every integer

that the map induced by

p,

we have

f:

(1)

is an isomorphism. In fact, for

d = 1,

equation (1) says that

isomorphism, which is condition (2). tion is established for the integer

If d,

d"::'l,

G (f)

p

is an

and the asser-

then we have the commu-

Section 6

256

tative diagram with exact-r0WS:

for each integer isomorphism.

p.

By the inductive assumption,

By the case

fore by the Five Lemma,

d = 1,

B

a

tration from (2)

A.

F A=

P

FpA

(3)

and each

A

is complete,

is complete for the induced fil-

Therefore

lim (FpA/FplA). p'~p

Considering equation (1), it follows that

l~m

(F B/F

I

B)

p' >p p P But then by Definition 4 applied to the filtered ob-

FpB,

object

p,

d > 1.

by Lemma 1 we have that

ject

There-

is an isomorphism, completing the

Since by hypothesis the filtered object

exists.

is an

is an isomorphism.

inductive proof of equation (1), for all integers integer

y

we have that the completion

F B P

(F B)A p

of the filtered

exists, and that

(F B)A P

=

lim

pl+-~p

(F B/F ,B). p P

Passing to the inverse limit of the isomorphisms (1) in the category

A, we obtain an isomorphism

(4)

in the category

A,

commutative diagram:

all integers

p.

But then we have the

Filtered Objects

257

(F A) 1\ _ ... ____ 2'_____ .____ ._> (F B) 1\

(5)

~l

/1'

F A

) F B

P

in the category phism.

A.

P

Since

F A

is complete,

p

is an isomor-

Therefore, from the commutative diagram (5), it follows

that the natural map:

F p B'" (F p B)

1\

is an epimorphism.

That is,

we have that the natural map: (6)

F B ... [lim (F B/F ,B»)

p' ~p

p

is an epimorphism.

p

p

Since inverse limit preserves kernels, the

n F ,B. p' >p p thesis Hausdorff, we have-that

kernel of this map is

the map (6) is an isomorphism. the commutative diagram (5).

FA'" F B

P

p

tegers

n F ,B

p'~p p

and

B

is by hypo-

is zero.

Therefore j

in

Therefore, from the diagram (5), f

is an isomorphism in the category

A,

all in-

p.

Next, consider the filtered objects A

B

But this map is the map

we have that the map induced by (7)

But, since

in the dual category

filtered piece of

AO,

resp.:

AO. BO,

AO

and

BO

Explicitly, the

p'th

is the quotient object

respectively of resp. : B/F -p B, of A, A, -p category A. Then the map of filtered objects in

B,

A/F

obeys the hypotheses of this Theorem.

dual to

in the

AO:

Therefore we have equa-

tion (7), which tells us that the map induced by

f

from the

258

Section 6 O

p'th filtered piece of

B

A,

A

or, translated back to

is an isomorphism in the category the category

O

into the p'th filtered piece of

that the map induced by

f

in the category

A: A/F

(8 )

-p

A -+ B/F

all integers

-p

B

is an isomorphism in the category

A,

for

p.

For any fixed integer exact rows:

p,

we have a commutative diagram with

y

o-_....

O---+)F B

p

By equation (7), A.

is an isomorphism in the abelian category

By equation (8) applied to the integer

A.

morphism in

Therefore by the Five Lemma,

morphism in the category all integers Remark 1.

A,

A

then define (A) = A/ (

-00

tion

AA

LF

pE;}' t:l

of

A

(resp.:

iff

G+ (A) = A)

Q.E.D.

n F A

pE;}' P

whenever it exists, and

whenever it exists.

(E. g,

exists, then by Corollary 3.1

exists).

Thus,

co-Hausdorff)

G _00 (A)

is an iso-

This fact, and equation (7) for

Dually, if the co-completion co-comp(A) of G_oo(A)

f

is an iso-

is a filtered object in the abelian category

00

G

A.

E

implies condition (1).

p,

If

-p,

the filtered object iff

G+oo(A)

exists and is zero).

tered objects, and if the induced map in

A,

G+oo(A)

and

A A

if the compleG+oo(A)

exists.

exists, then is Hausdorff

exists and is zero (resp.: If

f:A -+ B

G+oo(B)

is a map of fil-

exist, then we have

G+oo(f): G+oo(A) -+G+oo(B),

by restriction.

Filtered Objects Similarly, i f

Remark 2.

and

G_00 (A)

G_00 (B)

259

exist, then we have

The proof of Theorem 5, also provffithe following

somewhat stronger result: Corollary 5.1. B

A

Let

be an abelian category, and let

A,

be filtered objects in

G_oo(A),

G_oo(B)

exist.

such that

G+oo(A),

and that the filtered object: tration induced from

and

G+oo(B),

Suppose also that the filtered object

(with the filtration induced from

A/G+ oo (A)

A

B)

Ker(B

+

A)

is complete,

G_00 (B))

is co-complete.

If

(with the filf:A

+

B

is any

map of filtered objects, then the following two conditions are equivalent: (1)

f

is an isomorphism of filtered objects,

(2)

Gp(f)

is an isomorphism in the category

(The inequality on for

p = +00

and

p

meaning, "for all integers

A,

p,

and also

-00" ) .

The proof is very similar to that of Theorem 5. Remark 3.

Suppose that we have the hypotheses of Theorem 5,

and that condition (2) of Theorem 5 holds.

Then by Theorem 5,

we also have that condition (1) of Theorem 5 holds. it follows, in this case, that

A,

and also

B,

Therefore

are both com-

plete and co-complete. Remark 4.

Similarly, if the hypotheses of Corollary 5.1 hold,

and if condition (2) of Corollary 5.1 holds, then it follows that, if

K = Ker (A

and co-complete. Remark 5.

+

G A), -00

And that

is both complete B

also has this property.

Corollary 5.1, as has been indicated, can be proved

Section 6

260

directly.

However, it can also be deduced as a corollary of

Theorem 5. Sketch of Proof:

Let

K = Ker (A .... B_ooA),

K=Ker(B .... G_ooB),

B' =B/B/G+ooB.

A' = K/G+ oo (A) ,

Then if condition (2) of Corol-

lary 5.1 holds, one proves readily that condition (2) of Theorem 5 holds for the map of filtered objects induced by f,A' -7B'.

But

A'

is comPlete and co-Hausdorff, and

co-complete and Hausdorff. A' +B'

induced by

f

B'

is

Therefore by Theorem 5 the map:

is an isomorphism of filtered objects.

One then deduces readily, from two applications of the Five Lemma, since the map

G+oo(f)

f:A + B

and

G_oo(f)

are also isomorphisms, that

is an isomorphism of filtered objects.

Let us now state the dual of Corollary 2.2. Corollary 2.2°. category ists.

A.

Let

A

be a filtered object in an abelian

Suppose that the co-completion, co-comp(A),

Then for every integer of

p,

if we regard the quotient-

as being a filtered object with the fil-

A

tration induced from

A,

then exists,

(1)

co-comp(A/FpA)

(2 )

the natural map:

co-comp (A) -7 co-comp (A/F A) P is an epimorphism, and

(3)

the kernel of the epimorphism in (2) is naturally isomorphic to

Proof:

ex-

Corollary 2.2°

FpA,

all integers

p.

immediately follows from Corollary 2.2

by passing to the dual category. Before stating the next theorem, we note that, dual to Definition 5, if gory

A,

A

is a filtered object in the abelian cate-

such that co-comp(A)

exists, then we define a filtra-

Filtered Objects tion on co-comp(A)

by defining

of the natural mapping integers

F (co-camp A)

to be the kernel

p

¢p: co-comp(A) +co-comp(A/FpA),

(By Corollary 2.2°,

p.

261

all

conclusion (1), co-comp(A/F A) p

exists; and by Corollary 2,2°, conclusion (2), the mapping is an epimorphism, all integers object in the abelian category and if

AO ,

O

A

p.)

Then, if

A

¢p

is a filtered

A such that co-comp(A)

exists,

is the dual filtered object in the dual category

then the dual filtered object: O

is the completion of

A

co-comp(A)o

in the dual category

of co-comp(A)

AO .

So that,

if we prove any theorem about completions (or, respectively, co-completions) regarded as filtered objects, then we automatically have established the truth of the dual theorem, a theorem about co-completions (or, respectively, completions). Remark:

Notice that, by the definition above of the filtration

on co-comp(A)

(that

Fp (co-comp A) = Ker (co-comp (A) +

co-comp(A/FpA»,

and by conclusion (3) of Corollary 2.2°, it

follows that, if

A

gory

A

is any filtered object in an abelian cate-

such that co-comp(A)

co-comp A + A

exists, then the natural map:

induces isomorphisms,

F (co-comp A) ':; F (A), p p Theorem 6.

Let

A

filtered object in and (co-comp A)A category

A

all integers

p.

be an abelian category, and let A,

such that

AA,

A

co-comp (A ),

A

be a

co-comp(A)

exist (e.g., it suffices that the abelian

be closed under denumerable direct products and

denumerable direct sums of objects).

Then there is induced a

canonical isomorphism of filtered objects

262

Section 6

Proof:

By Definition 5, we have that F (AA) =F (A)A, p p

all integers

p.

By Corollary 2.2, we have that the natural mapping:

A-+AA

in-

duces isomorphismS:

The dual of this last observation (see the

Remark just pre-

ceding this Theorem) is that, the natural mapping:

co-comp(Al-+A

induces isomorphisms: F (co-comp A) p

'! F p A,

all integers

p.

Therefore Fp(CO-COmp(AA»

(1)

(2)

and

Fp((co-comp A)A) = (Fp(co-comp A)lA", (Fp(A»A, gers

p.

tegers ( 3)

"'Fp(A/\) =(Fp(A»/\,

all inte-

Combining (1) and (2), we see that, for all inp,

we have the natural isomorphisms

F (co-comp (AA) ) '" F ((co-comp A)/\) "" (F (A»A. p p p By Proposition 4, conclusion(l), we have that the filtered

object

AA

is complete.

is co-complete.

The dual of this is that co-comp(A) A

Therefore, co-comp(A )

is co-complete.

is, the natural mapping is an isomorphism: co-comp (A) ~ l,;j,m F (co-comp (Al ) . p -p

(4)

The composite of the isomorphism (3) with the inclusion: F

A

-p

(co-comp (A » "" F

-p

((co-comp A)

/\

) c..... (co-comp A)

A

That

263

Filtered Objects is a map in the category p ++

A.

Passing to the direct limit as

and using equation (4), we obtain a natural map in the

00

category

A:

(5)

II B:co-comp(A )

By equation (3),

B

(co-comp A)

-+

1\

.

is a map of filtered objects.

The dual of equation (3), is that we have natural isomorphisms: 1\

1\

(co-comp(A ))/Fp(co-comp(A )) ~ II II (co-comp A) /F ((co-comp A) ) "" co-comp (A/F (A)),

(6 )

p

all integers Fix an integer B

PO

p

p.

(e. g., one can take

Po = 0).

Then the map

induces the commutative diagram with exact rows:

1

1

II II co-com (AI\) (co-comp(A ))---?co-comp(A ).....;.-p II ~ Po F ((co-comp (A )) a B Po II /\ , 1 \ ) , AJ (co-comp A) 0-;> F (( co-comp A) --»\ co-comp ~ 1\ ...... Q Po F ((co-comp A) ) Po O-F

yl

By equations (2) and (6) respectively, we have that the maps and

y

a

in the above diagram are isomorphisms. By the Five Lemma,

it follows that the map

B

is an isomorphism in the category

A. Finally, equation (2) the map

B

implies that, for all integers

II maps the p'th filtered piece of co-comp(A)

phically onto the p'th filtered piece of (co-comp A) that the map corollary 6.1.

B

1\

-

is an isomorphism of filtered objects.

p, isomor-

Le., Q.E.D.

The hypotheses being as in Theorem 6, for every

264

Section 6

integer

p

F (co-comp (AI\) ) "" F ((co-comp A)I\) "" F (AI\) "" (Fp (A) P P P

(1)

and

we have canonical isomorphisms )1\,

(co-comp(AI\))/F (co-comp(AI\)) ""

(2)

p

1\

(co-comp A) /Fp((co-comp A)

1\

) ""

co-comp (A ) /F p (co-comp A) "" co-comp (A/F p (A)). Proof:

Conclusion (1) of the Corollary follows from equations

(1) and (2) in the proof of Theorem 6.

Conclusion (2) of the

Corollary is the dual of conclusion (1).

(And also follows,

alternatively, from equation (6) in the proof of the Theorem and from conclusion (3) of Corollary 2.2 applied to the filtered object Remarks:

1.

co-comp(A)). From Theorem 6, it follows that if

tered object in an abelian category comp(AI\),

(co-comp A)

A,

and (co-comp A)I\

A

is a fil1\

A ,

such that

co-

all exist (this condi-

tion is automatic if, e.g., the abelian category

A

is closed

under denumerable direct products of objects and under denumerable direct sums of objects), then co-comp(AI\) and (co-comp A)I\ are both complete and co-complete (since by Theorem 6 they are isomorphic as filtered objects, and by Proposition 4, conclusion (1), the latter is complete, and by the dual of Proposition 4, conclusion (1), the former is co-complete.) 2.

Of course, from Corollary 4.1, and the dual of

Corollary 4.1, it follows that, if the abelian category

A

A

such that

AI\

is a filtered object in and

co-comp(AI\) exist,

then there are natural isomorphisms: 1\

G (co-comp (A )) "" G (A), p p

all integers

p.

Filtered Objects And, similarly, if co-comp(A)

265

and (co-comp A)A

exist, then

there are induced natural isomorphisms: G ((co-comp A) p

A

) '" G A, p

all integers

p.

We next prove a few general lemmas about inverse limits and

liml

Lemma

7.

over denumerable directed sets.

A

Let

be an abelian category such that denumerable

direct products exist and such that the functor "denumerable

AW into

direct product", from A

A,

is an exact functor.

be any complete filtered object in

Proof:

A.

Let

Then

First, notice that since denumerable direct products ex-

ist in the abelian category

A,

is exact, we have that

and

lim pE;r

and denumerable direct product liml

make sense on the cate-

pE;r

A indexed by the integers; and that

gory of inverse systems in

these form a cohomological exact connected sequence of functors, in fact, a system of derived functors, by taking the zero functor in dimensions

"I 0,1.

(See Intro., Chapter 1, section 7).

The short exact sequences:

o -+ F

A -+ A -+ Alp A -+ 0

P

for all integers

P

p

define a short exact sequence of inverse

systems in the category integers. functors

A,

indexed by the directed set of all

Throwing through the exact connected sequence of lim, liml, pE;r

pE;r

we obtain the exact sequence of six terms:

266

Section 6

(1)

dO 1 O->[lim F Al -+A ~ [lim (A/F A) l - [11m F A l P p p€:r p pE:r p

Ez.

[11ml A l - [lim pE:r pEp Since by hypothesis

A

l

(A/FpA) l--:> O.

Also, the inverse systems:

phism.

is an isomor-

is complete, the map and

(A)pEP

(A/F p A)pE:l7

are

quots. of constant inv. systems, so (Intro. Chap. 1, sec. 7, Thm. ' 1 have l ~m zero.

Substituting these facts into the exact se-

quence (1) implies conclusion of the Lemma. Remark.

The proof of Lemma 7 also shows that,

Corollary 7.1.

Let

A

be any filtered object, not necessarily

complete, in an abelian category theses of Lemma 7.

A,

where

A obeys the hypo-

Then we have an exact sequence of four

terms:

where

G+oo(A)

Remark:

is defined as in Remark 1 following Theorem 5.

In particular, it follows that, under the hypotheses

of Corollary 7.1, if [lj,m pE.;! A

l

F Al = 0 P

iff

A

is any filtered object in

A/G+ooA

is complete.

A,

then

Such a filtered object

may be called "complete but not Hausdorff".

Corollary 7.2.

The hypotheses being as in Corollary 7.1, if

is any filtered object in the abelian category

A,

then

complete iff

Remark:

Corollary 7.2 above is a converse of Lemma 7.

Lemma 7, Corollary 7.1 and Corollary 7.2 apply to an

A

A is

Filtered Objects

A,

abelian category

such that denumerable direct products ex-

ist, and such that the functor, from

W

A

"denumerable direct product"

A is an exact functor.

into

267

generalized, say, to abelian categories

Can these results be

A such that denumerable

direct products exist, whether or not denumerable direct product (An example of such an abelain category,

is an exact functor?

in which denumerable direct product is not exact, is the category of all sheaves of abelian groups on any topological space X

such that there exists a sequence of open subsets of

X

that the intersection is not open; see Example 3 below).

such

The

answer is "Yes", and we sketch the generalization in a series of exercises, most of the easy details left to the reader.

(No use

of these results will be made elsewhere in this book).

A be a category, let

Exercise 1.

Let

(*) and let

(Ai,aij)i,jED

direct system

(A, ,a, ,),

'ED

index

an object

A

iO ED,

ti:A .... Ai a

ij

0

for all

ti = t , j

for every

all

i ED

1.J

with

i,j ED with

and such that If

1.,J

iED

(Ai,aij)i,jED

be a directed class

be a direct system in the category

A indexed by the directed class 1.

D

D.

trivializes iff there exists an

A,

in

i2:.io;

such that

iO'::' i,

Then we say that the

and maps

TIi:A

such that iO'::'i '::'j;

there exists

JED

i

TIi

.... A, 0

ti = idA'

and such that, such that

o 'IT. ,

1.

is a direct system indexed by a directed

(*) A directed class D is a class ([K.G.l) together with a binary relatiorl"<" that is syrrunetric, transitive and such that x,y E D implies there exists zED such that x:5.z, y.::.z; and also such that D is non-empty.

268

Section 6

class

D

in a category

A

that trivializes as above, then it

is easy to see that

lim A. exists in the category A and is iED ~ canonically isomorphic to the object A. We sometimes say that the direct system trivializes to the object

A.

Also, if

is as above, then we sometimes say that the indicated i •

direct system trivializes past the index Exercise 2.

Let

A

and

S

be an additive functor from ject of where A.

A. B

Then let

MA

A

Define a pre-order in

Then (M A

M , A

A

into

S

and let

and

MA,

l:A .... B

f:B .... B'

in

A

F:A~>S

be an ob(l,B)

is a monomorphism in

by defining:

(l,B) '::'(l',B')

such that

f

0

1 = 1'.

together with this pre-order, is a directed class.

is a directed set iff the category

every

A

be the class of all pairs

is an object in

iff there exists a map

o

be abelian categories, let

A

is a set).

For

(1, B) E M ,

let B/A denote Coker (1). Then the assignA (l,B)Coker(F(B) .... F(B/A)) defines a direct system in

ment:

the abelian category

S

indexed by the directed class

We say that the satellite of

F

exists at

A

system trivializes in the sense of Exercise 1.

M . A iff this direct When this is

the case, by Exercise 1 the direct limit: lim

Coker (F (B) .... F (B/A) )

(1, B) EMA

exists in the category

S;

in fact, by Exercise 1, the direct

in the abelian cate(Coker(F(B) .... F(B/A))) (l,B)EM ' A S, indexed by the directed class MA even trivializes to

system: gory

an object of

S;

we denote this object by

it the satellite of ever

it exists.

F

(SF) (A),

evaluated at the object

A E A,

and call when-

Filtered Objects Exercise 3. F:A~>

8

Let

A

and

8

be abelian categories and let

be an additive functor.

category of

A

the satellite,

269

C denote the full sub-

Let

generated by those objects SF(A),

F

of

additive subcategory of

A,

exists at

A

in

A.

8,

such that

C

Then

and the assignment:

is an additive functor from

A

A~

is an (SF) (A)

C

into

A

has enough injectives (or even

which we denote by

SF. Example 1.

If the category

enough injective monomorphisms) then objects

A

in

A

8.)

Infact,if

from

(l,B)~MA

(l,B)

A

is

is a maximal element in the directed class

Therefore any direct system in any category indexed by the

directed class MA trivializes past the index the direct limit of any such direct (l,B).

(1

,B)

~

M ; A

and

system is the value at

Therefore

(1) where

F

is an injective monomorphism (*), then it is easy

to see that M . A

exists, for all

(and for all additive functors

into any abelian category such that

(SF) (A)

(SF) (A) = Coker (F (B) .... F (B/A» , (l,B)

is any element of

(or, weaker, such that

MA

such that

is injective

is an injective monomorphism).

Of course, if the abelian category

A

is such that

has enough injectives (or, weaker, is such that

(*) injective category such that always an

B

A

A

has enough

A monomorphism 1: A ... B in a category A is called an monomorphism iff whenever f: A .... C is any mono. in the A with domain A, then there exists a map g:C .... B g 0 f = 1. A monomorphism into an injective object is injective monomorphism.

Section 6

270

injective monomorphisms), then in SF

of

F

is defined.

[C.E.H.A.l

the right satellite

By our above observation, when

A is

such an abelian category (with enough injectives, or weaker, with enough injective monomorphisms), we have defined additive functor from above.

F

S,

an

as defined in Exercise 3

By equation (1) above, and by the definition in Chapter [C~l,

2, of

A into

SF,

we have that, whenever the right satellite (SF) of

exists in the sense of [C.E.H.A.l, then it exists in the

sense of Exercise 3 above and is defined on all of then these two definitions coincide.

A,

and

Therefore, the definition

in Exercise 3 above generalizes that of the right satellite as given in [C.E.H.A.l. Exercise 4. (*)

The hypotheses as in Exercise 3, let

f' f" 0-+ A I--~A -~A" -+ 0

abelian category

be a short exact sequence in the

A such that

is induced a natural mapping

S,

gory

(SF) (A')

0(*) :F(A") -+ (SF) (A')

which we call the coboundary.

0(*) 0 F(fll) = O.

Of course, if

exists.

V

Then there in the cate-

We have that

denotes the additive cate-

gory of all short exact sequences (*) in

A such that

in

is a natural trans-

C,

then the assignment:

(*)'V'v> 0 (*)

A'

is

formation of functors from the functor:

(*)'V'v> F(A") into the

functor:

AEC then (SF) (f') 00(*)=(

(*)'If'v> (SF) (A'). I f (*)EV with

Exercise 5. (*)

that F (T)

0

The hypotheses being as in Exercise 3, let

-.A'.....!.:.>A..f:~A" (SF) (A') ,

-+0

(SF) (A)

be a short exact sequence in and

(SF) (A")

A

exist. If the functor

is half exact, then the sequence: F(A') F(f')>>F(A)

F(fll»F(AII)~>(SF)

(SF) (A) (SF) (fll) ~ (SF) (A")

such

(A')

(SF) (f') >

Filtered Objects is exact in

B.

condition:

(If we weaken the hypotheses, by deleting the

"SF (A")

obtained from

271

exists",

then the sequence of five terms

by deleting the six'th term is exact; if we

(T)

further weaken the hypotheses, by deleting also the condition "SF (A) (T)

exists", then the sequence of four terms obtained from

by deleting the last two terms is exact).

Exercise 6.

Let

A

be an abelian category.

I(A) =A~

Then let

denote the category of all inverse systems in the category

A

indexed by all the integers, and all maps (of degree zero) of such inverse systems.

Then

pose that the category products exist.

means for

into

A = (A"

a, ,),

l

~~m l ~

lJ

A, 'E '

l, J

~

to exist.

(A)

1

left exact functor from

A

Sup-

is such that denumerable direct

I(A)

(S lim) iE~

use the notation:

is an abelian category.

Then the assignment:

is a functor from Given an object

A

I(A)

for

(A, ,a, ,), l

lJ

'E '\IU> lim A,

l,J ~

When

(S lim) into

l

by Example 1 we know what i t (S lim) iE~ (A) •

A

exists, we

(A)

Since

iE~

I(A)

iE~

and is a left exact functor.

~tm ~

is a

l

(and is therefore a

-

fortiori a half exact functor), by Exercise 5, whenever (*)

f' f" o +A'-A--!>A"

+ 0

is a short exact sequence in

I(A)

such that

liml A', iE~

and

t t l A"

all exist, then we have the exact sequence of six

terms in the abelian category

A:

Section 6

272 Example 2.

Let

A

be an abelian category such that denumerable

direct products of objects exist. category

A.

Let

A

be any object in the

Then, going back to the definition in Exercise

it is easy to see that the constant inverse system

I,

(A) iEJ' E I (A)

1 'ml exists at this constant inverse system, and JZ'l (In fact, i f A denotes the constant in fact that lim A = O. iEJ' (1, B) , then those elements of M , inverse system A in I (A) , A

is such that

such that

is an inverse system that is "constant past zero"-

B

i. e., such that the mappings for all integers (1, B) f MA

Coker

B) lEi! for any object

-T

-T

Bp

are identity mappings

p':: 0 - are confinal in

is such that

((~im

Bp+ 1

(~im

B

And, if

is constant past zero, then

B/A)) = 0.)

lEZ' A in

MA .

I(A)

The same argument shows that,

that is constant past zero, we have

that

liml A exists and is zero. Using this, it is also easy iEJ' to show that, if A is any filtered object in the abelian cate-

A,

gory

l' m1 F A 1 p pEZ'

then

lj,m pEJ'

l

always exists, and in fact

FA"" Coker (A P

-T

canonically, in the category of

A,

and where

A -T

i\

i\)

A,

where

AA

is the completion

denotes the natural map into the com-

pletion. Proof:

In fact, the object

P = (F A)

p

is such that, a cofinal subset of (l,B)

such that

B

Mp

is the constant inverse system

is the inclusion).

Mp

I (A)

consists of those pairs

is an inverse system constant past zero.

(An example of such a pair in A

in the category

E

p Z'

is the pair A,

And, for every such

(B/P)p=BO/F pA, P ~ 0, where

BO

and

(j,A),

j : (F pAl pEJ'

where -T

(A) pEl:'

(l,B), we have that

denotes the zero' th object in the

Filtered Objects inverse system regard

B -

-

(B

p' Bp, q ) p, qE;r' .

273

For every such

as a filtered object by defining and

B = BO '

(1, B) E Mp '

F BO = F A,

P

P

all

lim B/P = lim (BO/F pAl = pE;r' p~o

Therefore Coker ((lim B) pE;r'

(1)

+

(lim pE~

Next, by Corollary 2.1 (with

A

and

BO

interchanged), we

have the commutative diagram with exact rows:

(2)

and, also by Corollary 2.1 (with map

y

is an isomorphism.

A

and

interchanged) the

BO

Therefore, by diagram chasing in the

commutative diagram (2), we deduce that the natural mapping is an isomorphism: (3)

Coker (A

+

Since the elements

A1\ )

"" +

(1,

Coker (BO

B)

E=

Mp

+

1\ BO) •

such that

zero are cofinal in the directed class

B Mp '

is constant past it follows from

equations (1) and (3) that the direct system: {Coker ((lim B) pEl'

+

(lim B/P») pEl'

(1

B)EM 'p

trivializes, in the sense of Exercise 1, and therefore by Exercise 2 we have that (S lim) (P)

exists, and by equations (I) and (3)

pEl'

(S lim) (P) = Coker (A

pEl-

+

r!') .

Section 6

274

Changing notations as in Exercise 6, we therefore have that liml FpA pEJ'

exists, and that

liml F A = Coker (A ->- A") . pEJ' P

Q.E.D.

We are now ready to generalize Lemma 7, Corollary 7.1 and Corollary 7.2. Corollary 7.1'.

Let

A be an abelian category such that de-

numerable direct products exist, and let

A

be a filtered ob-

1

lim F p A exists, and we have pEJ' an exact sequence of four terms in the abelian category A: ject in the category

Then

A.

0->- (lim F A) ->-A->-A" ->- (liml FpA) ->- O. pEJ' P pEJ' Proof:

We have already observed, in Corollary 3.1, that the

kernel of the natural map

A ->- A"

is

n F A = lim F A. pEJ' p pEJ' P

Exercise 6, we have observed that liml FpA " pEJ' cokernel of the natural map A->-A . Corollary 7.2'.

lim pEJ'

Q.E.D.

A

is complete iff

lim F A = pEJ' P

F pA = 0 .

Proof:

Follows immediately from Corollary 7.1'.

Lemma 7'.

A be an abelian category such that denumerable

Let

direct products exist. in

exists and is the

The hypotheses being as in Corollary 7.1', we

have that the filtered object 1

In

A.

Proof: Remark.

Then

Let

A

be a complete filtered object

lim F A = liml pEJ' p pEJ'

Follows immediately from Corollary 7.2'. Let

A

be an abelian category such that

injectives (respectively:

A

has enough

enough injective monomorphisms) •

Fil tered Objects Then

I(A),

275

as defined in Exercise 6 above, has enough in-

jectives (respectively:

enough injective monomorphisms).

pose now that the abelian category numerable direct products. left exact functor

lim

p~;?

A

Sup-

is also closed under de-

Then, by Exercise 6, we have the

from

I(A)

enough injectives (respectively:

into

A.

Since

I(A)

has

enough injective monomorphisms),

derived functors R q lim, q':' 0, of the pE;? which are half exact functors from the category

we have the usual right functor

lim, pE;? into the category

I(A)

A.

By Example 1 above, we know also

that

(S lim) (P) exists, for all P E I (A), and that (S lim) pE;? pE;? is an additive functor from I(A) into A. Also, by Example

1, we know that

(S lim) pE;?

is the usual first derived functor

R 1 lim

of lim. Changing notations as in Exercise 6, we have pE;? p€;? exists in that, for every P E I (A) , that liml P = (S lim) (P) pE;? pE;? 1 from I (A) the sense of Exercise 6, and that the functor lim pE;?

into

A

as defined in Exercise 6 coincides with the usual first

riqht derived functor of

lim. pE;?

Therefore, in this case, the

1

"11m"

in the statements of pE;? Lemma 7', Corollary 7.1' and Corollary 7.2' above is simply the first right derived functor of "lim" PE:27 fined for example in [C.E.H.A.]. Example 3.

Let

X

be a topological space, such that there

exists a sequence of open subsets of tion is not open.

X

such that the intersec-

(Virtually all topological spaces that one

comes across obey this condition). all sheaves of abelian groups on jectives (see [GJ).

in the usual sense, as de-

Let X.

A

Then

be the category of A

It is trivial to show that

has enough inA

is closed

under inverse limits of arbitrary functors indexed by set-theo-

Section 6

276

retically legitimate categories. In particular,

A

products of objects. product" from

W

A

is closed under denumerable direct

However, the functor

into

A

"denumerable direct

is not exact (this is proved by a

modification of the argument in [RJ). Therefore, the hypotheses of the last Remark hold. I(A)

particular, by the last Remark, the category injectives; the functor:

lim

from

I(A)

into

A

In

has enough is left-

pE;r

exact; and the usual first right 1

defined in [C.E.H.A.]) is

derived functor of

lim (as pt;?l' (as defined in Exercise 6 above).

lim pEl' Therefore, in this Example, we have that Lemma 7', Corollary " lj,ml"

7.1' and Corollary 7.2' all hold, where the

in these

pE;r

results is the usual first right derived functor of

lj,m: 1'lE,7

I (A )'V'v> A

(as defined, e.g., in [C.E.H.A.]).

This is an example

where Lemma 7', Corollary 7.1' and Corollary 7.2' all hold and are non-trivial (but in which the hypotheses of the less general Lemma 7, Corollary 7.1 and Corollary 7.2 are not satisfied). Remark.

Let

X

be a topological space such that the intersec-

tion of denumerably many open subsets is always open. be the category of all sheaves of abelian groups on the functor "denumerable direct product ":

Aw'V'v> A

Let X.

A

Then

is exact.

We leave the proof as any easy exercise. More generally, by techniques similar to those in [RJ, it is not difficult to show that, if and

X

K

is any infinite cardinal

is any topological space, then the following three con-

ditions are equivalent: (1)

open;

The intersection of


open subsets of

X

is always

Filtered Objects (2) of

x

If

x E X,

then the intersection of

is always a neighborhood of (3)

If

I

277

AI'VU>A,

of all sheaves of abelian groups on

K

neighborhoods

x;

is a set of cardinality

"I-fold direct product",

<

where X,

< K,

A

then the functor is the category

is exact.

Section 7 The Partial Abutments of the Spectral Sequence of an Exact Couple

Definition 1. g ory and let

A

Let

be an (ordinary, ungraded) abelian cate-

(1) = (EP,q dP,q TP,q)

r

' r

' r

be a conventional bi-

p,qE~

r.::..rO graded spectral sequence starting with the integer

A.

abelian category

rO

in the

Then an abutment for the conventional bi-

graded spectral sequence

(1)

is a sequence

(Hn,TP,q)n,p,qE~,

where (1)

n H

is a filtered object in the abelian category

all integers (2)

( ~,q) 00

(p,q)~

n,

A,

such that

exists as defined in section 4, Defi-

nition 4'; and where (3)

is an isomorphism in the category

Remark:

A,

for all intp.gers

p,q.

From condition (2) of Defintion 1, it is clear that a

necessary condition for the conventional spectral sequence (1) to admit an abutment, is that the

Eoo-term,

(EP,q) 00

exists. gory

p ,qE~'

If this is the case, and if, e.g., the abelian cate-

A is closed under either denumerable direct sums of ob-

jects or under denumerable direct products of objects, then an abutment to (1) exists. exist , then take

Hn =

(E.g., if denumerable direct sums ED EP,q 00 , p+q=n

278

F H

p

n

ED E£' , q) • p'>p p'+q=n

=

The Partial Abutments

279

A is closed under both denumer-

Thus, if the abelian category

able direct products and under denumerable direct sums of objects (so that by Lemma 2' of section 4, exists), then an abutment exists for every conventional bigraded spectral sequence (1) in

A.

Of course, however, there are in

general many, non-isomorphic, abutments for a given such spectral sequence (1). Definition 2.

Let

(EP,q dP,q ,p,q Hn ,p,q) r ' r ' r " p,q,n,rE~

(1)

r~rO

and

(2)

('EP,q 'dP,q "p,q 'H n "p,q) r ' r ' r" p,q,n,rE~

be conventional

r~ro

bigraded spectral sequences with abutments starting with the same integer

A.

rO

in the (ordinary, ungraded) abelian category

Then a mapping of conventional bigraded spectral sequences

with abutments from (1) into (2) is a sequence

such that (a)

( fP,q) r

.

p,q,rE~'

(EP,q dP,q ,p,q) --i> r ' r ' r p,q,rE~

r~rO

r~rO

('EP,q 'dP,q "p,q) r ' r ' r p,q,rE~ r~rO

is a map (in the sense of Definition 5' of section 4) of bigraded spectral sequences in the abelian category A

of bidegree (0,0)

(in the sense of Proposition 0.3

of section 4) -i.e.,

( fP,q) r

p,q,rE~

r~rO

is a map of bidegree (0,0) in the bi-

Section 7

280

graded additive category

Spec. Seq. (r

and such that, for each integer (b)

fn: Hn->'H n

r>r - 0

) (A

-Z x-Z

)-

n,

is a map of filtered objects in the

abelian category

A,

such that, for every pair of integer (cl

0"

(r -r+l)

p,q, the diagram

'TP,q 'EP'q--------....,> G ('HP+ql

f~·q"l

P

1p(f

""

G

q)

P+

TP,q EP,q

;>

co

G (H P + q ) p

""

is commutative in the abelian category

A.

With this definition

' the conventional biO graded spectral sequences with abutments, starting with the

of mapping, for each fixed integer

fixed integer

r ' O

r

in the fixed abelian category

A,

additive, although in general not abelian, category. denote this category by

form an We will

CSs;O,O) (Al.

o Remark:

One can generalize U'e above definition of "mapping"

of conventional bigraded spectral sequences with abutments starting with the same integer additive category.

r ' O

so as to obtain a bigraded

Namely,

Definition 2.1.

If

bidegree

from the spectral sequence with abutment (1)

(h,k)

(h,k) E-Z x-z,

then we define a mapping of

into the spectral sequence with abutment (2) to be a sequence (3)

where (a)

is a mapping in

The Partial Abutments Spec. Seq. (

( _ +1) ) (A) r O' r, r > r_rO and, for every integer n,

281

of bidegree

is a map in the abelian category such that, for all integers

p

and

A,

n,

'H n + h + k p+h ' and such that, for every pair of integers p,q, (b2) fn

maps

into

(h,k),

F

the diagram

(c)

A,

is commutative in the abelian category the map induced by

f P+

q

q "f P+ "

where

is

on the subquotients, by virtue of con-

ditions (bl) and (b2). Then, with this notion of "mapping of bidegree we have that for every fixed integer

r

O'

(h,k)",

the conventional bi-

graded spectral sequences starting with the fixed integer in the fixed abelian category

(A). rO graded) category of maps of bidegree the additive category

CSS

CSS (0,0) (A) r

O

A form an additive, but in

general not abelian, bigraded category. tive bigraded category by

r

We denote this addi-

Then the (ordinary, un(0, 0)

of

CSS

r

(A)

is

O

defined in Definition 2

O

above. The next theorem is important, and is used extensively in mathematics. Theorem 1.

Let

A be an arbitrary (ordinary, ungraded) abelian

282

Section 7

category.

Let (1) and (2) above be conventional bigraded spec-

tral sequences with abutments starting with the same integer ro

in the abelian category

(i.e., objects in

A

(fP,q fn) r ' p , q , n , rEJ' r.:::rO quences (i.e., a map in the bigraded category

n H

object

(A)),

rO be a mapping of such spectral se-

and let

arbitrary bidegree

CSS

(A)) of rO Suppose also that the filtered

(h,k) E J' x J'.

CSS

is complete and co-Hausdorff, for all integers

and that the filtered object for all integers

n.

'H

n

n,

is co-complete and Hausdorff,

Then the following two conditions are

equivalent. (1)

For every pair of integers fP,q: EP,q rO rO

-+

'EP+h,q+k rO

abelian category (2)

p,q,

we have that

is an isomorphism in the

A.

The mapping

(fP,q fn) is an isomorphism of r ' p,q,n,rEJ' r.:::rO (h,k) in the additive bigraded category

bidegree CSS (A) • Note:

Condition (2) is equivalent to saying that: (2a)

For all integers fP,q. EP,q r

.

r

-+

p,q,r,

'EP+h,q+k r

abelian category

we have that

is an isomorphism in the

A,

and (2b.l)

For every integer n fn: H

-+

category

n,

n h k 'H + + A,

n,

we have that the map

is an isomorphism in the abelian

such that, for all integers

p

and

The Partial Abutments (2b.2)

Proof:

fn

maps

F

'H n +h + k

p+h

FpH n

283

isomorphically onto

•

Clearly, condition (2) implies condition (1).

Assume

condition (1); to prove condition (2). Then by Proposition 4' of section 4, we have that condition (2a) of the above Note holds.

Therefore by Corollary 4.1'

of section 4, we have also that f~,q: E~,q .... 'E~+h,q+k

is an isomorphism in the abelian category p

and

q.

A,

for all integers

But then, from the commutative diagram (c) of Defi-

nition 2.1 above, it follows that, for all integers the map induced by

fn

in the abelian category

p

and

n,

A by passing

to the subquotients, is an isomorphism (3)

For each integer abelian category object of 'H n + h + k , tegers fn

"Hn

n,

define a filtered object

"H

n

in the

A as follows, by requiring the underlying to be the same as the underlying object of

F (IIHn) =F ('H n + h + k ) for all inp p+h ' Then by condition (bl) of Definition 2.1, the map

and by defining p,n.

in the abelian category

A is a map

of objects in the abelian category of Definition 2.1, the map

A;

and by condition (b2)

284

Section 7

is a map of filtered objects in the abelian category

A.

The

fact that the map in equation (3) above is an isomorphism in A,

is equivalent to: (3' )

is an isomorphism in the category

By hypothesis, the filtered object

A,

for all integers

n H

p,n.

is complete and co-

Hausdorff, and the filtered objects 'Hn-and therefore also "Hn-are co-complete and Hausdorff.

Therefore, by equation (3')

and Theorem 4 of section 6, it follows that

is an isomorphism of filtered objects. ment back in terms of

'Bn

Translating this state-

gives conclusions (2b.l) and (2b.2)

of the Note. Remark 1.

Q.E.D.

Suppose that the hypotheses of Theorem 1 above hold,

and that also condition (1) of Theorem 1 holds. rem I, condition (2) holds. objects, where n Since H n and that H 1.

"H

n

Therefore

Then by Theoas filtered

is as constructed in the proof of Theorem

is complete and "H n

Hn~ "H n

"H n

is co-complete, i t follows

are both complete and co-complete-and there-

fore also

'H n

is both complete and co-complete, for all n E?'.

Remark 2.

Let

A

be an abelian category.

Then in section 6

we have defined the category of filtered objects in additive category.

A

and

an

We generalize, by defining, for each integer

d, the notion of a map of filtered objects of degree Namely, if

A,

B

are filtered objects in

fine a map of filtered objects of degree

d

from

A,

d. then we de-

A

into

B

The Partial Abutments to be a map category

f:A

A,

->

B

285

of the underlying objects in the abelian

such that, for every integer

p,

f

With this notion of maps of degree

into integers

d,

maps

d,

F A P

for all

the filtered objects in the abelian category

form a singly graded additive category gory of maps of degree zero of

Filt(A).

Filt(A)

A

Then the cate-

is the (ordinary, un-

graded) category of filtered objects as defined in section 6. Of course, using this definition of the singly graded category

Filt(A),

one can pose more easily the definition, Defini-

tion 2.1, of the bigraded additive category avoid the explicit mention of

"H

n

CSS(A),

and also

in the proof of Theorem 1.

(For example, in terms of shift isomorphisms, in the proof of "H n ='H n +h +k h '

Theorem 1,

shifted down by Let

A

.

1.e.,

. t 'lS,

1

II

'H n +h +k

with degrees

h").

be an abelian category such that denumerable direct

products and denumerable direct sums of objects exist.

Then in

general, if (1)

is a conventional bigraded spectral sequence with abutment in the (ordinary, ungraded) abelian category integer

r

' O

A

starting with the

then by Theorem 6 of section 6, we have that

canonically as filtered objects; and by Remark 1 following Corollary 6.1 of section 6, these isomorphic filtered objects are complete and co-complete. section 6, we have that

Also, by Remark 2 following 6.1 of

Section 7

286

G

P

(co-comp(H

111\

)) ""

G

n

P

H

canonically, for all integers inverse of this isomorphism.

p,n.

Let

eP,n-p

denote the

Then it follows that, if

A is

any abelian category such that denumerable direct products of objects and denumerable direct sums of objects exist, then if

(1) above is any conventional bigraded spectral sequence with

A starting with the integer

abutment in the abelian category r

o'

(H

if we replace the abutment:

n TP,q)

,

p,q,nEZ"

of

(1)

with the completion of its co-completion, or equivalently with the co-completion of its completion, then the spectral sequence:

(EP,q dP,q TP,q) itself is unchanged, but r ' r ' r p,q,r~ r.::.rO the abutment is replaced by a complete and co-complete abutment.

A is closed under denumer-

Therefore, if the abelian category

able direct products and under denumerable direct sums of objects, then every object (1) in

CSS

(A) can be replaced by rO an object having the same spectral sequence, but having a com-

plete and co-complete abutment, simply by replacing the abutment with the completion of its co-completion. ject (1) in the category

For every ob-

(A) where A is such an abelian rO category, we will denote the object thus obtained in CSS (A) rO 1\ by co-comp«l) ). Corollary 1.1.

Let

A

CSS

be an abelian category such that de-

numerable direct products and denumerable direct sums of objects exist.

Let (1) and (2) above be spectral sequences with abut-

ments in the abelian category

A

starting with the same inte-

(fP,q fn) be a map of spectral r ' p,q,n,rEz r.::.rO sequences with abutments of bidegree (h,k) in the sense of Let

287

The Partial Abutments Definition 2.1. above.

Suppose that, there exists

can be either an integer or else fP,q: EP,q ->- EP+h,q+k r r r l l l

(3)

integers

co-comp(f

IlI\

n,

~rO

(r

l

such that

is an isomorphism for all

the induced mapping

): co-comp(H

is an isomorphism of degree Filt(A)

l

p,q.

Then, for every integer (4)

+00)

r

h

DI\

) ->-co-comp('H

n-+ h+kA

)

in the singly graded category

of all filtered objects in the abelian category

A,

as defined in Remark 2 following Theorem 1. Proof:

Suppose first that

r

l

is an integer.

Then replacing

the spectral sequence (1) starting with the integer its restriction past

r

l

,

r O'

and the abutment of (1) by the com-

pletion of its co-completion, we obtain the object in (co-comp(l

with

CSS

r

(A) : l

A

r

) ) l

where

is the natural isomorphism, for all integers

p,q.

Performing

the same operations to (2) yields similarly an object )A

(fP,q fn) yields p,q,n,rEJI' r ' r>r 1\ 1\ (A) a map in CSS from co-comp(l ) into co-corhp(2 ) r r r l l l and all of the hypotheses of Theorem I above hold, with r rel placing rOo In addition, hypothesis (3) of this Corollary imco-comp(2

r

l

in

CSS

r

(A) •

But then

l

plies condition (l) of Theorem 1.

Therefore by Theorem I we

Section 7

288

have condition (2) of Theorem 1, and in particular both conditions (2b.l) and (2b.2) of Theorem 1.

These are equivalent to

conclusion (4) of this Corollary. If

r

l

= +00,

by co-comp (2) if

f~,q

/\

then replacing (1) by co-comp(l)

/\

and (2)

the argument in the proof of Theorem 1- that

,

is an isomorphism for all integers

then we have

p,q

conditions (2b.l) and (2b.2) of the Note to Theorem l-applies, completing the proof of the Corollary in this case. Remark 1.

If in Corollary 1.1, we delete the hypothesis that

the abelian category

A

is "such that denumerable direct

products and denumerable direct sums of objects exist," then if n

is any fixed integer such that

co-comp(H n )

HnI\,

co-comp(HnI\),

(co-comp Hn)/\, 'H n + h + k /\, co_comp('H n + h + k /\),

and

co_comp('H n + h + k )

and

(co-comp 'Hn+h+k)/\

all exist, then the

proofs of Corollary 1.1 and of Theorem 1 show that co-comp (f

nI\

): co-comp (H

nI\

)

+

co-comp ( 'H

is an isomorphism of filtered objects in h,

for the fixed integer

Remark 2.

n+h+k/\

Filt(A)

) of degree

n.

The significance of Theorem 1 above is that, if we

have a conventional bigraded spectral sequence with abutment in an abelian category

A,

then it is important to know whether

or not the abutment is complete and co-complete.

In practical

examples, the abutment is often an interesting object, of some significance.

If we should replace the abutment with the com-

pletion of its co-completion, (assuming, e.g., for simplicity that the category

A

is closed under denumerable direct products

and denumerable direct sums of

object~,

then this might not be

The Partial Abutments as interesting an object.

289

This is why it is important to check,

given a specific conventional bigraded spectral sequence that has an interesting abutment, whether or not that abutment is complete and co-complete.

If not, then one should try, if pos-

sible, to compute explicitly, and determine, if possible, the significance of the completion of the co-completion of the abutment (which, as noted just before Corollary 1.1 is then another abutment for the given spectral sequence). Of course, the most cornman conventional bigraded spectral sequences are those that corne from conventional bigraded exact couples.

It is not true that every spectral sequence of a con-

ventional bigraded exact couple comes endowed with a "natural" interesting abutment (not even if the abelian category

A

is

the category of abelian groups); special assumptions are needed for this.

However, as we shall see, at the fullest level of

generality, the spectral sequence of a conventional bigraded exact couple does induce two sequences of filtered objects, which together sort of sUbstitute for an abutment.

This obser-

vation is the reason for the next definition. Definition gory.

3.

Let

A be an (ordinary, ungraded) abelian cate-

Let

(1)

be a conventional bigraded spectral sequence starting with the integer

rO

E~-term

(EP,q)

-

in the abelian category 00

(p,q)EJ'Q

exact sequences:

A.

Suppose also that the

exists, and that we are given short

Section 7

290

(2)

, q "" 'EP,q

,g
O
v

0

0)

A for all integers

in the abelian category

p

and

q.

(We

will call such a set of short exact sequences (2) a partition ~+'

...J.,U..

(EP,g) co (p, g) EZ' x)' • )

Then bv a set of partial abutments .1 _ sequence (1)

conventional bigraded to

(E~,g) (p,qlE)'xZ'

partition (2) of

respect

we mean a sequence

(3 )

where (al

(b)

'Hn

and

"H

category

A,

'" p, q:

' E~ ,q

n

are filtered objects in the abelian all integers

-+

G

p

(' H P +

n,

g)

is an epimorphism in the abelian category all integers

(e)

"TP,g:

G

p

p,q,

("HP+g)

+

for

A,

for

and "EP,g co

is a monomorphism in the abelian category all integers Suppose that

A,

p,g.

('H n 'TP,q "H n "TP,g) ,

"

p,g,nE~

is a set of

partial abutments for the conventional bigraded spectral seguence (1) with respect to the partition of (2).

Then we call

'Hn,

nEZ',

or the right abutments, and abutments, or the ~~:

If

"H n ,

the direct limit abutments, n

t;;;[ ,

the inverse limit

abutments.

('H n ,'T P ,g,"H n ,II TP,g)

is a set of partial abutments

as above, then it is of course in general false that n , p ,g) ( 'H , T p, q, nE;[

is an abutment, in the sense of Definition

The Partial Abutments 1, for the spectral sequence (l).

291

The terminology "direct limit

abutments," or "right abutments", therefore should not be taken to imply such a statement. butments Remark 1.

Similarly for the inverse limit a-

("Hn,"TP,q). Notice that the notion of a set of partial abutments

for a conventional bigraded spectral sequence (1) with respect to a given partition of

(2)

is self dual.

More precisely, if (1) is a conventional bigraded spectral sequence in the abelian category

A,

and if we have short

(E~,q) (p,q)E~x~

exact sequences (2) defining a partition of

as above, then (2) defines similar short exact sequences (2)0 in the dual abelian category

AO

(with

'EP,q

and

"EP,q

00

00

interchanging significance in passing to the dual category). If

('H n 'TP,q "H n "TP,q) , " p,q,nE~

is a set of partial abutments

for (1) with respect to (2), then

("H n "TP,q 'H n 'TP,q) , " p,q,nE~

is a set of partial abutments for the dual spectral sequence (1)0 with respect to (2)0 in the dual abelian category

AO •

Notice, therefore, in passing to the dual category, that the direct limit abutments and the inverse limit abutments interchange roles. The significance of Definition 3 is made clear by the following Theorem. Theorem 2.

(The partial abutments of the spectral sequence of

a conventional bigraded exact couple.) ungraded) abelian category, and let

Let

A be an (ordinary,

Section 7

292

(1)

be an arbitrary conventional bigraded exact couple starting with the integer

rO

in the abelian category

A.

Let

(2)

be the (conventional, bigraded) spectral sequence starting with the integer

rO

of the exact couple (1), using the indexing

convention given in Example 3.1 at the end of section 5. Let

t= (tp,q) (p,q)E,'lx,'l

and

V= (Vp,q) (p,q)E,'lxl".

Suppose

that the (t-torsion part), the (infinite t-torsion part), the (t-divisible part) and the (infinitely t-divisible part) of the bigraded object

Vall exist, as defined in Definitions I'

and 2' of section 5 (this condition is automatically satisfied if, for example, the abelian category

A

is closed under de-

numerable direct products and denumerable direct sums of objects, see Example l' following Definition 2' of section 5). Theorem 4' of section 5, we have that:

(E~,q) (p,q)E,'lx,?'

The

Then, by

Eoo-term,

of the spectral sequence (2) exists, and that

we have the short exact sequence of bigraded objects: (*)

0+ [(Ker t)

n (t-divisible) 1 ~'k" (EP,q) part of

V/(t-torsiOn)] [ t. (v/torsion)

0 +

,

V

00

(p,q)E,'lx,?

;'h"

The Partial Abutments where the maps

"h"

and

Ok"

are maps of the bigraded objects,

induced respectively by the maps

hand

(1) by passing to the subquotients. be the image of

"k",

IIhll,

and

("EP,q) 00

"h"

then since

293

k

in the exact couple ( 'EP,q)

If we let

00

(p,q)E'Z'x'Z'

is a monomorphism and

(p,q) E'Z'x71

be the co image of uk"

is an epimor-

phism, we have natural isomorphisms of bigraded objects in

V/(t-torsion)l~ ('EP,q)

(* .1)

[ t·NI (t-torsion))J

co

A:

(p,q) E'Z'X'Z'

and ("EP,q)

(*.2)

OJ

(p,q)€?'x,z

n (t-divisible) 1 part of

V

'

induced by

hand

quotients.

(These isomorphisms are not, in general, of bidegree

(0,0».

respectively, by passing to the sub-

And then we have the short exact sequences in the

abelian category P,q O+-" E 00

(*' )

k

':; [(Ker t)

all integers

A, -<-

P , q +- 'E P , q +E 00 00

p,q E 71.

a

,

(*') is a partition of

in the sense of Definition 3 above, which we will call the partition of

(E~,q) (p,q)E'Z'x71

induced £y the exact couple (1).

Then there is induced a natural set of partial abutments: (3)

for the spectral sequence (2) with respect to the partition (*') of (E~,q) (p,q)E'Z'xij" below:

•

And

'To(*.l), (*.2) °T"

induce (4),(5)

For the associated graded of the direct limit abutments, we have the canonical isomorphism of bigraded objects in

A,

294 (4)

Section 7 (

Gp

('HP,q))

"" [ VI (infinite t-torsion) ] (p,q)EZ'xZ''''- t· (Vi (infinite t-torsion))

and for the associated graded of the inverse limit abutments,

A,

we have the canonical isomorphism of bigraded objects in

(5)

(G ("HP,q)) . --> (p,q)EZ'xZ' p [(Ker t) n (infinitely vt-divisible)l. part of We have that the direct limit abutment

'H

n

is always co-

complete as filtered object, and that the inverse limit abutment "H

n

~ote

is always complete as filtered object, for all integers 1.

The bidegrees of the map

of the map

"h"

h

n.

in the exact couple (1),

in the short exact sequence (*), of the canoni-

cal isomorphism (*.1) and of canonical isomorphism (4) are all the same. the map

Also, if k

(ex, S) E Z' x Z'

denotes this bidegree, then

in the exact couple (1), the map

"k"

in the short

exact sequence (*), the canonical isomorphism (*.2) and canonical isomorphism (5) all have the same bidegree (r

o-

a,-r + 1- S). O

Note 2. (p,q)

For each integer such that

p +q

=

in the abelian category (6 )

n

t P + 2 ,n-p-2

.::....------;!>

V

n,

n.

consider all pairs of integers

Then we have the infinite sequence

A,

p+l n-p-l '

vp-l,n-p+l

P+l,n-p-l --=t _____ !> Vp,n-p

tp,n-p

!>

tp-l,n-p+l --"-----!>

The objects that actually occur in this sequence are precisely n those vp,q such that p +q = n. Let 'K be the direct limit of the sequence

(6 ), n

The Partial Abutments

295

'Kn=lim Vp,n-p.

(7.0)

p""-oo

Introduce a filtration on

'K

n

by defining

to be the vP,q""'K n ,

image of the natural map into the direct limit, where

p +q =n , (7.1)

Then we define the direct limit abutments to be the filtered n 'K ,

objects

with some shifting of degrees.

(7.2) where

F

(a,B)

'H

n

= F

p-a

'K n - a -

is the bidegree of the map

(1), as in Note 1 above. limit of the sequence (8.0)

p

Similarly, let

B

'

h "K

Explicitly,

in the exact couple n

be the inverse

(6 ), n

"K n = lim vp,n-p. P",,+OO

Then introduce a filtration on

"K

n

by defining

to be

the kernel of the natural map from the inverse limit, "Kn ....vp-l,q+l,

where

p+q=n,

(8.1)

Then we define the inverse limit abutments to be the filtered objects

"Kn,

with some shifting of degrees.

Explicitly,

(8.2)

where

is the integer such that the conventional bigraded

exact couple (1) starts with the integer

rO

in the sense of

Example 3.1 of section 5 (and therefore also the spectral

Section 7

296

sequence (2) starts with the integer Proof:

Let

n

r )' O

be an integer, and consider the sequences (6 ) n

of Note 2, defined for every integer

n.

Then by Definition 2'

of section 5, the (infinite t-torsion part of

V)

exists, if

and only if the direct limit of the sequence (6 ) exists for n every integer

n.

By the hypotheses of this Theorem, the (in-

finite t-torsion part of

V) exists.

Therefore the direct limit,

of the sequence (6 ) exists for every integer n. Next, n n we introduce a filtration on 'K by equation (7.1) of Note 2, 'Kn,

so that A,

'K

n

becomes a filtered object in the abelian category

for all integers

n.

By definition of "infinite t-torsion", see Definition 2' of section 5, we have that (infinite t-torsion part of for all integers

p,n.

V)p,n- p

Equation (9) above and Equation (7.1) of

Note 2 imply that the natural map

Vp , n-p ... 'K n

induces, by

passing to the subquotients, an isomorphism (10)

[V/(infinite t-torsion)]p,n-p"'F p 'K n ,

for all integers

p,n.

From the sequence (6 ) and the definition n

(7.1) we deduce the commutative diagram

(11)

Equation (10), for the pairs of integers

p,n

and

p + l,n,

the diagram (11), imply that we have the commutative diagram

and

The Partial Abutments

297

[V/(infinite t-torsion)lP,n-p----------~ "" >F p 'K n

J

"tP+l,n-p-ll

(12)

[V/(infinite t-torsion)lP+l,n-p-l ____ "" ~0> F in which the map

"tP+l,n-p-ll

passing to the quotients.

is induced by

p+l

'K n

'

tP+l,n-p-l

by

The commutative diagram (12) implies

that we have a canonical isomorphism (13) [

V/(infinite t-torsion) ]p,nt ·CV/(infinite t-torsionD

for all integers

p,n.

p

"" G 'K n p ,

Otherwise stated, we have a canonical

isomorphism of bigraded objects in the abelian category

A,

of

(0,0) ,

bidegree (14)

V/ (infinite t-torsion) ] ':; lG 'KP+q ) p (p,q)E:7! x 71. [ t ·f.J/(infinite t-torsion))

Let

(a,S)

be, as in Note 1, the bidegree of the map

in the exact couple (1).

h

Then, since the exact couple (1) is

a conventional bigraded exact couple, if

(al,Sl) =deg k,

then,

see Example 3.1 of section 5, we must have that a+S+a

+Sl =1, a+a l =rO· Therefore a l =rO -a, Sl =-rO+l-S. l Therefore deg(k) = (al,Sl) = (rO-a,-rO+l- S ), as asserted in Note 1. (*)

Also, since the map

and the map

in the short exact sequence

(*.1) are induced from

subquotients, it follows that Similarly,

"h"

h

by passing to the

deg("h") =deg(*.l) =deg(h) = (a,S).

deg("k") =deg(*.2) =deg(k) = (rO-a,-rO+l-S),

as

asserted in Note 1. If we define the filtered object

'H

n

in the abelian cate-

298

Section 7

gory

A

by requiring that Equations

(7.2) hold, then we have

that G 'H n = G

p

p-a

for all integers

p

n a- S

'K -

and

isomorphism of bidegree

n.

Therefore we have the canonical

(a,S),

"" ( ,p+q) ( G ,p+q) (p,q) E71 x 'l -> Gp H (p,q) E71x'l p K

( 15 )

Composing the isomorphisms (14) and (15), we obtain the isomorphism (4) of bidegree (a,S). Next, notice that by Example 4' of section 5, we have that (t-torsion part of

Vic (infinite t-torsion part of

V).

Therefore, we obtain the natural epimorphism of bidegree

vI (t-torsiOn)] [ t -(vi (t-torsion)}

(16)

[ vi (infinite t-torsion) ] -;:. t.(ljl(infinite t-torsion)} •

The isomorphisms (*.1) and (4) have bidegree (a,S), epimorphism (16) has bidegree (0,0). (' p,q) T

(p,q) E71x'l

(0, 0)

to be composite of:

and the

Therefore if we define the inverse of the iso-

morphism (*.1), followed by the epimorphism (16), followed by the isomorphism (4), then ( ' T p,q) (p,q) E71x71""

(17)

is an epimorphism of bigraded objects in the abelian category A

of bidegree (0,0). If we define the filtered objects "K

integers

n

n

and

"H

n

for all

as in Note 2, then the dual of the arguments just

given establishes the isomorphism (5), establishes that the bi-

The Partial Abutments

299

degree of the isomorphism (5) is as asserted in Note 2, and also defines a monomorphism (18)

of bigraded objects in the abelian category

A

of bidegree

(In fact, these observations about the inverse limit

(0,0) •

n "H ,

abutments

n E?!,

can alternatively be deduced directly

from the corresponding observations about the direct limit abutn 'H ,

ments

the sequence

n E?!,

by passing to the dual category).

('H n 'yP,q "H n "yP,q) ,

"

p, q, nE?!

But then

is a set of partial

abutments for the spectral sequence (2) with respect to the partition given by the short exact sequences

as asserted in the Theorem. To complete the proof of Theorem 2, it remains only to show that the filtered objects 'H the filtered objects

"H

n

n

are co-complete, and that

are complete, all integers

n.

By equation (8.2) of Note 2, we have that the filtered objects

"H

n

and

l a "K n + - -

singly graded category rem 1)

n.

integers n

are canonically isomorphic in the

Filt(A)

(see Remark 2 following Theo-

through an isomorphism of degree

integer

"K

B

n

Therefore, to prove that

"K

for every

is complete for all

it is equivalent to prove that the filtered objects

are complete for all integers n

"H

rO - a, n

n.

is by definition the inverse limit of the sequence

300

Section 7 (8. 0)

II

Kn = lim Vp , n-p • p++oo

Therefore we have also that (19)

"Kn=li m [Im("Kn+vp,n-p)l. (1) p++oo

But by equation (8.1) we have that (20)

in the abelian category

Substituting into equation (19),

A.

we have that the natural map (21)

is an isomorphism.

Therefore by Definition 2, part (iv) of

section 6, we have that "K

n

is complete as filtered object.

The proof that the direct limit abutments plete for all integers

n

'H n

are co-com-

is similar; and in fact can be de-

duced from the corresponding assertion about the inverse limit abutments by passing to the dual category.

Q.E.D.

(1) The reason for this is that, in general, if in an arbitary category if

Bi

A,

where

D

is any directed set, and

are sub-objects of vi, for all i E D, such that the i i K+V maps into the subobject B , for all iED,

natural map:

and such that for all i, JED such that j ~ i, the map in the ij inverse system a maps Bi into Bj ; then we have also that i K = lim B , since K then satisfies the necessary universal iED mapping property.

The Partial Abutments Definition 4.

Let

(1)

301

be a conventional bigraded exact couple

in the abelian category

A

such that the hypotheses of Theorem

2 are satisfied-i.e., such that the (t-torsion part of

the (t-divisible part of and the

V)

V),

the (infinite t-torsion part of

(infinitely t-divisible part of

Then the set of partial abutments

~

abutments of the spectral sequence (2) induced We will refer to

limit abutment, resp.:

I Hn,

all exist.

V)

(IHn ,TP,q "H n "TP,q) , " p,q,nE7

constructed in Theorem 2, will be called the

couple (1).

V),

resp.:

of partial

~

the exact

"Hn,

as the direct

the inverse limit abutment,

induced

Qy the exact couple (1).

Notice that the filtered objects, the direct and inverse limi t abutments,

I Hn,

uHn,

n E:2,

depend (up to a shifting

of degrees) only on the bigraded object

V -

-

and the map of bigraded objects of bidegree t = (tp,q) (

p,q

)C.~x:2

from

'-U-

(Vp,q)

(p,q)E:2 xZ

(-1,+1),

into itself occurring in the exact

V

couple (1), and not at all on the bigraded object

Definition 5.

Let

(EP,q dP,q TP,q) be a conventional r ' r ' r r~rO bigraded spectral sequence in the abelian category A starting

with the integer (EP,q) (p,q)E:2 xZ

(1)

rOo

Suppose that the

Eoo-term

exists for the spectral sequence (1), and that

00

we have short exact sequences (2)

o+

"E P , q 00

+

EP , q

in the abelian category is a nartition of ...

I EP , q

+

00

(EP,q) 00

+

0

00

A

for all integers

(p,q)E:2xZ'

p,q,

so that (2)

and that we have also

Section 7

302

(3)

a set of partial abutments for the spectral sequence (1) with respect to the partition (2) of say that the direct limit abutments limit abutments

"Hn)

(EP,q) We will co (p,q)E;.I',,?,· n 'H (resp.: the inverse

are perfect iff the epimorphisms

are isomorphisms, for all integers

p,q,

(respectively:

iff

the monomorphisms

are isomorphisms, for all integers Suppose

p,q.)

that (1) above is a conventional bigraded spectral

sequence such that the Eco-term exists, and that we have the partition (2) above of the Eoo-term of (1), and also the set of partial abutments (3) above with respect to the partition (2). Then we define the right defect, or the direct limit defect, to be the

kernels of the epimorphisms

This is a bigraded object in the abelian category

a

subobject of

A,

and is

Similarly, we define the

left defect, or the inverse limit defect, to be the cokernels of the monomorphisms q ) -+ "EP,q "TP,q: Gp ("H P +r o'

p,q E 'J'.

This is also a bigraded object in the abelian category is a quotient-object of the bigraded object

A,

and

Eoo= (E~,q) (p,q) EJ'x?!.

The Partial Abutments

303

Clearly, from the definitions, the direct limit (resp.:

in-

verse limit) abutments are perfect iff the direct limit (resp.: inverse limit) defect is zero. RemarkJ:. Under the hypotheses of Theorem 2, using equations (*.2) and (5) of Theorem 2, it follows that, we have the canoni-

A,

cal isomorphism of bigraded objects in

[(Ker t) (in ver se 1 imi t de f ec t) '"

:>

n

(t-divisible)] part of V

-------:--=-;--.,-:-~-..,--~;__.,._~_=_­

[(Ker t)

n

(infinitely t-divisible)] part of V

the isomorphism being induced by the map

k

in the exact couple

by passing to the subquotients, and therefore having bidegree bidegree(k)

equal to

(r

=

o

- a, - r

0

+ 1 - 13).

Similarly, under the hypotheses of Theorem 2, using equations(*.l) and (4) of Theorem 2, we have the canonical isomorphism of bigraded objects in

(direct limit

'"

defect)~

A, (infinite t-torsion in V) .(infinit~ t-torsion n t 1.n V

(t-~orsionH-[ 1.n V

the isomorphism being induced by the map

h

in

'

the exact couple

by passing to the subquotients, and therefore having bidegree equal to bidegree (h)= (a, 13) • Remark 2. In [LL.], if

A

is any abelian category such that

denumerable direct products exist, then given any inverse systern

C = (C

category object,

i

,

A,

a

ij

indexed by the integers in the abelian ). . E 1.,J 'l' j.:.i we define the deviation of C to be the'l'-graded

Section 7

304

Dev C _ a-divisible part of C - a-infinitely divisible part of where

C

~-graded

C '

on the right side of the equation is regarded as the

i (C )iE~'

A,

object in

morphism of degree -1 of this

and where

~-graded

a

object,

is the endo_ i, i-I a-(a )1' E 'Z' '

In terms of this definition, if the hypotheses and notations are as in the last Remark, then for each integer

n

we have

the inverse system of Note 2 to Theorem 2 t P + 2 ,n-p-2

-=-----0> V

cP n

= Vp,n-p

'

A.

p+l n-p-l tp,n-p ' 0> vp,n- p ....;;;--->

.....::;----_0> tP

'n

='

tp,n- p ,

therefore have an inverse system category

t

tp-l,n-p+l

vp-l,n-p+l

Defining

p+l n-p-l

for each integer C* n

=

(C

p P t ) n'npE;r

n

we

in the abelian

Then it follows immediately from the equation a-

bout the left defect deduced in the last Remark, that if

t

denotes the endomorphism of degree -1 of the graded object Dev(C*n)

(tP,n- p )

induced by

then we have that the biPE-r' graded object, the left defect of the part. abuts. of the spectral sequence of the exact couple (1) of Theorem 2, is canonically isomorphic, as bigraded objects, to the bigraded object in (precise t-torsion part of

Dev C*

p+q

)

(P,q)6~

the isomorphism being induced by the mapping

k

A,

q'

in the exact

couple (1) .. [left defect of

"k"

(l)~ ~

[precise t-torsion in Dev C* ] p+q (p,q)E;, x'Z'·

Therefore this isomorphism is of bidegree equal to bidegree

The Partial Abutments (k) = (r

O

- a,-r

corollary 2.1.

O

+1 -

305

s) • (*)

Under the hypotheses of Theorem 2, a sufficient

condition for the direct limit abutments to be perfect is that (t-torsion part of

V)

(infinite t-torsion part of

V).

A sufficient condition for the inverse limit abutments to be perfect is that (infinitely t-divisible part of Corollary 2.2.

V)

(t-divisible part of

V).

Under the hypotheses of Theorem 2, necessary

and sufficient conditions for the direct limit abutments to be perfect is that, the two subobjects of (t-torsion part of

V,

V)c (infinite t-torsion part of

have the same images in

V),

V/tV.

A necessary and sufficient condition for the inverse limit abutments to be perfect is that (Ker t)

n

(t-divisible part of

V)

=

(Ker t) n (infinitely t-divisble part of Proofs of Corollaries 2.1 and 2.2: prove Corollary 2.2.

V).

Obviously, it suffices to

By equations (*.1) and (4) of Theorem 2,

the direct limit abutments are perfect iff the natural epimorphism:

(*) Here, as elsewhere in this book, if C is an object in an abelian category, or in a graded abelian category, and t is an endomorphism of C, then by the precise t-torsion part of C we mean Ker t.

Section 7

306 V/(t-torsiOn)] [ t·cvI (t-torsion)) --~

l

V/(infinite t-torsion) ] t·CV/ (infinite t-torsion))

of bigraded objects of bidegree (0,0) is an isomorphism. equivalent to say, iff the (t-torsion part of (infinite t-torsion part of Also, by equations

V)

V)

It is

and the

have the same images in

V/tV.

(*.2) and (5) of Theorem 2, we have

that the inverse limit abutments are perfect iff the natural monomorphism: [(Ker t)

n

(infinitely t-divisible) l~ part of V

[(Ker t)

n

(t-divisible) 1 part of V

is an isomorphism.

Q.E.D.

A consequence of Corollary 2.1 is corollary 2.3.

The hypotheses being as in Theorem 2, suppose

that the abelian category

A is closed under denumerable dir-

ect sums of objects, and is such that the denumerable direct limit over the directed set of positive integers is an exact functor.

Then the direct limit abutments

for all integers Proof:

'H

n

are perfect,

n.

By Proposition I' of Example 5' of section 5, we have

that (t-torsion part of

V) = (infinite t-torsion part of

Corollary 2.1 finishes the proof. Example.

V). Q.E.D.

If we have any conventional bigraded exact couple in

the category of abelian groups, then the hypotheses of Corollary 2.3 hold.

Therefore, the direct limit abutments of every

The Partial Abutments

307

conventional, bigraded exact couple in the category of abelian groups, are perfect.

(The same is not true in general for the

inverse limit abutments, since inverse limit over denumerable directed sets is not an exact functor in the category of abelian groups.

See section 8 below for counter-examples.)

Remark 1.

Let the hypotheses be as in Theorem 2. 'H n ,

that, both the direct limit abutments verse limit abutments,

"Hn,

nE;?',

Then notice

n E~ ,

and the in-

as filtered objects,

de-

pend, up to a shifting of degrees, only on the bigraded object V= (Vp,g) of

V

(-1,+1)

of bidegree

also, the right part,

00

(p,g)E~x~'

(p,g)E~x~

in the exact couple (1) •

[ V/(t-torsion) ] to(Vi (t-torsion))

[ (Ker t) n (t-divisible part of ( EP,q)

t - (tp,g) -

and the endomorphism

(p,g)E~x~

V) ],

,

And

and the left part,

of the Eoo-term

as defined in Remark I' following Theorem 4'

of section 5, depend only on

V

and

t. (Vp,g) (p,g)E~X~

Graphically, the bigraded object with the endomorphism

(tp,g) (P,g)E~xZ

of bidegree (-1,+1)

can be pictured by a diagram, a portion of which is:

~l -21 V

-11

.

tal' ""'tIl '

~-11 ~Ol V

'~-20

"(10 v- lO

~-2-1

,t-l-l

tOO

~V00 0-1

~v-2-1 ~v-l-l ~\10-1

''(-2

~1-2

~-2

~l

\,11

~O v

lO

,(-1 vl - l ,,(-2

together

v 21

~

'(0 v

20

~-l

v 2- 1

~

~

~3-1

"""

~2

308

Section 7 Each of the lines

(Vp,q tp,q) , p+q=n

of slope -1, deter-

mines one of the groups of the direct limit abutment and one of the groups of the inverse limit abutment; namely, the direct limit of the n'th line, object

'Kn,

image of

(Vp,q tp,q) , p+q=n'

vp,q

in

'K n ;

n

is the filtered obJ'ect "K

n ('H ) .

and the inverse limit abutments n E.Z

and

n ("K )

"Kn,

where

is the kernel of the natural

"K

n ('K )

vp,q.

n

map from

from

into

is the

and the inverse limit of the n'th

the p'th filtered piece of

n~

'K n

where the p'th filtered piece of

(Vp,q tp,q) , p+q=n'

line,

is the filtered

And then the direct limit abutments

nEd'

("H n )

n~~

are obtained

by making the dimension shifts,

Equations (7.2) and (8.2), respectively, of Note 2 to Theorem 2. Remark 2.

Let

be an integer, and let

t (1)

(1)

be a conventional bigraded exact couple starting with the integer

rO

in an arbitrary abelian category

A.

It is equivalent

to say, "Let (1) be an object in the additive bigraded category EC

(rO,-rO+l) , (-1,+1)

section 5",

(A~x~)

where as usual

as defined in Definition 5.2 of

i' x

'l'denotes the abelian bigraded

category of all bigraded objects in

A.

Then (1) is canonically

isomorphic to a conventional bigraded exact couple

(1' )

The Partial Abutments starting with the integer

rO

309

in the abelian category

A,

the

isomorphism being of bidegree (0,0) and such that the bidegree of the map

h'

V' = V

let

in

(-a,-S)

(-a,-S) "- where of

and let

E,

(I')

is

- that is,

(0,0). "V

(a, S) = deg h. g=

(n

kt,..e»

-1

t' = got

0

g

,

h' = hog

-1

,

E' =E,

and

with degrees shifted down by Also, let

k' = g

0

f

be the identity

g:V~V'

Then

phism of bigraded objects of degree -1

Namely, let

(a,S). k.

is an isomor-

Define

Then

(I' )

is a conventional bigraded exact couple starting with the integer

r0

deg(hog

-1

in the abelian category

A,

and

deg (h') =

) =deg(h) -deg(g) = (a,S) - (a,S) = (0,0),

deg(k')

deg(gok) =deg(g) +deg(k) = (a,S) + (rO-a,l-rO-S) = (rO,l-r O). And the pair tive category

(g,id ) E

is an isomorphism in the bigraded addi-

EC(ro,-ro+l),(-l,+l) (A;!'Sl)-Le., in the cate-

gory of all conventional bigraded spectral sequences starting with the integer category

rO

in the (ordinary, ungraded) abelian

A -of bidegree (0,0)

(see Corollary 5.0.1 of section

5), as asserted. Therefore the spectral sequences of the exact couples (1) and (1') are canonically isomorphic.

In fact, if we trace the

construction, we see that the spectral sequences of the exact couples (1) and (1') are even identical (not merely canonically isomorphic). Therefore, if we wish, given any conventional bigraded

310

Section 7

exact couple starting with the integer gory

A,

rO

in an abelian cate-

we can always replace it with one canonically isomor-

phic in which the mapping

h

is of bidegree

(0,0) •

notation of Note 1 to Theorem 2, this means that

(In the

(a,S) = (0,0).)

If we were to make this convention, then in Theorem 2, we would have that deg h = (0,0)

and for the direct limit abutments, Equation (7.2) of Note 2 to Theorem 2 would then simplify to: "The filtered objects

'K

n

and

'H

n

coincide".

And for the inverse limit abutments, Equation (8.2) of Note 2 to Theorem 2 would then simplify to: for all integers n,p." However, we will not make the notational simplification of this Remark.

(The advantage of not insisting on such conven-

tions, is that one then does not have to re-index certain conventional bigraded exact couples that one comes across in applications). Remark 3.

As we have observed in the Note following Definition

3, a set of partial abutments for a conventional bigraded spectral sequence is not, in general, an abutment in the sense of Definition 1.

The next Theorem tells us, under the hypotheses

of Theorem 2, when the direct limit partial abutments are an ( honest) abutment in the sense of Definition 1.

The Partial Abutments Theorem 3.

311

Let the hypotheses be as in Theorem 2.

Then the

following two conditions are equivalent. (I)

n 'H ,

The direct limit abutments 'TP,q,

P,qEJI"

nEP,

together with

are an abutment for the conventional

bigraded spectral sequence (2) of Theorem 2 in the sense of Definition 1. (II)

(a)

The direct limit abutments

n 'H ,

nEJI',

of the

spectral sequence (2) of Theorem 2 are perfect (this condition is automatically satisfied if in the abelian category

A,

denumerable direct

sums of objects exist and the functor "denumerable direct limit" is and Example.

(Ker t) n (t-divisible part of

(b) If

tion (II) (a)

~),

A

V)

{o }.

is the category of abelian groups, then condi-

always holds.

equivalent conditions

Therefore, in this case, the two

(I) and (II) of Theorem 3 are each

equivalent to (II) (b), which in this case can be written (II) (b')

If

p,qEJI',

vEvP,q, ~

0,

tp,q(v) =0,

each integer

i

w. E vp+i,q-i

such that

and if for

there exists an element (tP+l,q-lo ••• 0

1

tP+i-l,q-i+1

Proof:

0

tP+i,q-i) (w) =

V1

then

v = O.

By definition of "a set of partial abutments", see

Definition 3 above, we have that

is an epimorphism, for all integers

p,q.

Condition (I) is

Section 7

312 equivalent to saying

is an isomorphism from

that

Therefore, for condition (I) to hold, it is

onto

necessary and sufficient that (a)

'TP,q

is an isomorphism, for all integers

p,q;

and that (b)

'EP,q = EP,q 00 00

for all integers

p,q.

Condition (a) is the definition of what it means for "the direct limit abutments to be perfect", see Definition 5 above. Condition (b) is equivalent to asserting that, "the subobject 'EP,q 00

of

EP,q 00

is the whole objectff; or, equivalently, that

"the corresponding quotient-object integers

p,q".

"EP,q 00

But by the isomorphism

is zero, for all (*.2) of Theorem 2,

this latter statement is equivalent to Condition (II) (b) above. Finally, note that, by Corollary 2.3, Condition (II) (a) of this Theorem always holds if denumerable direct sums of objects exist and the functor

A.

the abelian category Corollary 3.1.

"denumerable direct limit" is exact in Q.E.D.

Let the hypotheses be as in Theorem 2.

Then

the following two conditions are equivalent. (I)

The inverse limit abutments with

"TP,q,

p,q E 71'

"H n ,

nE;}',

together

are an abutment for the con-

ventional bigraded spectral sequence (2) of Theorem 2 in the sense of Definition 1. (II) (a)

(Ker t)

n

(t-divisible part of

V) =

(Ker t) n (infinitely t-divisble part of

V)

The Partial Abutments

(b)

Example.

If

v/(t-torsion) to(v/ (t-torsion)} =

313

o.

A is the category of abelian groups, then condi-

tion (II) (a) can be rewritten: (II) (a')

For every pair of integers tp,q (v) = 0

if

in

every integer wi

E vp+i,q-i

i,

p,q,

vp-l,q+l ,

if

v E vp,q,

and if, for

there exists an element

such that

'

(t P+I,q-1 o ••• 0 tP+i-l,q-i+1 0 tP+i,q-i) in

vp,q;

then there exists a sequence of

elements a nd such that integers Also, if

such that

i .:. 0,

i >

tP+i+l,q-i-1 (v

i+l

)

=

v. for all ~

o.

A is the category of abelian groups, then con-

dition (II) (b) can be rewritten: (II) (b')

For every pair of integers

p,q,

if

then there exist elements v

2

E v p + l ,q-l

v E vp,q, and

such that

v = v

I

+I q-l +tP ' (v) 2

'

and such that there exists an integer

i >0

such that (tp-i+l,q+i-l

0

t p - i + 2 ,q+i-2

0

•••

0

tp,q) (vI) = 0

in

vp-i,q+i.

Proof:

By the dual of Theorem 3, we have that condition (I)

of this Corollary is equivalent to the two conditions:

Section 7

314 (a)

The inverse limit abutments

"Hn,

n E'l',

are perfect,

in the sense of Definition 5, and (b)

The right part of

(EP,q) p , q) E'l' x'!' ' 00

as defined in

(

Remark l' following Theorem 4' of section 5, is zero. But (b) above is exactly equivalent to (II) (b) of this Corollary.

And the second part of Corollary 2.2 tells us exactly

that (a) above is equivalent to (II) (a) of this Corollary. Q.E.D. Remark.

In many expositions about spectral sequences, before

discussing abutments, enough hypotheses are made on exact couples to force condition (II) (b) of Theorem 3 to hold. if the category

A is the category of abelian groups, by

Then, Theo-

rem 3 the direct limit abutments form an abutment in the sense of

Definition 1.

For this reason, the reader will find, in

many references, that a single abutment only is discussed-the dir,ect limit abutment.

We do not wish to do this; partly

for

reasons of generality; partly because we think that the more general situation is easier to understand because of its selfdual nature; and partly because there

~

spectral sequences

in which the inverse limit abutments are important-e.g., some of the Adam's spectral sequences obey the hypotheses of Corollary 3.1 rather than those of Theorem 3, and therefore the inverse limit abutments are an abutment in the sense of Definition 1 for those spectral sequences (the direct limit abutments in fact being zero).

Also, the Bockstein spectral sequence (see

Chapter 1 below) is another example,of a singly graded spectral sequence, in which the left part of

(E~)nE'!'

is often important,

The Partial Abutments

315

and in particular is often non-zero. Suppose that we have a conventional bigraded exact couple as in Theorem 2; then it is obviously important to know, e.g., when the direct limit abutments (respectively:

the inverse

limit abutments) are complete; co-complete; discrete; co-discrete; or finite.

These results are easily read off from Theo-

rem 2 and the diagram in Remark 1 following Corollary 2.3. Proposition 4. (1) p,q

Let

n

such that

such that

tp-l,q+l

0

p +q = n

i >0 tp,q

'H n + a + S

=

and such that

such that 0).

tp-i+l,q+i-l

that

p I ,q I

t P + 2 ,q-2

t p - i + 2 ,q+i-2

0

p':::.p =

•••

0

•••

0

is discrete and co-complete; and the (n-l+a+S) 'th a n "H - l + + S

is co-discrete and complete.

Suppose, for every pair of integers

= p+q, t

0

Then the (n + a + S) 'th direct limit abut-

0;

and such that

v

PI

,q'

=0

or, weaker still, such that

otP'-l,q'+l otP',q' =0).

(t-torsion part of

V)

have the same images in

(p,q),

(p' ,q'), such that

that there exists a pair of integers p'+q'

(~, weaker,

vp,q = 0

or, weaker still, such that there exists

inverse limit abutment (2)

Then

be an integer such that there exist integers

tp,q = 0;

an integer

ment

Let the hypotheses be as in Theorem 2.

(££, weaker, such t

p+ 1 q 1 ,-

0

Suppose also that, the

and the (infinite t-torsion part of V/tv

cally if the abelian category

V)

(a condition that holds automati-

A is such that denumerable di-

rect sums exist and denumerable direct limit is exact).

Then:

The direct limit abutments are discrete and co-complete, and are an abutment for the spectral sequence in the sense of Definition 1. (3)

Suppose that, for every pair of integers

(p,q),

that

Section 7

316

there exists a pair of integers p+q=p' +q',

P':'P

' tP

such that

(p',q'),

and such that

,q' = 0;

vp',q' =0

(or, weaker,

or, wea k er Stl. 11 , sue h t h at

q+l p q tP '+l,q'-l o ••• op-l t' ot'=O).

Then:

such that

t P ' , q'

0

Suppose also that,

(Ker t)

n

(Ker t)

n (infinitely t-divisible part of

(t-divisible part of

V) = V).

The inverse limit abutments are co-discrete and complete,

and are an abutment for the spectral sequence in the sense of Definition 1. Proof:

(1)

By Theorem 2, the direct limit abutments are always

c-complete and the inverse limit abutments are always complete. By Equation (7.1) of Note 2 to Theorem 2, (7.1) Taking

p

as in part (1) of this Proposition, we have

and therefore

F 'K p

n

= 0

(QE,

then, since the natural map: t

P-i+l,q-i+l

0 .....

if

tp-i+l,q+i-l o •••

Vp,n-p

+

'K n

0

vp,n-p=O,

tp,q = 0,

factors through

otp,q, then once again we have that F 'K n = 0). p

By equation (7.2) of Note 2 to Theorem 2, F Therefore

p+a.

'Hn+a.+S=F 'Kn=O. p

'Hn+a.+S

is discrete, as asserted.

By duality, it

follows, likewise, that the inverse limit abutment

"Hn-l+a.+S

is co-discrete and complete.

integer

(2)

The hypotheses of (2) imply those of (1) for every

n.

Therefore the direct limit abutments are discrete

and co-complete.

The Partial Abutments

317

On the other hand, for every pair of integers have that there exists a pair of integers p' +q'==p+q, t,

t

p'-p

p' :.. p,

v p ' ,q'

,

->

(1m tP'-p)p,q == 0,

and such that the

vp,q,

such that

Hence

(t-divisible part of

This being true for all pairs of integers (t-divisible part of

we

(p '-p) 'th iterate of

v p ' ,q'

is zero on

and therefore

(p',q')

(p,q)

(p,q),

V)p,q==O.

we have that

V) = O.

But therefore the left part of (Ker t) n (t-divisible part of That is, Condition (II) (b)

V) = O.

of Theorem 3 holds.

By the hypo-

theses in (2), we also have ConditIon (II) (a) of Theorem 3. Therefore, by Theorem 3, we have Condition (I) of Theorem 3that is, the direct limit abutments ment in the sense of Definition 1.

n E~,

'Hn,

are an abut-

That completes the proof

of part (2) of the Proposition. (3)

Part (3) of the Proposition follows from Part (2)

by duality. Remark.

Q.E.D.

Under the hypotheses of Theorem 2, if

n

is any inte-

ger, then necessary and sufficient conditions for the (n+a+B) 'th group of the direct limit abutment, discrete, is that there exist an integer is entirely infinite t-torsion.

p

a 'H n + + B,

such that

to be vp,n-p

Similarly, necessary and suffi-

cien t conditions for the (n - 1 + a + B) 'th group of the inverse limit abutment,

a n "H - l + + B,

exist an integer

p

of

Vp,n- p )

is zero.

to be co-discrete, is that there

such that the (infinitely t-divisible part

318

Section 7

Proposition 5.

Let the hypotheses be as in Theorem 2.

be any fixed integer and let

p

vp,n- p ,

duce a filtration on

be any integer.

Let

n

Then we intro-

by defining the i'th filtered

piece to be:

j

the whole object, 1m (t

p+l,n-p-l

if

i ~ p;

p+2,n-p-2

0

t o •••

if

i~p.

Then the following two conditions are equivalent: (1)

The

(n+a+S)'th object of the direct limit abut-

'H n + a + S ,

ment,

(2)

is complete.

The quotient object vp,n-p/(infinite t-torsion)

is complete for the filtration induced from Proof:

Vp,n-p.

By definition of "infinite t-torsion",

(Definition 2'

of section 5), we have that [Vp,n-p/(infinite t-torsion)] =Coim(Vp,n-p .... 'K n ). By Equation (7.1) of Note 2 to Theorem 2, we have that

Therefore the factorization map, of the natural map Vp,n-p .... 'K n (3)

into the direct limit, is an isomorphism,

[V p , n-p / (infini te t-torsion)] ':;. F 'Kn. p

Also, by Equation (7.1) of Note 2 to Theorem 2 and the definition of the filtration on

Vp,n-p

given in this Proposition, we

The Partial Abutments

319

have that the isomorphism (3) is an isomorphism of filtered objects. F 'K

n

Therefore Eguation(2) of this Proposition holds iff is complete as filtered object.

p

6, this latter holds iff

'K

n

By Lemma 1 of section

is complete as filtered object.

And by Equation (7.2) of Note 2 to Theorem 2, this latter holds is complete as filtered object. corollary 5.1.

Q.E.D.

Let the hypotheses be as in Theorem 2.

n

be any fixed integer and let

p

duce a filtration on

by defining the i'th filtered piece

vp,n-p

be any integer.

Let

Then we intro-

to be:

!

(the

~ero. subobject of

Ker(t~,n-~ oti+l,n-i-l

if

i > p;

if

i,::p.

o •••

Then the following two conditions are equivalent: (1)

The

(n-l+a+B)'th object of the inverse limit abut"H n - l + a + B,

ment, (2)

.

~s

co-comp 1 e t e.

The subobject (infinitely t-divisible part of

Vp,n- p )

is co-complete for the filtration induced from vp,n-p.

Proof:

Corollary 5.1 follows by applying Proposition 5 to the

dual category. Remark 1. tions on

The reader should of course note, that the two filtravp,n- p ,

the one defined in Proposition 5 and the one

defined in Corollary 5.1, are, most often, totally different,

320

Section 7

and bear in general no interesting relationship to each other. Remark 2.

It is of course easy to give conditions under which

a direct limit abutment is co-discrete. Proposition 6. n

Videlicit,

Let the hypotheses be as in Theorem 2, and let

be a fixed integer.

Then a sufficient condition for the 'H n + a + 6 ,

(n + a + 6)' th object of the direct limit abutment, be co-discrete, is that there exist an integer for every integer

p'

t P ' ,n-p':

vp '

with

p'

~p,

p,

to tha~

such

we have that the map

,n-p' .... VP'-l,n-p'+l

is an epimorphism. A necessary and sufficient condition for the

n a 'H + + 6

object in the direct limit abutment is that there exist an integer p'

with

p'

~P,

p,

(n +

Ci

+ 6) 'th

to be co-discrete,

such that, for every integer

we have that the mapping induced by

' ,n-p', t P

[Vp',n-p'/(infinite t-torsion)]~ [Vp)-l,n-p'+l/(infinite t-torsion)]

is an epimorphism.

(Note:

This latter mapping is always a

monomorphism; so the condition is equivalent to asserting, that that mapping is an isomorphism, for all

pI

~

p.)

The proof is easy, and follows immediately from the diagram of Remark 1 following Corollary 2.3, and from Equations

(6 ), n

(7.1) and (7.2) of Note 2 to Theorem 2. Similarly, by duality, we obtain Corollary 6.1. n

Let the hypotheses be as in Theorem 2, and let

be a fixed integer.

Then a sufficient condition for the

(n - 1 + a + 6) I th obj ect of the inverse limit abutment,

The Partial Abutments n l a "H - + + S ,

321

to be discrete is that there exist an integer

such that, for every integer

p'

with

p' '::'p,

p,

we have that

the map p' n-p' p'-l n-p'+l P '' n-p' : V t ' +V ' is a monomorphism. A necessary and sufficient condition for the (n-l+a+S) 'th object in the inverse limit abutment,

"H n - l + a + S ,

crete, is that there exist an integer

p,

every integer restriction of

(

p'

with

p'

~p,

to be dis-

such that, for

we have that the mapping, the

' ,n-p' tP

infinitely t-divisible part) ___~(infinitelY t,-diVis~,'ble) of vp ',n-p' part of Vp - 1 , n - p +1 (~:

is a monomorphism.

This latter mapping is always an

epimorphism; so the condition is equivalent to asserting, that that mapping is an isomorphism for all Remark 3. hold, then

p' '::'p.)

If the hypotheses of Proposition 6 of Remark 2 above Fp+a'Hn+a+S

= 'H n + a + S ;

and similarly, if the hypo-

theses of Corollary 6.1 of Remark 2 above hold, then F

p -r n+a

l a "H n - + + S - 0 - •

These results are best possible- that

is, for any fixed integers conditions that

F

p+a

"H n - l + a + S - 0 Fp -r +a - ,

p

n a 'H + + S

and

n,

= 'H n + a + S

'

necessary and sufficient respectively:

is that for every integer

()

spectively:

p' ~p,

that the map induced by

p'

that

~P,

tp',n-p'

re-

,

[Vp',n-p'/(infinite t-torsion)]--> , [V p'-l,n-p'+l/(,~n f'~n~te

'

t-tors~on

)]

be an epimorphism (or, equivalently, isomorphism),

respec-

Section 7

322 tively:

(

that the restriction of

t

PI ,n-p' ,

infinitely t-diVisibl, -\infinitelY t-divisible) p'-l n-p'+l ' n-p' part of V P ' part of V '

be a monomorphism (or, equivalently, isomorphism). Remark 4.

We leave it to the reader, to put together Proposi-

tiorn4 and 6, and the Remark following Proposition 4, to write down sufficient; and also necessary and sufficient; conditions for the direct limit abutments of the spectral sequence in Theorem 2 to have finite filtrations; and similarly for the inverse limit abutments, with Corollary 6.1 replacing Proposition 6. Remark 5.

The partial abutments of the spectral sequence ofa

conventional bigraded exact couple were first introduced by

O. A. Laudal in an earlier unpublished version of IO.A.L).

He

uses a considerably different notation than ours. Remark 6. In Section 9, Corollary 4.1, we will show that, under the hypotheses of Theorem 2, if the abelian category

A is such

that denumerable direct products exist and such that functor, "denumerable direct product":

AW'\fI,> A is exact, then, if the

cycles stabilize in the spectral sequence (see section 9, Definition 1), then the left defect is zero, and therefore the inverse limi t abutments

I

Hn, n E 7",

are perfect in the sense of Defini-

tion 5. Remark 7.

In Remark 1 after Definition 5, under the hypotheses

of Theorem 2, we have a characterization of the left defect entirely in terms of

"V" ,

"t"

and inverse limits.

However, in

[O.A.L.), assuming additional very mild hypotheses on the given abelian category,

a (more complicated)

formula is given, in a

very different notation than ours, for the left defect in terms

The Partial Abutments . 1 .

of higher inverse limits (i.e., systems

depending only on

323

's) of certain lnverse

l~m

"V"

and

"tn.

Perhaps we should

note this here, somewhat generalized. Proposition 7.

([O.A.L.l, Theorem 2.2, pg. 18).

theses be as in Theorem 2 above. gory

A

Let the hypo-

Suppose that the abelian cate-

is such that denumerable direct products of objects

exist, and such that the functor "denumerable direct product": AW'V\,> A

is exact.

Then for every pair of integers

the inverse system

p,q

)

we have

p+i,q-i

r+l],q-l[preCiSe Pj,q l precise t + -;. rprec~se ~ -+- t-:-torsion . . . -; [ torsion in V -;. ••• ltorslon In V In V

1

i

(6

p,q,

This inverse system is a subsystem of the inverse system (6 ) n n = p+q.

of Theorem 2, Note 2, where p,q,

Also, for all integers

we have that the inverse system (6

of the inverse system (6

p,q

1

1) is a sub-system

p+ ,q-

).

Then, for every pair of integers natural isomorphism in the category (0)

p,q,

there is induced a

A, --;>

Ker [(liml ..... 0 l~

Notes: 1.

~i_] p+i ,q-i)_> (1. ml[prec~se [prec~se torslon In .to torslon V V l_

An equivalent statement is, that:

isomorphism of bigraded objects in

We have a natural

A,

"" 1· ~i_)P+i ,q-i (left defect) -;. (Ker( [lim (prec~se . · .... 0 torslon In l~

V

. l.m 1 (prec~se [ ·~o torslon l_

V

~ni+l-)P+il

q-i ))

J

(p,q)E7 x 7

324

Section 7

of bidegree equal to that of the mapk in the exact couple, i.e., in the notations of Theorem 2, Note 2, of bidegree l-r

(r

o

- a,

8).

O

2.

The proof shows in fact that the indicated isomorphism;

are induced by the mapping k in the exact couple, followed by a coboundary in an exact sequence of six terms of

liml.

I

The proof also computes the cokernel of the map of

3.

limIts

lim

induced by the inclusion

(6p+l,q_l)

->-

i>O

(6 p ,q)

as being

I

Ilml[«tiV) II (Ker t»Plq].

i>O Proof:

As noted in the statement of the Proposition, the inverse

system

(6p+l,q_l)

is a subsystem of (6

Plq i I th spot of the quotient inverse system is

(

).

The object in the

. se t i + l - t ors~on . ti-torslon)P+i ,q-i . )P+i Iq-i; ( prec~~e prec 7 ~

V

~n

~n

The mapping

V

induces an isomorphism from this latter object

onto

)

]

Plq

.

Therefore, we have the short exact sequence of inverse systems (1)

0-+ (6

p+-

1

1)'" (6

,q-

P,q

)

:ri.(I(preCiSe t-torsion)n in

V

(1m

)

p,q)

I

.. 0 •

i>O

The inverse limit of the rightmost inverse system is

(2)

[(precise t-torsion in

V) n (t-divisible part of

V)]p,q.

Also, the inverse limit of the middle term in (l} is lim (6

i~O

) p,g

=

lim [precise ti+1-torsion in V]p+i,q-i. i >0

The Partial Abutments The (infinitely t-divisible part of

325

V)p,q

is by definition the

image of the mapping [lim vP+i,q-i] .... vP,q. i>O It follows readily that (3)

[(precise t-torsion in

n

V)

(infinitely t-divisible part of

V)]p,q

is the image of the natural mapping, p:

' ( preclse ' t i + l -torslon "V)p+i,q-i] [1,lm In .... l>O [precise t-torsion in

V]p,q.

But the inverse limit of the rightmost mapping followed by the inclusion, is the mapping Im(lim TI) =Imp = the object (3). i>O (4)

p.

TI

in (1),

Therefore

That is,

[Im{lj;m TI)] "" [(precise t-torsion in i>O {infinitely t-divisible part of

V) n

V)]p,q.

Throwing the sh0rt exact sequence of inverse systems (1) through the cohomological, exact connected sequence of functors liml

lim,

of length two, we obtain, using the computation (2) of

Ij;m of the rightmost inverse system in (1) and using equation (4), that [(t-divisible part of V) n (I<er t) ]p,q [(infinitely t-divisiblE) n (Ker tJP,q part of V

Section 7

326 But, by Remark

I followinq Definition 5, we have that (left

defectl P- r O+ o ,q+r O+8-1 by

k,

is isomorphic, by the mapping induced Q.E.D.

to the left side of this equation.

Remark:

The preceding proposition can be used in the proof of

section 9, Theorem 4, below; but it is perhaps easier to prove that result directly, rather than using

liml

The Partial Abutments of the Spectral Sequence of an Exact Couple in

AD

Much of the results of this section can be generalized to exact couples in the D-graded category Definition 6. TED

Let

D

be an additive abelian group and let

be an element that is not an integer torsion element- i.e.

such that

n E if,

n >I

define the lines in il'T

AD.

= {n'

T:n Eil}.

of all lines in

D

n. T 'lOin

to be the cosets of

We will sometimes let D.

D. D

L

the y-intercept of the

YR.

of the line

~

Notice that the set

R.,

Then let us by the subgroup

stand for the set

Suppose that, for each line

we choose an element

Remark:

implies

R.

(!

L

that

which we shall call

R.. L

of all lines coincides with

I

The Partial Abutments the quotient group sum of two lines.

D/(~·

,).

Also, if

then we have the translate d.

Therefore, we can speak of the 9,

is a line and

9, + d

of the line

(This is simply the sum of the line

d + tZ . ,

in

Example. il x ~

D/

(~

9,

d ED 9,

is a degree,

by the degree

and the line

• ,) ) •

D=ilx~

If

327

and if

,=(-1,+1),

then the lines in ~

are the lines in the usual sense of slope -1 in

Therefore the lines in

il x il

are in natural one-to-one corres-

pondence with the integers, where to each integer ciate the line cept

y9,

9,n = { (p,q): p + q = n}.

of the line n

integers nology

9,

n

x Z.

n

we asso-

Let us take the y-inter-

to be the element

(O,n)E 9, ,

all

n

(The choice in this Example explains the termi-

n.

ny-intercept" in Definition 6 above).

The remainder of this section is parallel to, and more general than, definitionsand theorems given earlier in this section. Remark: if

A

Accordingly, we use a corresponding numbering. In general, if

D

is an additive abelian group, and

is an (ordinary, ungraded) abelian category, then we

have the abelian D-graded category section 3.

as in Example 1 of

In Example 2 of section 4, we have used the term

D-graded spectral sequence in

A

the abelian D-graded category

AD.

term

AD,

for a spectral sequence in Let us similarly use the

D-graded exact couple in the (ordinary, ungraded) abelian

category

A,

for any exact couple in the D-graded abelian cate-

gory Definition 3'.

Let

D

be an additive abelian group, let

,E D

be an element that is not an integer torsion element, and suppose that we have chosen a representative element

y 9, E 9"

(the

328

Section 7

ny-intercept"), for all lines Let

A

1 EL

as in Definition 6 above.

be an (ordinary, ungraded) abelian category.

Let

(1)

be a D-graded spectral sequence starting with the integer

rO

(as defined in Example 2 of section 4) in the (ordinary, ungraded) abelian category n

(Eoo)nED

Suppose also that

A.

E -term 00

are given short exact sequences:

exists, and that we

(2)

in the abelian category

A

for all degrees

nED.

(We call

such a set of short exact sequences (2) a partition of n

(Eoo)nED')

Then by a set of partial abutments for the D-graded

spectral sequence (1) with respect to the partition (2) of n

(Eoo)nED'

we mean a sequence

(3)

('H 1 , p,l "Hl" p,l)

(a)

'Hl

,

T

,

,

T

pEJ', 1 a line

where and

"Hl

category (b)

A,

are filtered objects in the abelian for all lines

1 Y +pT 'T P , : 'E 1 -+G p ('Hl) 00

abelian category lines

1,

A,

1 E L.

is an epimorphism in the

for all integers

p

and all

and is a monomorphism in the

(c)

abelian category lines Suppose that

1.

A,

for all integers

p

and all

The Partial Abutments

329

of partial abutments for the D-graded spectral sequence (1) with respect to the partition of (E~)nED

'H~,

~

(2).

Then we call

a line, the direct limit abutments, or the right abut-

"H~,

ments, and

~

a line, the inverse limit abutments, or

the left abutments. Theorem 2'.

(The partial abutments of the spectral sequence

of a D-graded exact couple.) Let of

0

0

be an additive abelian group, let

T

be an element

that is not an integer torsion element, and suppose that

we have chosen elements line

~

Let

E L

y~ E~,

the "y-intercepts", for every

as in Definition 6.

A be an (ordinary, ungraded) abelian category, and

let

(1)

be an arbitrary D-graded exact couple in the abelian category such that

n

deg ( (t ) nED)

into

for all

(2)

n dn n) (E r' r,T r nED r>O

=

T•

nED) •

(Thus,

is a map in A from

Let

be the spectral sequence of the exact couple (1). n

t = (t ) nED'

n

V = (V ) nED.

Let

Suppose that the (t-torsion part),

the (infinite t-torsion part), the (t-divisible part) and the (infinitely t-divisible part) of the D-graded object

Vall

exist, as defined in Definitions l' and 2' of section 5 (this

Section 7

330

condition is automatically satisfied if, for example, the abelian category

A

is closed under denumerable direct products

and denumerable direct sums of objects, see Example l' following Definition 2' of section 5). we have that:

Then, by Theorem 4' of section 5, n

The Eoo-term,

(Eoo)nED'

of the spectral sequence

(2) exists, and that we have the short exact sequence of 0graded objects in

o ...

(*)

A:

[(Ker t)

n (t-diviSible)] part of

vi (t-torsion) ] [toryl (t-torsion» where the maps

"h"

and

spectively by the maps

of "h"

"h",

and

"k"

are maps in

hand

("En) 00 nED

<:

«- 0,

"k"

passing to the subquotients.

v

k

If we let

natural isomorphisms in

"k"

induced re-

in the exact couple (1) by ('E~)nED

be the coimage of

is a monomorphism and

AD,

"k",

be the image then since

is an epimorphism, we have the

AD:

(*.1)

vI (t-torsion) ] ~ [ to(V/(t-torsion) ~

(*.2)

("En)

n ('E',)nED

and

00

nED

~[(Ker

induced by

hand

quotients.

(These isomorphisms are not, in general, of degree

zero.)

k

t) n (t-diviSible)] part of V

respectively, by passing to the sub-

And then we have the short exact sequencsin the abelian

category

A,

(*' )

all degrees

nED.

(*')

is a partition of

The Partial Abutments

331

sense of Definition 3' above, which we call the partition of n

(EoolnED

induced

£l the exact couple (ll.

Then there is induced a natural set of partial abutments: (3)

for the spectral sequence (2) with respect to the partition (*')

of For every

such that

dE D,

(p,~)

there exists a unique pair

d=y~+pT.

(Pd'~d)'

Call this pair

Eiflx L

Then, for the

associated graded of the direct limit abutments, we have the canonical isomorphism in G (Pd(

(4)

'H~d

AD,

~

»dED-

[

V/(infinite t-torsion) ] t'(V/(infinite t-torsionl)

and for the associated graded of the inverse limit abutments, we have the canonical isomorphism in (G

(5)

Pd

("H ~d»

dED

~>

AD,

(Ker t)n [(infinitelY t-divisiblel1 part of V •

We have that the direct limit abutment

'H~

is always co-

complete as filtered object in the abelian category that the inverse limit abutment

"H~

The degrees of the map

of the map

"h"

h

and

is always complete as

filtered object in the abelian category Note 1.

A,

A,

for all lines

~.

in the exact couple (1),

in the short exact sequence (*), of the canoni-

cal isomorphism (*.1) and of the canonical isomorphism (4) are all the same.

Let

cS

ED

Similarly, the map "k"

denote that degree. k

in the exact couple (1), the map

in the short exact sequence (*), the canonical isomorphism

332

Section 7

(*.2) and the canonical isomorphism (5) all have the same degree.

Let

Note 2.

K

denote that degree.

ED

51, ~

For every line

y -2T

~

> VY 5l,-T t

we have the infinite sequence

A,

in the abelian category t

L,

Y -T 51, >

Y

vy5l,~Vy5l,+T t

Y +T 51,

>v Y 5l,+2T t

Y +2T 51,

> •••

The objects that actually occur in this sequence are precisely those

v

d

such that

d

is on the line

51,.

Let

'K5I,

be the

direct limit of the sequence (651,)' 51,

(7.0)

'K

Y 5I,+PT = lim .... V p .... +oo

.

Introduce a filtration on

be defining

to be the

image of the natural map into the direct limit, (7.1)

for all integers

p.

Then we define the direct limit abutments

to be the filtered objects

'K5I"

with some shifting of degrees.

Explicitly,

(7.2) where

<5 =

deg h

and every line

(as in Note 1), and where for every degree 51"

a!(',d

denotes the unique integer such that in

Similarly, let

"K!(,

Y -pT

(8.0) "K5I, = lim V!(,

p .... +oo

d

D.

be the inverse limit of the sequence

•

The Partial Abutments Then introduce a filtration on

333

by defining

to be

the kernel of the natural map from the inverse limit, t (8.1)

F P "K

t = Ker ( "K

for all integers

p.

Yt-(p-l), ->- V

)

,

Then we define the inverse limit abutments

to be the filtered objects

t "K ,

with some shifting of degrees.

Explici tly,

is the integer defined by equation (7.2.1) above.

where Proof:

The proof of Theorem 2' is the same as that of Theorem

2 above.

Remark 1.

Definitions 4 and 5, Corollaries 2.1, 2.2 and 2.3,

and the Example following Corollary 2.3 all generalize immediately to the situation of Theorem 2'.

Similarly, Definition 1

generalizes to D-graded spectral sequences. Definition 1'.

Let

D,T,L,

and

Yt'

tEL,

Namely, be as in Defini-

tion 3', let

A be an (ordinary, ungraded) abelian category

and let

n n n (Er,dr,T r) nED

(1) =

be a D-graded spectral sequence

r.::ro in the (ordinary, ungraded) abelian category

A.

Then an abut-

ment for the D-graded spectral sequence (1) is a sequence (8

t

,T

p, t)

E

where

HQ,

is a filtered object in the abelian category

P 6' tEL (1)

for all lines (2)

(E~) nED

t,

A,

such that

exists as defined in section 4, Definition

4 '; and where t y +PT

(3)

TP

,

:

E t 00

":;. G (Ht)

P

is an isomorphism in the cate-

334

Section 7 gory 9,

A,

pE~

for all integers

and all lines

E L.

Then, also, the Remark following Definition 1; Definition (The generalization of Definition 2.1 has

2; Definition 2.1

to be carefully phrased. D

It defines an additive D-graded cate•

(A), such that the objects are all rO D-graded spectral sequences with abutments in the (ordinary,

gory, Spec. Seq. Abut.

A); Theorem 1; Remarks 1 and 2

ungraded) abelian category

(and the material following) after Theorem I: Corollary 1.1 and Remarks 1 and 2 following Corollary 1.1; all generalize to the situation of Definition l' above. And also, Remark 3 following Corollary 2.3; Thm. 3 and the assoc. Example; Cor. 3.1 and the assoc. Example; Proposition

4;

the Remark following Proposition

4;

Proposition 5; Corollary

5.1; Remarks 1 and 2 following Corollary 5.1; Proposition 6; Corollary 6.1; Remark 3 following Corollary 6.1

(In the gener-

alization of Remark 3 following Corollary 6.1, the indexing has to be done carefully, as in Note 2 to Theorem 2'); and Remark 4 following Corollary 6.1; Rks. 6 and 7; Prop. 7; and Rks. 1 and 2 after Defn. 5; then all generalize to the situation of Thm. 2'. Remark 1.

Let

D

be an additive abelian group, let

A be an

(ordinary, ungraded) abelian category and let

V

t

>V

~/"

(1)

E

be an arbitrary D-graded exact couple in the (ordinary, ungraded) abelian category

A.

(That is,

(1) is an exact couple in the

The Partial Abutments abelian D-graded category

AD).

335

Then when do the considera-

tions of Definition 3' and of Theorem 2' above hold?

,= deg (t),

In Definition 3' and in Theorem 2', if we must have that additive group

,

is not an integer torsion element of the What if the element

D.

then

integer torsion element?

,= deg (t)

ED

is an

Then is there still a way to make use

of Definition 3' and of the results of Theorem 2'?

The answer

is "yes". Namely, let ¢ :DO

->-

that E

D

DO

be an additive abelian group and let

be an epimorphism of additive abelian groups, such

Ker ¢ is not entirely integer torsion.

are objects, and

category n

h = (h )nED' K=deg(k),

t,h

and

we have that

k

V

and

are maps, in the D-graded

n V = (V )nED'

n

E = (E )nED'

n

t = (t )nED'

n

o=deg(h) and k = (k ) nED I f ,=deg(t), n n n then t (resp.: hn,k ) are maps from V (resp. :

Choose an element

into

into such that

ger torsion element and such that since by hypothesis

Ker(¢)

¢(00) =0, ¢(K ) =K. O

'0

¢ h 0) = ,

(resp.:

Vn + K). is not an inte(this is possible

has at least one element that is

not an integer torsion element), and let that

Since

0 , K0 E DO 0

Then define

Then and are objects in the Do-graded category

Define

be such

336

Section 7

degree t

TO'

o

resp.:

= (t¢ (m),

= (h¢ (m))

h

'!nED'

mED O'

o

o

Then we have the Do-graded exact couple in the (ordinary, ungraded) abelian category

A

is, by construction, not an integer torsion element of the additive abelian group (2)

DO.

Let

(E~,d~,T~)nED r>O

be the spectral sequence of the D-graded exact couple (1).

Then

(2) is a D-graded spectral sequence in the abelian category For each degree

all integers

r > O.

m E DO'

A.

define

Then

is a Do-graded spectral sequence in the abelian category It is easy to see that (2 ) 0

A.

is the spectral sequence of the

Do-graded exact couple (2 ). 0 Since the degree

TO = deg (to)

of

to

is by construction

not an integer torsion element, Definition 3' applies.

There-

The Partial Abutments fo:rewe can choose DO'

"y-intercepts"

with respect to

TO'

337

y 9. E 9. for every line

as in Definition 3'.

J1.

in

Then by Theo-

rem 2' we have the partial abutments of the Do-graded spectral sequence (2 ) in the (ordinary, ungraded) abelian category 0

A.

In this way, Definition 3' and Theorem 2' can be applied to every D-graded exact couple in every (ordinary, ungraded)

A,

abelian category replacing

with

DO

and

torsion element of

DO

as above.)

Remark 2.

D

for every additive abelian group T

with

TO

D.

(By

that is not an integer

The construction of Remark 1 above is related to

that of Exercise 4 ("change of grading group, other direction") following Corollary 5.1 of section 3. Example.

Let

A

be an (ordinary, ungraded) abelian category,

and let

(1)

be a conventional singly graded spectral sequence in the (ordinary, ungraded) abelian category

A.

Then, by definition of a

"conventional singly graded spectral sequence", see section 4, Example 3, we have that n

deg ( (t ) nE?) = O. Therefore, Definition 3' and Theorem 2' cannot be applied to the exact couple (1) without using the technique of Remark 1 above.

338

Section 7 Let

CP::?l xZ-+Z

TO = (-l,+l)EZ xZ.

be the sum map, If

CP(p,q) =p+q.

n

Let

n

and S = deg(k )~Z'

a = deg (h ) nE Z

then, by definition of a "conventional singly graded spectral sequence", we must have that a+S=+l Define

a

O

=

(O,a)

in

:?l.

and

So

= (O,S)

tion of Remark 1 above applies.

in

:?l x:?l.

Then the construc-

The exact couple

associated to (1) by the construction of Remark 1, is a conventional bigraded exact couple starting with the integer

0.

Also,

if (20) is the conventional bigraded spectral sequence starting with the integer zero associated to the exact couple (10)' and if (2) is the conventional singly graded spectral sequence associated to the exact couple (1), then it is easy to see that the bigraded spectral sequence (20) comes from the singly graded spectral sequence (10) by the construction of Remark 2 following Example 5 of section 4. Therefore, in this way the results of Definition 3 and of Theorem 2 can be applied to any conventional singly graded exact couple (1) in any abelian category

A-or, more precisely, to

the associated conventional bigraded exact couple (1°) starting with the integer zero constructed in Remark 1 above.

Section 8 The Spectral Sequence of a Filtered Cochain Complex

Let

A

be an abelian category and let

cochain complex in

A.

Suppose that we

ha~e

C*

p,

such that

FpC*::::l Fp+1C*,

Thus, equivalently, in the abelian category for each integer

n,

and for all integers

n

such that F p+l C n n n+l , d (F C ) C F C p

n d +l

0

p

d

n

= 0,

C

an object nand

a subobject

p

F p C* ,

all integers

n C

(7-indexed)

a filtration on

C*--that is, that we have subcochain complexes tegers

be a

and a map

all in-

p. (*) A,

we have,

dn:C n

->

cn + l

;

of

n

FpC, for all integers p,n; such that . all ~ntegers p,n; and such that

all integers

n.

We will call such a collection

of data a filtered cochain complex in the abelian category

(Spectral Sequence of a Filtered Cochain Complex).

Theorem 1. Let

(F p C*)pE7

category

A.

A.

be a filtered cochain complex in the abelian Then there is induced a conventional, bigraded

spectral sequence starting with the integer 1 such that

(*)An equivalent way of formulating this data is: "Suppose that we have a filtered object in the abelian category CoCA)". (.Another equivalent formulation is: "Suppose that we have a w-indexed) cochain complex in the additive category Filt(A) of all filtered objects in the abelian category A." 339

340

Section 8

This spectral sequence comes from a conventional, bigraded exact couple starting with the integer one,

(2)

A,

in the abelian category

in which

EP,q=HP+q(G p «C*» , tp,q

HP+q(inclusion:

=

C*) hP,q = HP +q (natural epip-l' kP,q = the (p+q) 'th coboundary in the

F C*

F

+

p

morphism: Fp C* .... Gp C*) and cohomology sequence of the short exact sequence (3), vide infra. In particular, the bidegrees of h

(hP,q) (p,q)E.lx.l

and (+1,0) Proof:

and

k

=

t

= (tp,q)

(p,q)EzxZ'

(kP,q) (p,q)E.lxz

are (-1,+1), (0,0)

respectively.

For every integer

P,

we have the short exact sequence

A,

of (.l-indexed) co chain complexes in the abelian category (3)

A+ Fp+lC* inclusion>F PC*

natural> G C* epimorphism p

+

0

,

which yields the long exact sequence of cohomology n-l

(4)

2--> Hn (F pH C* ) n

H (G C*)

P

d

n

-=--!>•••

n n H (inclusion» Hn (F C*) H (natural p epimorphism)

Filtered Cochain Complex

341

statement of the Theorem, then we obtain the conventional bigraded exact couple (2) starting with the integer 1.

Then let

(1) be the conventional, bigraded spectral sequence starting with the integer 1 of the exact couple (2), using the indexing convention given in Example 3.1 of the end of section 5. Q.E.D. The following diagram may aid in visualizing the exact couple of Theorem 1.

In this diagram, Corollary 1.1.

n = p+q

throughout.

The hypotheses being as in Theorem 1, suppose

in addition that the abelian category

A

has denumerable di-

rect products of objects and denumerable direct sums of objects. Then the spectral sequence (1) of Theorem 1 comes equipped with a set of partial abutments.

The direct limit abutments are

'Hn=limHn(F C*) -+ -p' p-++oo

(5)

and the inverse limit abutments are n "H = lim Hn + l (F C*) ofp' p-++oo

(6)

all integers Proof:

n.

Follows immediately from Theorem 1 and from section 7,

Theorem 2. Corollary 1.2.

The hypotheses being as in Corollary 1.1,

342

Section 8

suppose in addition that the functor, "denumerable direct product":

AW'VV> A n C

objects

is exact.

are complete, for all integers

verse limit abutments Note:

Suppose also that the filtered

"H

n

n.

Then the in-

are zero, for all integers

The proof shows, more generally, that if

n

Cn

n c -l

cular integer, such that

is Hausdorff and

n.

is a partiis "com-

plete but not Hausdorff" as filtered objects, then the (n-l) 1st n l "H -

inverse limit abutment Proof:

By Chapter

3

is zero.

of the main text of this book below,

Proposi tion 1, and the N:> te following, if we define for all integers n

objects

An

i, in

C!

=

F i C* ,

then there are induced for each integer A,

and short exact sequences,

(2)

and a long exact sequence (3)

is Hausdorff, then

lim cr: = n c~ = o. i>O 1 i>O

If

is "complete but not Hausdorff", then by section 6, Corollary 7.1,

(and the Remark immediately following), we have that

l l' ,l:.m1 Cr:- = O.

1> O

1

Therefore

Hn (lim C'l<) = Hn - l (liml C~) = O. i>O 1 i>O 1

from the long exact sequence (3), we have tha t

An = O.

fore, from the short exact sequence (2), we have that

Then There-

Filtered Cochain Complex

343

or, changing notations back, that n

l~m H (F C*)

p-,+oo

P

=

o.

Comparing with equation (6) of Corollary 1.1 gives the result. Q.E.D. Corollary 1.2.1.

Under the hypotheses of Corollary 1.2, we

have also that

for all integers Note:

n.

If the hypotheses are as in the Note to Corollaryl.2,

then the proof of this Corollary shows that

for the particular integer Proof:

n.

Proceedings as in the proof of Corollary 1.2, the proof

showed that

An =

o.

Equation (2) in the proof of Corollary 1. 2

therefore completes the proof. Remark.

Q.E.D.

In the Introduction to an earlier unpublished version

of the paper [O.A.L.], Laudal poses the problem:

Given a con-

ventional bigraded exact couple in an abelian category, such that denumerable direct products and denumerable direct sums exist, and such that the functors, "denumerable direct product" and "denumerable direct sum" from

AW into

A are both exact;

then does it follow that there necessarily exists some decreasing filtration on

344

Section 8

liml Vp,n-p p++ro such that the associated graded is isomorphic to the bigraded object, the left defect, as defined in section 7, Definition 5?

The answer to this question is "No".

(F p C*) p E""

In fact, let

be any filtered cochain complex of abelian groups

(I.

such that the left defect of the spectral sequence (1) of Theorem 1 is non-zero. section).

(We give such Examples at the end of this

Then, as shown in Corollary 2.1 below, the left

defect of the spectral sequence of

C*

is isomorphic to the

left defect of the spectral sequence of the completion C*.

of

If we consider the exact couple of the filtered cochain

complex

1,

C*A

C*A as in Theorem 1, then by equation (2) of Theorem

VP,q=HP+q(F (C*». p

Therefore

and this latter is zero by Corollary 1.2.1.

Therefore, given

any complete filtered cochain complex with left defectizero, it is a counterexample to the problem of Lauda1 (since the associated graded of any filtration on

liml Vp,n-p + p++ro

is zero, yet the

left defect is not zero). Corollary 1.3.

The hypotheses being as in Corollary 1.1, if in

the abelian category

A we have that the functor "denumerable

direct limit" is exact, then the direct limit abutments are perfect in the sense of section 7, Definition 5, and moreover (7) Proof:

n n 'H = H (co-completion C*),

all integers

If denumerable direct limit is exact in

A,

n. then by

Filtered Cochain Complex

345

section 7, Corollary 2.3, the direct limit abutments perfect, all integers "4m"

n.

in equation (5),

Moreover

n "H "

are

commutes with

giving equation (7).

Q.E.D.

p++oo

The next corollary is rather special (in comparison to the preceding two), but is often useful. Corollary 1.4.

The hypotheses being as in Theorem 1, suppose

that the abelian category

A

is such that denumerable direct

sums of objects exist, and such that denumerable direct limit is exact.

Suppose also that the filtered cochain complex

(F p C*)pE7.

is such that, for each integer

n

object

(FpC )pE7.

n,

the filtered

is discrete.

Then the direct limit abutments,

I

n H ,

n E 7. ,

defined by

equation (5) of Corollary 1.1, obey equation (7) of Corollary 1. 3, and

are an abutment for the conventional bigraded spectral

sequence (1).

Moreover, this abutment is discrete and co-com-

plete. Proof: teger

By hypothesis, p' = p' (n),

~or

each integer

depending on

then for all integers

p

~

pI

F ,C

n

p

we have likewise that

stated, for the fixed integer such that

there exists an in-

such that

for all integers

Therefore

p' = p' (n)

n,

n

Vp,n-p = 0

n,

P ~p'.

=

o.

But

F Cn=O. p

Otherwise

there exists an integer whenever

p ~pl (n).

Therefore

the hypotheses of section 7, Proposition 4, part (2), hold. Q.E.D. Remark 1. A

Of course, by Corollary 1.2, if the abelian category

obeys all the hypotheses of Corollary 1.4, and if the fil-

tered objects

n C

are complete, for all integers

n,

then

Section 8

346

necessary and sufficient conditions for the direct limit abutments to be an abutment (in the sense of section 7, Definition 1), is that the inverse limit defect be zero.

That is not

always true under these hypotheses, not even if

A is the cate-

gory of abelian groups, as we shall see by an example below. Remark 2.

Of course, many more observations can be made in the

situation of Theorem 1.

For example, section 7, Theorem 3, gives

necessary and sufficient conditions for the direct limit abutments to be an honest abutment in the sense of section 7, Definition 1; similarly for section 7, Corollary 3.1, and the inBy section 7, Theorem 2, the direct

verse limit abutments.

limit abutments are always

co~omplete.

Section 7, Proposition

5, gives necessary and sufficient conditions for the direct limit abutments to be complete.

Section 7, Proposition 6

necessary and sufficient conditions for the direct limit abutments to be co-discrete; and section 7, the Remark following Proposition 4, gives necessary and sufficient conditions for the direct limit abutments to be discrete.

Similar conditions

for the inverse limit abutments can be given. Let

A

be an abelian category, and let

C* = (F (C*)) p

A.

be a filtered cochain complex in the abelian category

,,~

P'"4'

Then

by Theorem 1, we have the conventional bigraded spectral sequence (1) starting with the integer 1, which we shall call the spectral sequence of the filtered cochain complex (FpC*)pE~.

=

Under the hypotheses of Corollary 1.1, we shall call

the partial abutments

'Hn, "H

filtered cochain complex of

C*

(E~,q) (p,q)E~x~

n

the partial abutments of the

C* = (F C*) P pE~·

The induced partition

will be called the partition of

Filtered Cochain Complex (EP,q)

(p, q)

00

E7! x:r

347

of the filtered cochain complex

-

C * = (FpC * ) pE:r • Remark 1. completion exists

Under the hypotheses of Theorem 1, suppose that the C*A of the filtered cochain complex

(F C*) P pE:r (a condition that is automatically satisfied if, e.g.,

the abelian category products).

C*

=

A is closed under denumerable direct

Then we have a map:

of filtered cochain complexes, which induces a map:

of the associated exact couples, where (2A) denotes the exact couple of the filtered cochain complex

C*A.

By section 6,

Corollary 4.1, the natural mapping of filtered cochain complexes induces an isomorphism on the associated gradeds,and therefore,

EP,q 1 •

by Theorem 1, on 4', the mapping

6

Therefore, by section 4, Proposition

of exact couples induces an isomorphism on

the associated spectral sequences. chain complexes

C*

tral sequences.

The exact couple

~

called ~

and

C*A

Therefore, the filtered co-

have canonically isomorphic specA

(2),

when defined, will be

exact couple of the completion of the filtered co-

C* = (F C*) • Similarly, the partial abutments P pE:r A of (2 ), when defined, will be called the partial abutments of complex

the completion of the filtered cochain complex

C*.

However, it is not difficult to see by examples (see the Examples at the end of this section) that in general the exact couples

(2) and (2A) have different direct and inverse limit

348

Section 8

abutments, and also in general induce different partitions of

(E~,q)

(p,q) E'J'x'l'.

Remark 2.

The hypotheses being as in Theorem 1, suppose that

co-comp(C*) exists.

Then we have a natural mapping:

co-comp (C*) .... C* that induces a mapping on the associated exact couples. ever, by Theorem 1, we have that HP+q(G C*) p

for all integers

How-

vp,q = HP +q (F p C*) ,

p,q

in the exact couple (2).

By the dual of section 6, Corollary 1.2, conclusion (3), we have that the natural map: F (co-comp C*) .... F C* P P is an isomorphism of cochain complexes, for all integers

p.

Similarly, by the dual of section 6, Corollary 4.1, the natural map: G (co-comp C*) .... G (C*)

p

p

is an isomorphism, for all integers

p.

It follows that the

natural mapping of filtered cochain complexes: co-comp (C*) .... C* induces an isomorphism, not merely on the associated spectral sequences, but even on the exact couples. cochain complexes, co-comp(C*)

and

C*,

That is, the filtered have canonically iso-

morphic exact couples, and therefore also canonically isomorphic: spectral sequences; partial abutments tions of

E •

00'

(if they exist): parti-

direct and inverse defects; etc.

Filtered Cochain Complex Remark 3.

349

Suppose, under the hypotheses of Theorem 1, that the

filtered cochain complex co-comp(C*)

C*

=

(F C*) p pE2'

1'1

and co-comp(C* )

all exist.

is such that

c*1'I ,

(A condition that is

automatically satisfied if, e.g., the abelian category

A

is

such that denumerable direct products and denumerable direct sums exist).

Then the construction of Theorem 1 applies, to

yield, at first glance,

~

distinct exact couples, namely the

exact couples of the filtered cochain complexes co-comp(C*)

and co-comp(C*I'I).

By Rk. 2,

C*

C*, c*l'I,

and co-comp(C*)

have the same exact couple; similarly C*I'I and co-comp(c*l'I) have the same exact couple. Therefore, these four filtered cochain complexes yield only two in general different exact couples, namely, the exact couple (2) of of the cochain complex

C*

1'1

and the exact couple (2 )

c*l'I.

The exact couple (2) constructed in Theorem 1 is not selfdual, although the spectral sequence (1) of that exact couple is self-dual. Theorem 10.

More precisely, (Dual Exact Couple of the Spectral Sequence of a

Filtered Cochain Complex). 1.

Let the hypotheses be as in Theorem

Then there is induced a conventional, bigraded exact couple

starting with the integer one,

in the abelian category

A,

in which

350

Section 8

°vp,q

=

HP + q (C*/F

°EP,q

=

HP +q (G C*),

°tp,q

=

HP + q (natural epimorphism:

°hP,q

=

the (p+q+l) 'st coboundary of the short exact sequenc

°kP,q

=

HP +q (inclusion:

p+l

C*)

'

P

G C* p

In particular, the bidegrees of h = (hP,q) (p,q)EZ'xZ' ~1,0)

and

->

C*/F

C*/F

p+l

p+l

C*-+C*/FC*) p'

C*).

t = (tp,q) (p,q)Ei!xi!'

k = (kP,q) (p,q)EZ'xZ'

are (-1,+1),

and (0,0) respectively.

The spectral sequence of the exact couple (2

0

)

coincides

with the spectral sequence (1) of the exact couple (2) of Theorem 1. Note:

The proof of Theorem 1

0

also constructs an explicit map

(g,f) of bidegree (0,0) of conventional bigraded exact couples starting with the integer one, from the exact couple (2 0

)

into

the exact couple (2) of Theorem 1, such that the indicated map induces the identity mapping on the associated spectral sequencee Remark:

Although the exact couples (2) of Theorem 1 and (2 0

)

of this Theorem have the identical spectral sequence (1) of Theorem 1, nevertheless as we shall see later, they are in general very different exact couples. Proof:

For each integer

of cochain complexes in 0-+ G C* -+ C*/F C* P p+l

p

we have the short exact sequence

A ->

C*/F C* -> 0 p'

which yields the long exact sequence of cohomology

Filtered Cochain Complex n- l

Hn(,

1

'

351

)

d --~ Hn (G C* )~nc US1.on ~Hn (C* IF C*) P p+l

n

H (natural ~ epimorphism)

n

Hn(C*/F c*)_d __ >

p

The rest of the construction of the exact couple (2°) is similar to (and dual to) that of Theorem 1. Let

n

respectively:

d(3) ,

denote the n'th co-

boundary in the cohomology sequence of the short exact sequence of cochain complexes this Theorem.

(3) of Theorem 1, respectively:

Let

denote the n'th coboundary in the

short exact sequence of cochain complexes

o . ,. F

C* ..,. C* ..,. C* IF C*..,. O.

P

P

Then we have the commutative diagram with exact rows, (6 )

•••

(The upper row in (6) is the long exact sequence (4) of the Proof of Theorem 1; the bottom row is the long exact sequence (4

0

)

of this Theorem.)

Commutativity of the displayed three

squares, from left to right, follows, respectively, from naturality of the coboundary, with respect to the map of short exact sequences:

from (3) of Theorem 1 into (5 + ); respectively p l

from (5 + ) into (5 ); respectively from (5 ) into (3°). p l p p

There-

fore, it we define gP,q = d P +q

(pH)

then

and

fP,q

g = (gP,q) (p,q)E7Q'

=

identity of respectively:

HP +

q (G C*) P

,

f = (fP,q) (p,q)E;rq'

Section 8

352

°v -_

is a map of bigraded objects from V = (Vp,q) (p,q)Ezq' into

E = (EP,q)

respectively:

(p,q)E~XJl:

(0 p, q )

V

from

(p,q)E~x~

°E - (oEP,q) -

. t 1n

(p,q)~·x~

of

of bidegree (1,0), respectively:

'

0

bidegree (0,0); and the pair (g,f) is a map of exact couples, as defined in section 5, Definition 5', in the bigraded abelian category

A~x~,

Corollary 5.0.1.

of bidegree

(0,0) in the sense of section 5,

Therefore, by section 5, Proposition 6, we

have an induced map of spectral sequences of bidegree (0,0) from the spectral sequence of the exact couple (2 spectral sequence of the exact couple (2). identity of

°EP,q=EP,q,

sequences is the identity phism.

Since

0

into the

)

fP,q

is the

it follows that this map of spectral 0

and is therefore an isomor-

n

Therefore by section 4, Proposition 4', with

r 0 = 1,

it follows that this map of spectral sequences must be an isomorphism of spectral sequences.

Finally, by section 4, Remark

l' following Proposition 0.3 (after Definition 51), this map of spectral sequences, along the

E~,q,s,

to the subquotients along the map of of

fP,q

is

is induced by passing El,q,s.

Since the map

which is the identity, all integers

p,g,

it follows that this map of spectral sequences must be the identity map. Remarks.

1.

Q.E.D. Under the hypotheses of Theorem 1, the filtered

complex

C*= (F C*) in the abelian category A deP pE.:?i' fines a filtered cochain complex in the dual category AO, which

cochain

we might denote plex.

°c*

and call the dual filtered cochain com-

Explitictly, the n-cochains in the dual filtered cochain

complex are tered piece

on -n C =C ,

for all integers n. And the pi th filo on of C is the subobect in A , that is

353

Filtered Cochain Complex the quotient-object in integers

n,p.

A,

for all

With this definition of the "dual filtered co-

chain complex;. we have of course that the dual of the dual of C*

is

C*,

°C* = C*,

for all filtered cochain complexes

Of course, the exact couple (2°) of Theorem 1°

C*.

is identical

to the exact couple of Theorem 1 applied to this dual filtered cochain complex

8*

in the dual category

AO, with a few minor

notational changes (the most notable notational shift is an interchange of the letters

"h"

and

"k"

for convenience; the

other notational shift is are-indexing). 2.

As in the case of the exact couple of Theorem 1,

a diagram of the exact couple (2 category

0

)

of Theorem 1

0

in the abelian

A may aid in visualizing this exact couple: Hn (

(Hn(C*/Fp+lC*))

natural

(epimorphism)~p,q)E~X~

(p,q)E~x~

) (Hn(C*/FpC*)) (p-l,q+l) Eil'xil'

n (H (inclusion) ) (p,q) E7Lq

In this diagram,

n = p+q

n (d ) (p-l,q +1) E6'xil'

throughout.

By duality, the corollaries to Theorem 1 each imply analogous corollaries to Theorem 1°.

We record a few of these,

correspondingly numbered. Corollary 1.1°.

The hypotheses being as in Theorem 1°, suppose

in addition that the abelian category

A has denumerable direct

products of objects and denumerable direct sums of objects. Then the spectral sequence (1) of the filtered cochain complex

354 C*

Section 8 comes equipped with a set of partial

abutment~

which we

will call the dual partial abutments, and which we denote

°

'''H n ,

n E £.

,~n,

The dual inverse limit abutments are

,,~n

n

= lim H (C* /F C*) p~+oo p

and the dual direct limit abutments are

°

(6)

on = l,l;m . Hn- 1 (C*/F 'H p ... +oo

-p

C*). (E P,q)

We will call the induced partition of

00

(EP,q)

dual partition of

00

°

Corollary 1.2 •

n C

The hypotheses being as in Corollary 1.1°,

is exact.

"denumerable direct sum":

Suppose also that the filtered objects

are co-complete, for all integers

limit abutments ~:

the

(p,q) EZXl','

suppose in crldition that the functor

AW~>A

(p,q) EZx.?

°

'H

n

n.

Then the dual direct

are zero, for all integers

n.

The proof shows, more generally, that if

cular integer, such that

n C

n

is co-Hausdorff and

is a partiCn + l

is

"co-complete but not co-Hausdorff" as filtered objects, then the (n+l) 'st dual direct limit abutment Corollary 1.2.1°.

,~n+l

is zero.

Under the hypotheses of Corollary 1.2°, we

have also that

for all integers Note:

n.

If the hypotheses are as in the Note to Corollary 1.2°,

then the proof of this Corollary shows that

355

Filtered Cochain Complex for the particular

n.

Proofs of Corollaries 1.1°, 1.2° and 1.2.1°:

Follow immediately

from Corollaries 1.1, 1.2 and 1.2.1 respectively applied to the dual exact couple (2°)

(which is the exact couple of the dual

filtered cochain complex Remark.

°

C*

in

AO)

in the dual category AO.

Of course, Corollaries 1.3 and 1.4, and Remarks 1 and

2 following Corollary 1.4, have similar dual statements which follow immediately by duality. For later use, we record a Lemma. Lemma 2.1.

Let

(g,f)

denote the map of exact couples de-

scribed in the Note to Theorem 1°. (8)

g -1 (1m t s )

(9)

°v

s

= 1m (( °t) s ).

Also, if the t-divisible part of ible part of

Then for all integers

V

exists, then the

°t-divis-

exists, and in fact

g-l(t-divisible part of

V) = (Ot-divisible part of °V).

(10) for all integers

n,P.

And, if the abelian category

A

is

closed under denumerable direct products and is such that the functor, "denumerable direct product", is exact, then (11)

g-l(infinitely t-divisible part of (infinitely

For all integers

°t-divisible part of

s,

And, if the t-torsion part of

V

exists,

V)

=

°V).

356

Section 8 (13)

(t-torsion part of

V)

C

1m g.

Also, we always have that (14)

9 -1 (Ker t) = Ker(°t) + Ker (g) •

We first prove equations (8), (10), (12) and (14).

Proof:

the Exact Imbedding Theorem,

By

([I.A.C.]) it suffices to prove

these equations in the case that the abelian category

A

is

the category of abelian groups. The commutative diagram with exact rows

(6) can be re-

written as (6' )

To prove equation (8), we must show that if gP,q(x) Elm(t s ) then

iff

s gP,q(x) E Im(t ).

gP,q(x) E Im(t s ) tion on

s.

Case 1.

s = 1.

then

Then

of the top row in

xElm«Ot)s).

x ECVP,q,

Clearly, if

then

xElm«Ot)s),

Therefore, it suffices to show that, if xE Im«Ot)s).

This is proved by induc-

gP,q (x) E 1m (t).

Therefore by exactness

(6 ' ), °hP +1, q(gP,q (x»

=

tive diagram(6'), it follows that °hP,q(x)

o. =

O.

From the commutaTherefore

x E 1m (Ot) •

Case 2.

s > 2.

have that x E Im(Ot),

Then since

gP,q (x) E Im(t). say

x = (Ot) (y),

gP,q(x) E Im(t s ),

a fortiori we

Therefore by Case 1, we have that where

y E vP+l,q-l.

Filtered Cochain Complex Then since of (6')

gP,q(Ot(y»

=gP,q(x)

we have that

Im(t s ),

t(gP+l,q-l(y»

from commutativity

=ts(z), there exists some

( p+l,q-l s-l tg (y)-t (z»=O, so by exactness of the top row of (6'), there exists eEE P + l ,q-l such that zEV

P+s+l q-s ' •

E=

357

Then

kP+1,q-l(e)=gP+l,q-l(y)_tS-1(z).

°vp + l

,q-l,

But then

(Ot) (y') = (Ot) (y) _ 0 = x,

gP+l,q-l(YJ_kP+l,q-l(e)=ts-l(z).

and

Y'=y_ Ok P + 1 ,q-l(e) E gP+l,q-l (y') =

Therefore gP+l,q-l(Y')Elm(t S - 1 ).

s y' = (Ot - l ) (x )' there exists o (Ot) s (x ) = (OtX (Ot s - l ) (x »

By the inductive assumption, E v P + s ,q-s xo· (Ot) (y') =x,

But then so that

0

xE Im(Ot)s,

That proves equation (8).

0

as required.

To prove equation (lO),notice

that under the identification of the diagrams (6) and (6'), we Since

is the n'th coboundary

in the cohomology sequence of the short exact sequence of cochain complexes (5 ), it follows that p

which proves equation (10). (12')

Also, it follows likewise that

Im(gP-l,n-p+l) =rm(d n( » = Ker(H n + 1 (F C*) +Hn+l(C*». p p .

The (p,n+l-p)'th component of the map is

n l H + (inclusion:

F C* p

-+

(p,n+l-p)'th component of ' Hn+l (.lonc l uSloon:

F C*

of equation (12'). compon'ent of being

P

-+

F t

C*) •

C*).

S

of bigraded objects

Therefore the kernel of the

is contained in the kernel of

But this latter is the right side

Therefore by equation (12') the (p,n+l-p)'th

Ker t S

is contained in

true for all integers

Equation

p-s S

t

n,p,

rro(gP-l,n- p +l). This

we obtain equation (12).

(14) is easily proved by diagram chasing.

358

Section 8

This proves equations (8), (10, (12) and (14). prove equations

(9),(11) and (13).

equation (8), for all integers

s

It remains to

Equation (9) follows from

~

1, and from section 2, part

(1) of the Lemma, just preceding Theorem 3 following Definition

5. Next, let us prove equation (11). °t-divisible part of V).

Clearly,

(infinitely

°V)cg-l(infinitely t-divisible part of

Therefore it suffices to prove that

(11')

g-l(infinitely t-divisible part of

V)c

(infinitely °t-divisible part of Let

C*/\

C*.

Then the exact couples (2

denote the completion of the filtered cochain complex 0

of

)

identical.

While the exact couples

C*

C*"

and to

natural map

C*

denote (2) for

C*

and of

-+

C*'\

induces a map, call it

C*I\

C*

,

into (2)

let

..j\

for

denote

constructed in the Note to Theorem 1 C*/\.

are

(2) of Theorem 1 applied to

Then

log = g/\ •

0

of exact

1 ,

C*'\ •

Let

(2/\)

for

C*I\

and let

V

denote the map of exact couples from

complex

C*'\

are not in general identical, nevertheless the

couples from (2) for

(f ,g/\)

°V).

(2°) into

(2/\)

for the filtered cochain

Therefore, to prove (11') it

suffices to prove that (11'/\)

(g/\)-l(infinitely t/\-divisible part of

(~nfinitely °t-divisible part of That is, replacing

C*

by its completion

c*/\,

vA)c °V). it suffices to

prove equation (11') in the case that the filtered cochain complex

C*

is complete.

But then

Filtered Cochain Complex lim Vp,n-p = lim Hn(F C*) = "Hn - l , p':f-oo p':+oo P zero by Corollary 1.2. of

V) = O.

359

and this latter is

Therefore (infinitely t-divisible part

Therefore, to prove equation (11), it suffices to

prove that (II")

Ker gc (infinitely °t-divisible part of

But, by definition of "infinitely

°V).

°t-divisible part", we

have that (infinitely

°t-divisible part of

This latter clearly contains

=

n H (C* IF f*) 1. But by p+ Ker(gP,n- p ). Therefore

n 1m [H (C*)

equation (10), the last group is (infinitley

°V)p,n-q

->-

°V)~

°t-divisible part of

Ker g,

proving equa-

tion (11"), and therefore completing the proof of equation (11). Finally, it remains to prove equation (13). (t-torsion part of subobjects of

V,

V)

is by definition the supremum of the s

Ker (t ),

for

s > 1.

Therefore equation (13)

follows from equation (12). Theorem 2.

Q.E.D.

Let the hypotheses be as in Theorem 1.

addition that the

In fact, the

abeli~n

category

able direct products (respectively:

A

is closed under denumer-

denumerable direct sums)

and that the functor "denumerable direct product": "denumerable direct sum":)

Suppose in

AW'VU> A is exact.

(respectively:

Then the natural

mapping frQm the exact couple (2°) of Theorem 1° into the exact couple (2) of Theorem 1 induces an isomorphism on the left de-

360

Section 8

fects (respectively: Proof: 9

on the right defects).

Equations (9) and (11) of the previous Lemma imply that

induces a monomorphism from (Ot-divisible part of °V) (infinitely Ot-divisible part of (t-divisible part of V) (infinitely t-divisible part of

into

OV)

V).

Comparing with the isomorphism in section 7, Remark 1 following Definition 5, it follows that

g

induces a monomorphism from

the left defect of the exact couple (2 0 of the exact couple (2).

into the left defect

)

On the other hand, by equation (12)

of the preceding Lemma with

s=l,

we have that

Ker tc 1m g. Therefore (15)

(Ker t)

n

(t-divisible part of

V) c 1m g.

Fix an exact imbedding from some exact full subcategory

A'

of

A

such that

that is a set into the category of abelian groups Cn,F Cn ,

sible part of gers

p,q,n.

p

(t-divisble part of

°V)p,q Identify

V)p,q

are all objects in

A'

A',

and

(Ot-divi-

for all inte-

with its image under the imbedding.

Then if (16)

xE [(Ker t)

n

(t-divisible part of

then by equation (15) we have Then by equation (16)

V)]P+l,q,

x = 9 (y), for some

g (y) = x E Ker t,

so that

By equation (14) of the Lemma, we therefore have

y E °vp,q. YE 9

-1

(Ker t).

y=z+w,

Filtered Cochain Complex where

(Ot) (z) = 0

and

g (w) = O.

361

Therefore

x=g(y) =g(z+w) =g(z) +g(w) =g(z),

g(z) =x.

By equation (16), we have that z E [g-l (t-divisible part of

V) ]p,q.

But then by equation (9) of the Lemma, of °V) •

Since also

z E [(Ker °t) Since

g (z) =x,

n

(Ot) (z) = 0,

we have that,

(Ot-divisible part of °V) ]p,q.

and

x E [(Ker t) n (t-divisible part of

is arbitrary, it follows that [(Ker °t)

zE (Ot-divisible part

gP,q

induces an epimorphism

n (Ot-divisible part of °V) ]p,q

(t-divisible part of

V) r+l,q

onto

n

[(Ker t)

V) }p+l,q, for all integers

p,q.

Combining this last result with the isomorphism in section

7, Remark 1

following Definition 5, it follows that

g

in-

duces an epimorphism from the left defect of the exact couple (2

0

)

onto the left defect of (2). The parenthetical assertion follows by duality. Let

A

be an abelian category, and let

Q.E.D.

C*= (F C*) P pE;l

be a filtered cochain complex in the abelian category

A.

Then by-Theorem 1, we have the spectral sequence (1) of the filtered cochain complex

(FpC*)pE;l'

By Theorems 1 and 1

0

,

we

have two exact couples, the exact couple (2) of Theorem 1 and a a the exact couple (2 ) of Theorem 1 , both of which have the

~ spectral sequence (1).

By the Note to Theorem 1

a natural map of bidegree (0,0);

0

,

we have

362

Section 8

of conventional bigraded exact couples starting with the inte0

ger one.

By Theorem 1

this map

,

e

induces the identity map

on the associated spectral sequences.

However, the map

e

definitely is not in general an isomorphism of exact couples. In fact,

(2°) and (2) in general have different direct limit

abutments, different inverse limit abutments, and induce different partitions of (E~"q) (p,q)E 7 X7 of the spectral sequence. However, by Theorem 2, if the abelian category

A

is closed

under denumerable direct products and denumerable direct sums, and if the functors "denumerable direct product " and "den um-

e

erable direct sum" are exact, then the mapping

of exact

couples induces an isomorphism on the left defects and on the right defects of the exact couples (2

0

)

and (2).

We will call the exact couples (2) and (2 0 C* ,

tered cochain complex

)

of the fil-

the exact couple, respectively:

the dual exact couple, of the filtered cochain complex When the partial abutments of (2), respectively of (2 0

C*. ),

exist, we shall call these the partial abutments, respectively: the dual partial. abutments, of the filtered cochain complex (F C*)pE7. p

When they exist, we will refer to the partition,

respectively: Remark 1.

the dual partition, of

(E~,q) (p,q)E7 x, .

Under the hypotheses of Theorem I, suppose that the

co-completion, co-comp(C*), of the filtered cochain complex C* = (F C*) p pE;r

exists.

Then, as in the sequence of Remarks,

Remark 1 preceding Theorem 1 co-comp (C*)

-+-

C*

0

,

we have the natural map:

Filtered Cochain Complex

363

of filtered cochain complexes, which induces a mapping:

on the associated dual exact couples.

Here, co-comp(20) de-

notes the dual exact couple of the filtered cochain complex co-comp(C*),

i.e., the exact couple obtained by the construc-

tion of Theorem 1° applied to co-comp (C*).

By the dual of

Remark 1 preceding Theorem 1°, we have that the map

SO

in-

duces an isomorphism on the spectral sequences, but in general does not induce an isomorphism on the partial abutments, if they exist; and in fact different partitions of

(2°) and co-comp(20) in general induce (EP,q)

(p,q) EIUl' •

00

We will call the

exact couple co-comp(20), the dual ~ couple of the co-completion of the filtered cochain complex Remark 2.

If

C*A

C*.

exists, then by the dual of Remark 2 pre-

ceding Theorem 1°, we have that

C*A

and

C*

have canonically

isomorphic dual exact couples. Remark 3.

If

C*A,

1\

co-comp (C*) and co-comp(C* )

all exist,

then they all have dual exact couples by Theorem 1°. by the dual of Remark 3 preceding Theorem C*

1°,

in fact

However, C*A

and

have canonically isomorphic dual exact couples, both iso-

morphic to (2°); and similarly co-comp(C*) and co-comp(C*/\) have canonically isomorphic exact couples, both isomorphic to co-comp (2°). Remark 4.

Therefore, if

such that

C*A

C*

is a filtered cochain complex

and co-comp (C*) both exist (condition auto-

matic if the abelian category

A has denumerable direct prod-

ucts and denumerable direct sums), then we have the four, in

364

Section 8

general different, exact couples, in the following diagram: (16)

S° ° )->(2)->(2 e 13 A ). co-comp(2 ° )-->(2

These exact couples are, from left to right: the dual exact couple of the co-completion of C*; the exact couple of pletion of

C*.

C*;

the dual exact couple of

C*; and the exact couple of the com-

(The mappings

13

S°

and

of exa~t couples

are given by Remark 1 preceding Theorem 1°, and by Remark 1 above, respectively. Theorem 1°).

The mapping

Notice also,

e

is given by the Note to

(by Remark 3 preceding Theorem 1°,

and Remark 3 above), that the outermost exact couples,

A

(2 )

and co-comp(20), in the diagram (16), can be interpreted alternatively as being, respectively, the exact couple and the dual exact couple of the filtered cochain complex, the co-completion of the completion of (17)

C*:

(2A) "" exact couple of co-comp (C*A), co-comp(20) "" dual exact couple of co-comp (C*A).

We have seen (by Theorem 1°; by Remark 1 preceding Theorem 1°; and by Remark 1 above), that these four exact couples, in the diagram (16), all have canonically isomorphic spectral sequences (the isomorphisms being induced by

sO,e,

and

13).

The spec-

tral sequence that these four exact couples have in common is the spectral sequence of the filtered cochain complex

C*.

However, as we have already noted, these four exact couples are all, in general, no two isomorphic; and in general have, each pair, different partial abutments, and induce in general four different partitions of

(E~,q) (p,q)E~x~.

However, the next

Filtered Cochain Complex

365

corollary shows that, under very mild conditions on the abelian category

A, they all have canonically isomorphic

left and right defects. Corollary 2.1.

Let

A

be an abelian category, such that de-

numerable direct products (respectively:

denumerable direct

sums) exist, and such that the functor "denumerable direct product"

(respectively:

"denumerable direct sum") is exact.

Suppose that co-comp(C*) exists (resp.:

that

C*A

exists).

Then each of the three maps of exact couples:

induce isomorphisms on the left (respectively: ~:

The proof shows a little bit more.

the hypothesis that "co-comp (C*) the maps

8

and

S

(resp.:

(resp. : so

phisms on the left (respectively: Proof:

and

8)

right)

right) defects.

Namely, if we delete C*A)

exists", then

still induce isomordefects.

We prove the non-parenthetical part of the Corollary.

The parenthetical part then follows by duality. That

e

induces an isomorphism on left defects follows

from Theorem 2.

co-comp(C*)

and

C*

have the same exact

couple (see Remark 2 preceding Theorem 1). (exact couple of co-comp(C*»

Therefore

and (exact couple of

C*)

have the same left defect. But, by Theorem 2 applied to co-comp(C*), (exact couple of co-comp (C*» co-comp1C*»

and (dual exact couple of

have the same left defect.

Section 8

366

Also by Theorem 2, (exact couple of

C*)

and (dual exact couple of

C*)

have the same left defect. Combining these three results, it follows that (dual exact couple of co-comp (C*» of

C*)

and (dual exact couple

have the same left defect.

SO

Otherwise stated,

induces an isomorphism on left defects.

Similarly, since

C*

and

C*A

have the same dual exact

couple (by Remark 1 above); and since by Theorem 2 applied to C*A,

the (dual exact couple of

C*A) and the (exact couple of

C*A) have the same left defect; and since likewise by Theorem 2 applied to

C*,

(exact couple of

the (dual exact couple of C*)

and the

have the same left defect; it follows

likewise that (exact couple of C*A)

C*)

C*)

have the same left defect.

and the (exact couple of

S induces an iso-

That is,

morphism on left defects.

Q.E.D.

Corollary 1.2, respectively Corollary 1.2 Corollary 1.5.

A

Let

0

imply

be an abelian category such that de-

numerable direct products of objects exist and denumerable direct sums of objects exist.

Suppose also that the functor,

"denumerable direct product" (respectively: direct sum"):

A.

complex in

AW'VV>

A

is exact.

Let

C*

"denUIl'arable

be a filtered cochain

Then the inverse limit abutments (respectively:

the direct limit abutments) of the exact couple (2'\) of the completion of co-comp(2

0

)

C*

(respectively:

of the co-completion of

of the dual exact couple C*) are all zero.

Filtered Cochain Complex Proof:

367

Follows immediately from Corollary 1.2 (respectively:

Corollary 1.2°) (respectively:

applied to the filtered cochain complex to the filtered cochain complex

Thus, under the

C*A

co-comp(C*».

(extremely mild) hypotheses of Corollary

°

1.5, the exact couples co-comp (2 ) and (2 A ) on the extremes of the diagram (16) have the property that, "all the information is contained in the inverse limit abutment, resp.:

the direct

limit abutment"- in the precise sense that the direct limit abutment, resp.:

the inverse limit

abutmen~

of the exact

A couple co-comp(20), resp.: of the exact couple (2 ), is zero. Since, by Corollary 2.1, we know that all of the exact couples in the diagram (16) have the same left and right defects, it follows that the associated gradeds of the inverse limit abutment of

co-comp(20) and of the direct limit abutment of (2A)

are isomorphic.

(Both are isomorphic to

defect)P,q»)/(right defect)P,q).

[Ker (EP,q ... (left 00

This suggests attempting to

find a natural mapping between these filtered objects that induces an isomorphism on the associated gradeds. mapping of exact couples

8

0

8

0

8°:

8

0

co-comp(20) ... (2 A ),

8

0

8°

The natural

in the diagram (16),

simply induces the zero map on

both the direct limit and the inverse limit abutments

(since

the direct limit abutmentsof co-comp(20) and the inverse limit abutments of (2A) are both zero), so this mapping cannot accomplish that end.

However, as the next proposition and

corollaries (especially Corollary 3.2.1) below show, there is a natural map of filtered objects in the opposite direction that induces an isomorphism on the associated gradeds. Proposition 3.

Let

C*

be a filtered cochain complex in an

Section 8

368

abelian category

A.

Then, for every integer

Hn(C*)

(see section 6, the Remark following Definition 3).

equivalent description of this filtration on

for all integers (1)

If

n

is

and therefore inherits a filtration from

a subquotient of en

n,

p,n.

Hn(C*)

An

is,

Then

is an integer such that the n'th direct limit

partial abutment of the exact couple (2)

exists (condition automatically satisfied if the abelian category

A is closed under denumerable direct sums), then the

natural map

is a map of filtered objects for the integer

~(Fp('Hn»

=FpHn(C*),

for all integers

n.

In fact, then

p,

and the induced mappings

are epimorphisms, for all integers (2)

Similarly, if

n

dual inverse limit abutment

p.

is an integer such that the n'th

lI~n

(that is, the n'th inverse

limit abutment of the exact couple (2°) of Theorem 1

0

)

Filtered Cochain Complex

369

exists (condition automatically satisfied if the abelian category

A

is closed under denumerable direct products), then

the natural map

is a map of filtered objects for the integer

n.

In fact, then

and the induced mappings

are monomorphisms for all integers (3)

p.

If the hypotheses of (1) and (2) above both hold for

the fixed integer

n,

then we have the diagram

of filtered objects and maps of filtered objects. each integer

p,

Then for

the induced diagram

is the canonical factorization of the mapping G (11 0

p

0

graded

11):

G (' Hn) ->- G p p

G*(Hn(C*))

of

(,,~n). Hn(C*)

Otherwise stated, the associated is canonically isomorphic to

the image of the composite mapping

Section 8

370

Proof.

First, that the filtration on

garding

Hn(C*)

F (Hn(C*» p

as a subquotient of

=Im(Hn(F C*)~ Hn(C*»

Hn(C*) Cn

induced by re-

is such that

follows easily from an appli-

p

cation of the Exact Imbedding Theorem ([I.A.C.]).

(In fact,

by the Exact Imbedding Theorem, it suffices to prove that assertion in the case in which

A is the category of abelian

groups.

~

But then (see section

tion 3) an element of sen ted by a cocycle

Hn(C*) n u EC

the Remark following Defini-

is in

iff it is repre-

such that

this is so iff the element of

Hn(C*)

But clearly

is in the image of the

Hn (F C*) ~ Hn (C*) ) •

map:

p

Next, let us prove the assertions (2) of the Proposition. That is, suppose that

n

is an integer such that

"~n

exists.

Then, the definition of the filtration on

is (17)

(see section 7, Note 2 to Theorem 2, equation (8.1); and Theorem 1

0

of this section).

A portion of the long exact sequence

of cohomology of the short exact sequence of cochain complexes

o ~ Fp C* ~ C* ~

(C* /F C*) p

~

0

is the sequence n-l

(18)

n

d ... d _ H n ( F C*) + Hn (C*) + Hn (C*/F C*)--...y p

p

Filtered Cochain Complex

371

It follows from equation (17) and this last exact sequence that (In fact, fix an exact imbedding from an exact full subcategory

°f

the category

A'

of

A

abelian groups, such that

that is a set into "on , F p ( "Hon) , H

Hn(C*/F C*) are objects in A'. Then P n by equation (17), an element x E ,,~n is in F ("H ) iff the Hn(F p C*) , Hn(C*)

and

°

p

image of y EH y

n

x

(C*)

in

y EF

p

will map under

Hn(C*/F C*) p

happen iff (H

n

y

(C*) ) •

asserted) •

Therefore an element

is zero. lJ

°

is zero.

into

F

p

(,,6°)

iff the image of

--

By exactness of

(18~

Im[Hn(F C*)~ Hn(C*)], i.e., iff p -Therefore {"o)-l(F ("Jrn»=F (Hn(C*» ~ p p ,

is in

It follows in particular that

IJ

°

~ Gp (lI~n)

as

is a mapping of

filtered objects, and that the induced mappings G (H n (C*» p

this will

G (1J0): p

are monomorphisms for all integers

This proves the assertions in (2) of the Proposition.

p. The

assertions in (I) of the Proposition follow from those in (2) by applying part (2) to the dual filtered cochain complex in the dual category

O

A

•

And the assertions

(3) follow

Q.E.D.

immediately from (1) and (2). Corollary 3.1.

8*

Suppose that the abelian category

A

is such

that denumerable direct products exist, and such that the functor "denumerable direct product": AW'VV>A

is exact.

be a filtered cochain complex in the abelian category that the filtered object

C

n

Let (0 )

on n "H = lim H (C* IF C*) p"" ~oo p

Let A

C* such

is complete, for all integers

n.

372

Section 8

be the n'th object of the inverse limit abutment of the dual exact couple of

C*,

for all integers

n.

Then the natural

mapping of filtered objects:

is an epimorphism, and induces an isomorphism on the associated gradeds. of

In fact, as filtered objects,

Hn(C*).

"~n

It follows that

"~n

is the completion

is complete, and that

is "complete but not Hausdorff", as filtered objects. kernel of the natural mapping (2)

11

o

Hn(C*)

The

is

Ker(l1o) =G+oo(Hn(C*));>:lj,m l Hn-l(C*/FpC*), p++oo

for all integers Proof:

n.

By Chapter 3 of the main text below, Corollary 1.1 and

the Note to Corollary 1.1, we have the short exact sequence (3)

(Namely, take

C~

~

= C*/F C*

tion (0) above).

i

for all integers

It follows that the mapping

phism, as asserted.

i, 11

and use equa0

is an epimor-

By Proposition 3, part (2), we know that

(4)

for all integers

p.

From equation (4), and the fact that

11

o

is an epimorphism, it follows from the Third Isomorphism Theoo 0 n rem that Gp (11 ): Gp ("H n ) + Gp (H (C*)) is an isomorphism, for all integers

n,p.

In general, for any filtered object

H,

have by definition (see section 6, Remark 1 following Corol1ary 4.1) that

G+ H = 00

n F H. Therefore by equation (4) and pEi?' P

we

Filtered Cochain Complex

373

section 2, the Lemma following Definition 5, part (1), it follows that

Since

,,§n

is an inverse limit abutment associated to an exact

couple (namely, to the dual exact couple (2 the filtered cochain complex have that

,,~n

is complete.

and therefore Ker(~

o

G+00

(,,~n) = 0 •

n

) =G+oo(H (C*».

0

)

of Theorem 1

0

of

C*), by section 7, Theorem 2, we In particular,

,,§n

is Hausdorff,

Therefore equation (5) implies that

This observation and equation (3) imply

conclusion (2) of the Corollary. Since of

~o

Hn(C*).

quotient-object Hn(C*).

is a quotient-object

Equation (4) and the fact that

phism implies that

from

,,~n

is an epimorphism,

Fp

,,~n

(,,~n) = ~o (F p of

n H (C*)

n (H (C*») •

~o

is an epimor-

Therefore, the

has the:til tration induced

That is,

(6) as filtered objects. (7)

Since

,,§n

is complete, it follows that

is complete.

Therefore, by definition (see section 7, the Remark following Corollary 7.1), we have that Hausdorff".

Hn(C*)

Also, it follows readily from equation (7)

using section 7, Proposition 4) that completion of

is "complete but not

Hn(C*)

Hn(C*)/G+oo(Hn(C*»

as filtered object.

(e.g., is the

This observation

and the isomorphism (6) of filtered objects implies that

,,~n

is the completion of

Q.E.D.

Hn(C*)

as filtered object.

Section 8

374

Corollary 3.1

0

Suppose that the abelian category

•

A

is such

that denumerable direct sums exist, and such that the functor "denumerable direct sum":

AW'V'u> A

is exact.

Let

filtered cochain complex in the abelian category the filtered object

n C

C*

A,

be a such that

is co-complete, for all integers

n.

Let (0)

be the n'th object of the direct limit abutment of the exact couple of

C*,

for all integers

n.

Then the natural mapping

of filtered objects: jJ: 'Hn+Hn(C*)

(1)

is a monomorphism, and induces an isomorphism on the associated gradeds. tion of that

In fact, as filtered objects, Hn(C*).

Hn(C*)

The

is the co-comple-

is co-complete, and

cokernel of the natural mapping

jJ

is:

Cok(jJ) =G_oo(Hn(C*)) '''d!ml Hn+l(FpC*), p++oo

for all integers Proof:

n.

Follows by applying Corollary 3.1 to the dual filtered

cochain complex

8*

o Corollary 3.1 .1. lary 3.1°. in

'H

n

is "co-complete but not co-Hausdorff", as fil-

tered objects. (2)

It follows that

n

'H

A.

Let

in the dual category Let

C*

A

O

A

Q.E.D.

•

be an abelian category as in Corol-

be an arbitrary filtered cochain complex

Then for each integer

n, Hn(co-completion

C*)

is "co-

complete but not co-Hausdorff" as filtered object; and we have

Filtered Cochain Complex

375

a canonical isomorphism of filtered objects,

where

'H

n

is the n'th direct limit abutment of the spectral

sequence of the filtered cochain complex Corollary 1.1.

C*

as defined in

Also,

n n G-00 (H (co-comp C*)) ~ liml H (F _C*) -+ p' p-++oo for all integers Proof:

n.

As noted in Remark 2 preceding Theorem 1°, the natural

map of filtered cochain complexes co-comp (C*) -+ C* induces an isomorphism on the exact couples as defined in Theorem 1.

Therefore, replacing

C*

by co-comp(C*) if necessary,

the Corollary reduces to the caSE in which

C*

is co-complete.

Then the conclusions follow from Corollary 3.1°. Remark:

Under the hypotheses of Corollary 3.1°, suppose in

addition that the abelian category

A is such that the func-

tor "denumerable direct limit" is exact.

Then of course every

A is "co-comp. but not co-Hausd." Also,

filtered object in

in this case, obviously

l~ml:: O.

Therefore, in this case,

p++ro

co-comp C* =F_roC*, and the conclusions of Corollary 3.1 °.1 can then be written:

for all integers 1.3.

n.

This gives another proof of Corollary

Thus, in the special case that the abelian category

A

Section 8

376

is such that the functor "denumerable direct limit" is exact, 0

Corollary 3.1 .1 reduces to Corollary 1.3. Corollary 3.1.1. lary 3.1. in

A.

Let

Then

Let C*

A be an abelian category as in Corol-

be an arbitrary filtered cochain complex

Hn(C*A,

is "complete but not Hausdorff" as fil-

tered object, and we have a canonical isomorphism of filtered objects

,,~n

where

is the n'th dual inverse limit abutment as defined

in Corollary 1.10 chain complex

,

C*.

of the spectral sequence of the filtered coAlso,

canonically, for all integers Proof:

n.

Follows by applying Corollary 3.1 0 .1 to the dual fil-

tered cochain complex Corollary 3.2.

8*

AO •

in the dual category

Suppose that the abelian category

A is such

that denumerable direct sums and denumerable direct products of objects exist, and is such that the functors "denumerable direct sum" and "denumerable direct product", exact. category

Let

A

C*

are

be a filtered cochain complex in the abelian

such that

for all integers

Aw'VV> A,

n.

n C

is both complete and co-complete

Then for every integer

n,

Hn(C*),

as

filtered object, is both "complete but not Hausdorff" and "cocomplete but not co-Hausdorff". (1)

We have the natural mappings

377

Filtered Cochain Complex

which are mappings of filtered objects that induce isomorphisms on the completions of the co-completions.

o n Hn (C* YG +00 (H (C*) ) R> "Hn

(2)

Also Hn (C*)

is the completion of

as filtered object, and (2°)

F_ooHn(C*) ~ 'H n

is the co-completion of

filtered object, for all integers (3)

and

n.

Follows immediately from Corollary 3.1 and 3.1

Corollary 3.2.1.

as

In addition,

G+ooHn(C*)R> liml Hn-l(C*/F C*), p++oo p

for all integers Proof:

n.

Hn(C*)

Let

0

•

A be an abelian category obeying the

hypotheses of Corollary 3.2.

Let

C*

be an arbitrary filtered

cochain complex in the abelian category

A.

Then consider the

sequence (16)

o ",0 0 e S 1\ co-comp ( 2 ) ~> (2 ) + (2) + (2 )

of exact couples described in Remark 4 following Theorem 2 above. Then for all integers

n

there is induced a natural map of

filtered objects: (17)

(n'th direct limit abutment of

(21\»

+

(n'th inverse limit abutment of co-comp (2 in the opposite direction to the composite map exact couples in the sequence (16).

0

S

», 0

e

0

SO

of

The map of filtered objects

(17) induces an isomorphism of filtered objects on passing to the completion of the co-completion.

378

Section 8

Notes:

The proof shows that the map (17) naturally factors,

1.

as map of filtered objects, through

Hn(co-comp(c*A)); that this

latter filtered object is both "co-complete but not co-Hausdorff" and "complete but not Hausdorff"; and that the completion of

n A H (co-comp(C* )) is the right side of equation (17), while the co-completion of

n A H (co-comp (C* ))

is the left side of equa-

tion (17). 2.

The kernel of the map (17) is

A limit abutment of (2 )), morphic to is

and is also

liml Hn-l(C*/F C*). p-++oo p

G+ oo (n'th direct

G+ro(Hn(C*)),

The cokernel of the map (17) 1\

G_oo(n'th inverse limit abutment of co-comp(2 )), G_oo(Hn(C*)),

also Proof:

and is isomorphic to

As noted in Remark 4

and is iso-

and is

liml Hn+l(F C*). p-++oo p

following Theorem 2, the exact

o

couples (21\) and co-comp(2 ) are canonically isomorphic to the exact couple and, respectively, the dual exact couple, of the 1\

filtered cochain complex co-comp(C*). then follow from Corollary Remarks 1.

,,~n teger

3.:~.

Q.E.D.

Under the hypotheses of Proposition 3, if

both exist for a given integer n,

The Corollary and Notes

and for all integers G

p

EP,n-p 00

(lJ

o 0

p,

n,

n

and

then for the fixed in-

we have a diagram:

lJ)

«map induced bye) (identity map)

'H

EP,n-p 00

Filtered Cochain Complex

EP,n-p.

a subquotient of

00

R,

'

resp. :

379

S, denotes the thereby

induced relation (not in general a mapping) from subquotient Gp (,,~n) ,

,

map induced by identity map.

into

00

It is not difficult to see that this diagram of A

~

c*)/(infini~e) p t-tors~on

~

t oHn(F f*) / (infini ~e ) p+ t-tors~on

is commutative.

But since

= Hn (natural

A,

map induced by

Hn(F C* nat.> C*/F C*) ~~e7 t)n(ig] p map p+l >f~n~tely t di visible part of Hn(C*/Fp+lC*»

relation induced by hP,n-p

hP,n-p

is commutative.

It is equivalent to prove that the diagram of objects

and relations in the abelian category

Hn(F

The bottom map is the

whcih by the Note to Theorem 1 0 is the

G

relations in the abelian category Proof:

EP,n-p.

relation induced by ~p,n-p

~p,n-p = Hn (inclusion) and

projection), this latter follows from

commutativity of the diagram of objects and mappings in the abelian category

A,

Section 8

380

> Hn(C*/F

1 H

n

n H (G

2.

p+l

C*)

(indu,ionl

p

(C*)).

Q.E.D.

As noted in Remark 4 following Theorem 2, under the

hypotheses of Theorem 1, we have that the composite

S c80 SO

of the sequence of maps of exact couples (16) induces an isomorphism of the associated spectral sequences, and therefore also of the corresponding

E co -terms.

tion 3, this isomorphism of

As noted just prior to Propos i-

Eoo-terms induces, by passing to the

subquotients, an isomorphism from the associated graded of the inverse limit abutment of co-comp(2 0 of the direct limit abutment of

)

A (2 ).

onto the associated graded By commutativity of the

diagram in Remark 1 above, it follows that this just described isomorphism of associated gradeds is the inverse of the isomorphism of associated gradeds induced by the map of filtered objects (17) of Corollary 3.2.1.

A be an abelian category obeying the hypotheses of

Let

Corollary 3.2.1, and let complex in

A.

C*

be an arbitrary filtered cochain

Then, by Corollary 3.2.1, we have that (the A

direct limit abutments of the exact couple (2 ) of the completion of

C*)

and (the inverse limit abutments of the dual exact

o couple co-comp(2 ) of the co-completion of by very much.

C*)

do not differ

In fact, by Corollary 3.2.1, Note 1, these both

do not differ very much from the sequence of filtered objects

Filtered Cochain Complex (H n (co-comp(C* "

»)n~.

381

We call these latter filtered objects

the integrated partial abutments of the filtered cochain complex

C*,

and occasionally will denote them

Then, by Corollary 3.2.1, the completion (respectively: co-completion) of the n'th integrated partial abutment C*

Hn

of

is canonically isomorphic to the n'th direct (respectively:

inverse) limit abutment of (2")

(respectively: of co-comp(2 o ».

A portion of Corollary 3.2.1 can be summarized as Corollary 3.2.1.1. A

(The integrated partial abutments).

Let

be an abelian category such that denumerable direct products

and denumerable direct sums of objects exist, and such that the functors, "denumerable direct sum" and "denumerable direct product", AW'VV>A chain complex

G)

are exact.

C*

in

A,

Then, for every filtered co-

we have

A conventional bigraded spectral

sequence starting

with the integer one, the spectral sequence (1) of the filtered cochain complex

C*.

~ A subobject

(right defect), and a quotient object (left

defect), of the bigraded obJ'ect

(EP,q) 00

(right defect)

®

C

(p,q)E'!'x'!"

such that

Ker ((E~,q) (p,q) Ez x71 ... (left defect»;

A sequence

Hn,

nE7l,

of filtered objects, called

the integrated partial abutments, each of which is "complete but not Hausdorff" and "co-complete but not co-Hausdorff"; and ~ An isomorphism of bigraded objects from

onto the sub-quotient of

[Ker( (E~,q) (p,q)E'!'X7l "'(left defect»

(EP,q) 00

(p,q)E'!'x'!'.'

]/(right defect).

Given a perfectly general filtered cochain complex

C*

in

Section 8

382

A as in Corollary 3.2.1.1, the data, ~

an abelian category

Q),

and

GD

of Corollary 3.2.1.1 can be thought of as being

"the closest thing to an honest abutment that the spectral sequence of the filtered cochain complex thing, these data are clearly self-dual.

C*

has".

For one

(The partial

abutments of each of the exact couples co-comp(20),

(2°),

(2),

1\

(2 ) are not self-dual, but switch between themselves under duality).

Also, all of the data of the partial abutments of co-

comp(20) and (2") are contained in the integrated partial abutments, i.e., in the data

0, G), ®

(by the observation

immediately preceding Corollary 3.2.1.1 and by Corollaries 1.2 and 1.2°). co-comp(20),

The fact then, Corollary 2.1, that the exact couples: (2°),

(2) and

(21\)

all have the same left and right

defects can then be written equivalently as Corollary 3.2.1.2.

Under the hypotheses of Corollary 3.2.1.1,

let

respectively:

(IHn,"Hn)nEi.:'

(,§n,,,§n)nEi.:'

abutments of the exact couple (2), respectively: exact couple (2°), of the filtered cochain complex for all integers

n,p

be the partial of the dual C*.

Then

we have the short exact sequences:

(18)

The mapping

8:

(2°) ~ (2) of exact couples (of the Note to

Theorem 1°) defines, for all integers

nand

p,

a mapping of

short exact sequences from (18°) into (18), that is the identity on the middle terms. Remarks 1.

Corollary 3.2.1.2 of course explains the terminology

"integrated partial abutments" for the filtered objects

Hn,

Filtered Cochain Complex n EZ;

383

since by equation (18) they "tie together" the direct

limit and inverse limit abutments of the filtered cochain complex

C*,

while still classifying "exactly as much" of Hn

(Equation (18°) interprets

similarly viz-a-viz the direct

and inverse limit dual partial abutments). 2.

Corollary 2.1, or the equivalent formulation Corol-

lary 3.2.1.2, shows that, if a filtered cochain complex

C*

has

a non-zero left or right defect, then there is "no way" to ever

E~,q

classify this part or parts of "abutment".

by any kind of meaningful

From this point of view, the left and right defects

can be thought of as being the "cancerous part" of

EP,q 00

,

since

they can never be so classified; and on the contrary "every other part" of

is classified in the appropriate fashion

o by any of the four sets of partial abutments (of co-comp(2 ) , A of (2°), of (2) or of (2 )),

and also by the integrated partial

abutments. 3.

exact couple (2), resp.: (18), resp.:

(E~,q) (p,q)E~x~

The partition of

(18°),

(2°),

induced by the

can be described from equation

of Corollary 3.2.1.2 as follows.

First,

take the image of the composite of the monomorphisms: G (' HP +q ) .... G (H P +q ) .... EP,ql (right defect) p,q p p 00 ,

pre-image of this subgroup of to obtain

'E~/q.

E~,q/(right defect)P,q

Then of course,

°

a P , q "EP,q) The dual partition ('E 00

'

00

and then take the

(p,q)~~

in

"E P , q = EP ,q I' EP,q. 00

of

00

( EP,q)

00

(p,q)~~

can be determined similarly from (18°). 4.

From Corollary 3.2.1.1, it follows that necessary

and sufficient conditions for the integrated partial abutments of the filtered cochain complex

C*

to be an honest abutment

384

Section 8

(in the sense of section 7, Definition 1) is that the left defect and right defect both be zero. 5.

By definition of the integrated partial abutments,

and by Corollary 2.1, we have that, under the hypotheses of Corollary 3.2.1.1, that the ~

("abutment-like") data,

of Corollary 3.2.1.1 is invariant under replacing the fil-

tered cochain complex 1\

C*

co-comp(C* ).

pletion,

with the completion of its co-com(This is, of course, also true for

o the partial abutments of co-comp(2 ) and of

(21\)--which sets of

partial abutments are, in fact, completely determined by the integrated partial abutments, i.e., data

(]),

CD, 0,

as we

have already seen--but is, of course, not true in general for the partial abutments of (2) or of 6. and

0

o (2 )).

0,

Thus, again, to summarize, the data

(1),

is "the closest thing to an honest abutment" that

the spectral sequence

(1) of the filtered cochain complex

C*

has; and this data actually is an honest abutment iff both the left defect and the right defect are zero. Because of the special importance of the category of abelian groups, it is worthwhile noting that simplifications occur in Corollary 3.2.1, etc., in that case.

The notable simplifica-

tions all arise from the fact that the functor, "denumerable direct limit", is then exact. Corollary 3.2.1.3.

Let the hypotheses be as in Corollary 3.2.1.1,

and suppose in addition that the abelian category that the functor, (1)

A

is such

"denumerable direct limit", is exact.

The integrated partial abutments

Hn,

n E~ ,

Then are

co-complete, and "complete but not Hausdorff".

Filtered Cochain Complex (2)

The right defect vanishes.

(3)

For all integers

p,q,

385

we have the short exact

sequence 0-+ G HP +q -+ EP,q -+ (left defect)P,q

->-

O.

P

Note:

In this case, we have that the n'th integrated partial

abutment

n H

=

n H (co-comp (C*A»

is canonically isomorphic as

filtered object to the (n'th direct limit abutment of the exact couple (2A) of the completion of

C*),

for all integers

n.

On the other hand, the (n'th inverse limit abutment of the dual o exact couple co-comp(2 ) of the co-completion of

C*)

in this

case is both complete and co-complete, and is canonically isomorphic as filtered object to Proof: G

By Corollary 3.2.1, Note 2, we have that

(Hn(C*»

""liml Hn+l(F C*). Since "denumerable direct limit" p-++oo p n is by hypothesis exact, this latter is zero. Therefore H is _00

co-Hausdorff.

This and

(t)

of Corollary 3.2.1.1 implies con-

clusion (1). Next, by Corollary 1.3, we have that the direct limit abutment of the spectral sequence of the filtered cochain complex C*

is perfect.

Conclusion sion

(3)


Otherwise stated, the right defect is zero. of Corollary 3.2.1.1 then simplifies to conclu-

of this Corollary.

for all integers

n,

The fact that

n H

is co-complete,

and Note 1 to Corollary 3.2.1 then imply

the Note to this Corollary.

Q.E.D.

We conclude the theorems of this section with a proposition about induced mappings of filtered cochain complexes. Proposition 4.

Let

A

be an abelian category obeying the

Section 8

386

(very mild) hypotheses of Corollary 3.2.1.1. be filtered cochain complexes in

A,

and let

a map of filtered cochain complexes(*). integers

n,p,

Let

C*

and

f*:C*+ 0*

0* be

Suppose that, for all

we have that

(0)

A.

is an isomorphism in the category "suppose that

f*

(It is equivalent to say, of the

induces an isomorphism along

spectral sequences of the filtered cochain complexes 0*"). (1)

C*

and

Then The map:

(EP,q(f*» r p,q,rE'l' r> 1

induced by

f*

is an

isomorphism of spectral sequences, from the spectral sequence of (2)

C* f*

onto the spectral sequence of

0*.

induces an isomorphism on the left defects, and an iso-

morphism on the right defects. (3)

The mappings induced by

f*

on the integrated partial

abutments are isomorphisms of filtered objects, Hn (co-comp (f J\ »: Hn (co-camp (C 'Ii' » ~ + Hn (co-comp (0* A». Note 1.

In addition,

f*

couple of the completion of completion of

0*;

induces isomorphisms form the exact C*

onto the exact couple of the

and also isomorphisms from the dual exact

couple of the co-completion of

C*

onto the dual exact couple

t*J That is, a map in nthe category of filtered objects in Co(A). That is, fn:Cn+O is a map in A, for all integers n, such that fn(FpC n ) C FpOn, for all integers n,p, and n n n+l n . such that d o * 0 f = f 0 d for all 1ntegers n. C*'

Filtered Cochain Complex of the co-completion of 2.

387

0*.

If, in the hypotheses on the abelian category

A

in

the Proposition, we replace the hypothesis on the abelian cate-

A,

gory

that

"A

obeys the hypotheses of Corollary 3.2.1.1"

by the weaker hypotheses, that "the abelian category closed under denumerable direct sums (respectively:

A is products)

of objects and the functor "denumerable direct sum (respectively: product) is exact", then the proof of the Proposition shows that conclusion (1); sion i2);

the second (respectively: first) part of conclu-

and the second (respectively:

first) part of Note 1,

remain valid. Remark:

A theorem in [E.M.l is equivalent to the assertion that,

if we havt all of the hypotheses of the Proposition except perhaps hypothesis (0), then conclusion (3) of Proposition 4 holds if there exists a positive integer isomorphism, for all integers Proof:

r

is an

l p,q.

The spectral sequence; the left and right defects (by

Corollary 2.1); the integrated partial abutments (by definition); 0

and the exact couples (l') and co-comp (2

)

(by equations (17) of Re-

mark 4 following Theorem 2); of a filtered cochain complex

C*,

all actually depend, up to canonical isomorphisms, on co-comp(c~). Therefore, replacing

C*

and

0*

by the completion of their

co-completions, it suffices to prove the Proposition in the case that

C*

and

0*

are both complete and co-complete.

But then, for every integer

i

~

0,

f*

induces a mapping

from the long exact sequence of cohomology associated to the short eyact sequence of cochain complexes,

Section 8

388

O-+G+,(C*) p 1

(4)

F (C*)

{

Fp+~+i (C*)

1

--;;.0

into the corresponding long exact cohomology sequence with replacing i,

"C*".

"0*"

Therefore by the Five Lerruna and induction on

it follows that, for every integer

i..:: 0,

the mapping

f*

A,

induces isomorphisms in the category

Hn(F (C*)/F +' (C*))~Hn(F (O*)/F +' (0*)),

(5)

P

p

for all integers n

F (C ) p

1

n,p.

P

P

1

By Lerruna 1 of section 6, we have that

is complete for all integers

Therefor~ by Chapter

n,p.

3 of the main text below, Corollary 1.1 and the Note to Corollary 1.1, we have the short exact sequences

for all integers

n,p.

The map

f*

induces a mapping of these

short exact sequences into the analogous short exact sequences for

0*.

Therefore, by the isomorphisms (5) and the Five Lerruna,

it follows that the maps in the category

A induced by

f*

are

isomorphisms (6)

for all integers

~*: 8* -+ 8* category

n,p.

Applying equation (6) to the map

of the dual filtered objects in the dual abelian

AO,

we have likewise that, the maps induced by

are isomorphisms in the category

A,

f*

Filtered Cochain Complex for all integers Po = 0,

f*

n,p.

389

But, for any fixed integer

Po'

say

induces a map from the long exact sequence of coho-

mology of the short exact sequence of cochain complexes,

o -T F

C*

-T

C*+C*/F

Po

C* + 0, Po

into the corresponding long exact sequence with "C*".

Therefore, equations

fixed integer

Po'

the fixed integer induced by

f*

(6)

for all integers

and equations (6 -Po'

0

)

"0*" n

replacing and the

for all integers

nand

and the Five Lemma, imply that the maps

are isomorphisms in the category

A,

(7)

for all integers abutments", since tegers

n,

n. C

By definition of the "integrated partial n

is complete and co-complete for all in-

we have that (n'th integrated partial abutment of

the filtered cochain complex

nAn C*) (=H (co-comp(C*) )) = H (C*),

and (p'th filtered piece of the n'th integrated partial abutmentoT the filtered cochain complex for all integers "C*".

n,p; and similarly with

Therefore, equations

"0*"

replacing

(6) and (7) above imply conclusion

(3) of the Proposition. Next, notice that C*

and

vp,q

and

EP,q

in the exact couple of

are

EP,q:HP+q(G C*), P

for all integers

p,q;

and similarly with

0*

replacing

C*.

390

Section 8

Therefore, from the equations

for

C*

and

D*,

we have

and by the hypo-

induces an isomorphism along

f*

that

(6)

theses of the Proposition, f*

induces an isomorphism along

EP,q,

Therefore

for all integers

p,q.

f*

induces an isomor-

phism from the exact couple of the filtered cochain complex onto the exact couple of the filtered cochain complex implies conclusions

(1)

C*

D*.

This

and (2) of the Proposition, and also the

first partof Note 1 to the Proposition.

The second part of

Note 1 to the Proposition follows from the first part of that 000

Note applied to the induced mapping

f*: D*

tered objects in the dual abelian category Remark:

-+

C*

of the dual fi 1-

O

A

Q.E.D.

In the next section, we will show that, under the

hypotheses of Corollary 3.2.1.1, if the spectral sequence is such that the cycles stabilize (see section 9, Definition 1), then the left defect is zero.

(See section 9, Corollary 4.1).

(If the cycles stabilize "in a certain uniform manner", then the integrated partial abutments are also complete, see section 9, Proposition 5 and Proposition 5.1). Finally, we conclude this section with a few examples. Example 1.

Let

A

be a ring such that we have an element

in the center of the ring and is not a unit.

(E.g.,

A

such that

A=zr,

ule

M

is not a zero divisor

t = any non-zero integer.

A = 0, a discrete valuation ring, and meter).

t

t

Or

t =a uniformizing para-

Then it is easy to see that there exists a left A-modsuch that there exists an element

x EM

is t-divisible, but not infinitely t-divisible. such modules such that

M

tx = O.

such that

x

And in fact,

can be constructed such that the element (See Chapter 4 of the main text, below)

x

is

Filtered Cochain Complex Let

pO, pI

391

be projective left A-modules and

a homomorphism of left A-modules such that

M~Cok(dO).

denote the localization of the ring element

t,

n

and let

C = A [t

-1

1

® A

dO :pO

A

n

p ,n = 0,1.

-+

pI

Let

at the central

Then

C*

is a

cochain complex of left A-modules that is zero in dimensions ;' 0,1.

Define a filtration on

all integers

n,p.

plication by

t P"

C*

by defining

Then

C* = (F C*) is a filtered cochain p pEzr complex in the category of left A-modules. The mapping "multi-

plex

P*( =FOC*)

Cok(dO) ~M. chain complex

induces an isomorphism from the cochain comonto

FpC*.

Therefore, in the exact couple of the filtered coC*,

we have that

to multiplication by

p

-+

vp - l , 2-p

and that

corresponds

t.

But then, for all integers (left defect)P,-p~

vP ' I-p = HI (F C*) ~ M,

t P ' I-p :vP ' l-p

under these isomorphisms

y EM:ty '" 0,

I

that

By

Hl(FpC*) ~Hl(p*)

It follows that

y

p,

1l1

and such

is t-divisible

Y EM:ty = 0, and such } that y is infinitelY. t-divisible

hypothesis, the right side of this equation has the non-zero

element, the coset of all integers

x.

Therefore,

(left defect)P,-p;,0,

for

p.

Therefore, the zero'th inverse limit abutment

"HO,

and by

Corollary 2.1 also the zero'th dual inverse limit abutment of the filtered cochain complex

,,~o,

C* = (F C*)

is not perfect. pEl' By Remark 4 following Corollary 3.2.1.2, it follows also that the zero'th integrated partial abutment

p

(=HO(C*A)

in this

case) is not an honest abutment in the sense of section 7, Definition 1. Remark:

By section 7, Corollary 2.3, one cannot make a similar

Section 8

392

counterexample, about right defects and non-perfect direct limit abutments, or about non-perfect dual direct limit abutments, in any abelian category in which denumerable direct sums exist and such that the functor "denumerable direct limit" is exact. ever, in the dual of the category of left A-modules Example 1), the dual filtered cochain complex

8*

(A

as in

C*

of

How-

is

such that the zero'th right defect is non-zero, such that the zero'th direct limit abutment is not perfect, and also is such that the zero'th dual direct limit abutment is not perfect.

And

the zero'th integrated partial abutment is not an honest abutment in the sense of section 7, Definition 1. Example 2.

Let

left A-module

A,t M,

be as in Example 1.

such that

Then there exists a

M has no non-zero infinitely

t-divisible elements, such that there exists a non-zero divisible element xEM

x

E;

M,

and such that every non-zero divisible element

is not a t-torsion element

for all

i.::: 0).

left A-modules

(i.e., is such that

i

tx-j.O,

Then construct a filtered cochain complex of

C*,

concentrated in dimensions

Example 1, using this left A-module for the exact couple of

C*,

t).

Then, as in Example 1,

we have that

and that under these isomorphisms, (left multiplication by

M.

-j. 0,1, as in

tP,l-p

corresponds to

Then, in the notations of [I.L.]

(see also section 7, Remark 2 following Definition 5) we have that the deviation of the inverse system

(6 ),of sec1

tion 7, Note 2 to Theorem 2, applied to the exact couple of the filtered cochain complex

C*,

is such that

Filtered Cochain Complex

tP+2,-p-l

+1Dev( ... --------~> vP , p

I

~

tP+l,-p --------..;»

V

p -p+l

(divisible part of M) (infinitely divisible part of (divisible part of

for all integers

i.

'

393

tP,-p+l

--'---->... )

1i

M)

M) '10,

Therefore the deviation of the inverse

system (6 ) is non-zero, and also has no non-zero t-torsion. 1 By section 7, Remark 2 following Definition 5, we have (left defect) p ,- p = {x

~[Dev(61)

f+l: tx=O}

={x E(divisible part of

M): tx=O}

= 0,

for all integers

p.

Therefore in this example the (p,-p) 'th

left defect is zero for all integers

p,

so that

e.g. the

zero'th inverse limit abutment is perfect, even though the deviation of the corresponding inverse system,

(6 ) of Note 2 to 1

Theorem 2 of section 7, is non-zero. Example 3.

A,t

as in Example 1, let

P*

be a cochain complex

of t-torsion-free left A-modules such that

is not a t-divisible left A-module , and such that there exists an element tx = 0.

x

Define

in

HO(P*) that is t-divisible and such that

C* = A [t -1] ® P*,

all integers

n,p.

A

Then

C*

is a filtered cochain complex of left A-modules, and

using section 5, Theorem 4, it is easy to show that

'E~,l-P~Hl(p*)!(t-torsiOn) 'I t
{a},

Section 8

394

'" "EP,l-p ,=

1x

00

E HO (rp*) :tx =

x

°

and such that

I ;i{O}.

is t-divisible induced by

Therefore in this case the partition of the exact couple of the filtered cochain complex trivial.

C*

is non-

In this case, the first direct limit abutment is per-

feet; the first inverse limit abutment is perfect iff every element

XEHO(P*)

such that

tx =

ble, is infinitely t-divisible. cochain complexes

C*

°

and such that

x

is t-divisi-

(Examples can be given of such

that do, and that do not, obey this last

condition) . When this is so, both the first dire lim. abut. and

for all integers zero.

p;

therefore

'Hl

, l-p .,.. "E P ~ {O} 00

'E~' l-p,

the first inv. lim. abut. are perfect, and and

"Hl

,

are both non-

However, as always, the partition of

induced by the exact couple co-comp(2 A (2 ),

Theorem 2), resp.:

0

)

(of Remark 4 following

is such that the direct limit (resp.:

inverse limit) abutments are all zero, and is such that the right (resp.:

left) part of the partition of

(EP,q) 00

is zero.

Therefore, the three exact couples:

and (2A) must all induce different partitions of all integers

p,

(p,q) El'x'l'

co-comp(2 0 EP,l-p 00

(2),

) ,

'

for

and also induce different (non-isomorphic by

the natural map of exact couples) sets of partial abutments. Since the filtered cochain complex

C*

is co-complete, in this

case the exact couple (2°) is isomorphic to the exact couple co-comp(2 of

0

) ,

and these two therefore induce the same partition

(E~,q) (p,q)E'l' xl'

and have canonically isomorphic sets of

partial abutments. Remark.

Examples 1,2 and 3 above all come from Bockstein spec-

Filtered Cochain Complex

395

tral sequences--see Chapter 1 of the main text below.

All fil-

tered cochain complexes coming from Bockstein's are co-Hausdorff (and also Hausdorff). limi~

direct

Therefore, if the functor "denumerable

is exact

left A-modules,

A

(as it of course is in the category of

a ring), then all such filtered cochain

complexes are co-complete, and therefore are such that co-comp(2 and (2

0

)

are isomorphic exact couples, and therefore yield iso-

morphic sets of partial abutments, and identical partitions of (EP,q) 00

(p , q) Ei?' xJ1 •

Example 4.

Let

C*

be a cochain complex of abelian groups,

such that we have a sUb-cochain complex

E*

and an integer

n

o

such that the induced mapping: -1

n

H

0

-1

n

(C*)

-+

H

is not an epimorphism.

(C*/E*)

0

(It is very easy to construct such

examples for each integer

no'

since the functor

"nO'th coho-

mology group" of course does not preserve epimorphisms). Po

be any integer, and define

F C* == P

Then

Let

C*

couple of

0,

if

E* ,

if

p ~ PO + 1

is a filtered cochain complex, and for the dual exact C*,

we have

~p,n-p==Hn(C*/F

p+l

C*)

'

all integers

Using section 7, Theorem 2, we see that

n,p.

0

)

Section 8

396

Therefore G

(, oHnO)~~

Po

Im(HnO-l(C*/E*) -+Hno-l(C*/E*» Im(HnO- l (c*) -+ HnO- l (C*/E*» n -1 H 0 (C* /E*»

co-comp(2 co-comp(2

0

) ) ,

n -1 0

(C*)-+

{O} •

op n-p 'E 0' 0 0"1

It follows that 0

"I

= Cok (H

0

Since, for the exact couple

we always have that the direct limit abutments of are always

and the right part of

zero, it follows that, for the filtered cochain complex of abelian groups

C*

constructed in this Example, we have that

the dual exact couple co-comp(2 0 ) of the co-completion of C* and the dual exact couple (2 0

)

of

C*

must both have different

no'th direct limit abutments, and must also induce different p ,n -p partitions of E 0 0 0 00

Example 5.

Taking the direct sum of the filtered cochain com-

plex in Example 3 with the filtered cochain complex in Example 4, yields a filtered cochain complex, call it groups such that the four exact couples:

C*,

of abelian

the dual exact couple

of the co-completion; the dual exact couple; the exact couple; and the exact couple of the completion; of the filtered cochain complex

C*,

are such that they all yield four different, pair-

by-pair non-isomorphic,

sets of partial abutments; and such that

the four partitions of quence of the filtered cochain complex

of the spectral seC*

induced by these

four exact couples are all different. Example 6.

Let

0

be a discrete valuation ring containing

the ring of integers as a subring, such that there exists a rational prime

p

that is not a unit in

O.

Let

t

be a

Filtered Cochain Complex generator for the maximal ideal of 0=7

(p)

= {!!.l E (j):

p

n

O.

does not divide

0=7 p ,

Another example is

examples, one can take

nomial ring in one variable over

Let 0,

the usual derivative with respect to complex of

i C*

Hn(C*),

n},

where

~

C*

and let T.

i

and similarly with

"C*A"

tion if,e.g., i.::.O

then

~

is a cochain 0,1.

Intro-

that

-1,

replacing

"C*"

(since by defini-

Hn(c*))

Im(H n (tic*)

-+

Hn (C*)) = tiHn (C*) ,

Then it is shown, in Chapter 4, in an

Example, of the main text below, that ly complete for the t-adic topology.

Hl(C*A)

be

Then the filtration on

-+

follows that

cl

is the t-adic filtration, such

1

that the filtration on

C*

-+

n, i.

F,Hn(C*) = Im(Hn(F,C*)

n).

the poly-

dO: cO

Then

for all integers

Hn(C*A),

for all integers

In both these

by defining

-1,

1

is any prime.

= c = 0 [Tl,

cO

is a filtered cochain complex. and on

p

l

O-modules concentrated in dimensions

duce a filtration on

Then

(For example,

the p-adic integers.

= p. )

t

397

HLlc*A)

Hl(C*A)

is not t-adical-

Since we have just observed

is the t-adic filtration, it

is not complete as filtered object.

Since the filtered object

n C

is co-discrete, we have that

c

n

Section 8

398

is co-complete, for all integers

n.

Therefore co-comp(c~) =

is the n'th integrated partial abutment in the sense of Corollary 3.2.1.1 of the filtered cochain complex

C*.

Therefore we have an example of a filtered co-

chain complex of abelian groups in which the first

inte~.

part.

abutment is not complete, and therefore by Corollary 3.2.1.1,

conclusion~,

also not Hausdorff.

By Corollary 3.2.1, it

follows also that the first direct limit abutment of the exact couple (2A) of the completion of is also not Hausdorff.

C*,

is also not complete and

(In this Example, it is easy to see that

the first direct limit abutment

'Hl,

which since

complete is isomorphic by Corollary 1.3 to although not complete. n 'H "" Hn(C*),

n 1-1,

C*

is co-

Hl(C*), is Hausdorff,

The other direct limit abutments

are complete, in fact have finite filtra-

tions; in fact, are finitely generated as

O-modules.

(In fact,

~ ~, 1.) (And the inv. lim. abuts. are all zero) .)

they are: Example 7.

By Coroll'ary 3.2.1.3, itisimpossible to find a

filtered cochain complex in the category of abelian

groups

such that the integrated partial abutments are not co-complete. However, in the dual of the category of abelian groups, the dual

8*

of the filtered cochain complex of Example 6 is a

filtered cochain complex such that the (-1) 'st integrated partial abutment is not co-complete (and therefore by Corollary

Cl),

3.2.1.1, conclusion Example

8.

Let

A

is not co-Hausdorff.)

be the direct product of the category of

abelian groups and of the dual of the category of abelian groups.

Then

A is an abelian category obeying the hypotheses

of Corollary 3.2.1.1.

Let

o n +2 n n +2 D = (Cn,C )nE;f (=(Cn,C) nE;r)'

Filtered Cochain Complex the filtered cochain complex in

A

399

whose first co-ordinate is

the eochain complex of Example 6, and whose second co-ordinate is the eochain complex of Example 7 with degrees shifted up by +2.

Then

D*

is a filtered coehain complex in

first integrated partial abutment of is also not co-complete.

D*

A,

and the

is not complete and

Therefore by Corollary 3.2.1.1, con-

clusion CD,the first integrated partial abutment of also not Hausdorff and not co-Hausdorff.

D*

is

Section 9 Convergence

Given, say, a conventional bigraded spectral sequence, there are several things that are desirable. (1)

In order, they are

Does the spectral sequence have an "honest" abutment

(in the sense of Definition 1 of section 7)?

Or, at least, a

set of partial abutments with some nice properties? questions are answered in principle:

These

in section 7, if the

spectral sequence comes from a (conventional, bigraded) exact couple, see Theorems 2 and 3 and their Corollaries, and Proposition 4, parts (2) and (3); and in section 8, in the special case that the spectral sequence comes from a filtered cochain complex, see Theorems 1, 1 (2)

0

and 2 and their Corollaries.

Is the abutment (or partial abutments) complete and

co-complete?

This question, also, is answered in the case of

the spectral sequence of an exact couple, in section 7, see Propositions 4, 5 and 6 and their Corollaries and associated Remarks.

For the special case of the spectral sequence of a

filtered cochain complex, the question is answered in Theorems 1, 1

0

,

2 and their Corollaries (see especially Corollaries

3 . 2. 1 . 1, 3. 2 . 1 . 2 and 3. 2. 1 . 3) . It is noted (section 7) that a conventional, bigraded spectral sequence coming from an exact couple has a left defect and a right defect, which can be thought of as obstructions for the 400

Convergence

401

inverse limit abutments and the direct limit abutments, respectively, to be perfect (Definition 5 of section 7).

(In

the category of abelian groups, the right defect is always zero.)

In the case of a filtered cochain complex, section 8, it

is shown that, all of the four in general different exact couples that yield the spectral sequence have the same left and right defects -- and that even a map of filtered cochain complexes that induces an isomorphism along Ei,q induces an isomorphism on the left and right defects, see Proposition 4 of section 8.

Thus, when the spectral sequence comes from a fil-

tered cochain complex, necessary and sufficient conditions to have a reasonable abutment (in the sense of Definiton 1 of section 7) is that these left and right defects are zero, see section 8, Corollary 3.2.1.1 and the Remarks following.

(Again,

in the category of abelian groups, right defects are always zero.) A sufficient condition for a conventional, bigraded exact couple in an abelian category

A (such that denumerable direct

products exist, and such that the functor "denumerable direct product":

AW~~ is exact),

to have a left defect that is zero

(i.e., to have perfect inverse limit abutments), is for the spectral sequence to converge (see Remark 2 following Corollary 4.1 below).

Weaker still, it suffices that the cycles stabilize,

in the sense of Definition 1 below, with no assumptions at all on uniformity of this stabilization. lary 4.1 below.

This is proved in Corol-

Since it is important to know when the left

defect vanishes, this motivates the study of spectral sequences that converge (in the sense of Definition 1.1 below), or that

402

Section 9

at least are such that the cycles stabilize (in the sense of Oefinition 1 below).

These definitions are usually only consid-

ered for conventional bigraded, or perhaps conventional singly graded, spectral sequences.

But, as it is no more difficult

to pose some of these definitions in greater generality, we begin at such a greater level of generality. Proposition

1.

Let

A be an abelian category, let D be an addi-

m m m tive abelian group, let rO be an integer and let (Er,dr,Tr)mED r~rO

be a D-graded spectral sequence starting with the integer rO in the abelian category A (see the beginning of section 4, Example 2 following Definition 1).

r

l

be a fixed integer (A)

Let m E D be a fixed degree, and let

~rO.

Then the following three conditions are equivalent.

(1)

For every integer r

quotient object Em of Em r

tient-object of

(2)

r

Em

rl '

~

r

l

, we have that the sub-

in the abelian category

A is a quo-

l

so that we have a natural epimorphism:

Fix any integer ra such that rO 2 ra 2 r l

(for example, one can take ra integer r

~

=

rO; or rC

=

r l ).

Then for every

r l , we have that the (r-ra)-fold cycles and the

(rl-ra)-fold cycles in Em coincide, ra m

Zr-r' (E r ,) o 0 (3)

(B)

d

m

r

=

Z

(Em) r -r' r' , all r > r l 100

0, for all r

~

r

.

l

In addition, the following three conditions are

equivalent.

Convergence (1°) subquotient subobject, r

For every integer r

object Em of Em r

r

~

403 r

l

, we have that the

in the abelian category

A is a

l

so that we have a natural monomorphism:

>

(2°)

Fix any integer r6 such that rO 2 rO 2 r

= rO; or rO = r l ).

example one can take r6 teger r

~

(for l Then for every in-

r l , we have that the (r-r6)-fold boundaries and the

(r l -r6)-fold boundaries in

E;.

o

coincide,

(Em) r'

o

m-a

d

a

r: E r r

r -+

-+

Em r

is the zero map, where a

r

E D is

E* r

The proof of part (A) is an easy consequence of the definition of a D-graded spectral sequence, see section 4, Example 2 following Definition 1, and is left as an exercise (one can use the Exact Imbedding Theorem, which

[I.A.C.l, to reduce to the case in

A is the category of abelian groups if one wishes, al-

though this is not necessary).

And, of

cou~se,

part (B) follows

from part (A), by passing to the dual category. Defini tion 1.

The hypotheses being as in Proposition 1, if

the three equivalent conditions of part (A), hold (respectively: (B),

(1°),

(1),

(2) and (3),

if the three equivalent conditions of part

(2°) and (3°), hold), then we say that the cycles

(respectively:

boundaries) stabilize in degree m past the inte-

Section 9

404

(The terminology emphasizes condition (2) m m m If (Er,dr"r)mED

(respectively:

is a D-graded spectral sequence in an

r~O

abelian category A, and if m E D is a degree, then we say that the cycles (respectively: there exists an integer r

boundaries) stabilize in degree m iff l

such that the cycles (respectively:

boundaries) stabilize past r Corollary 1.1.

l

.

Let the hypotheses be as in Proposition 1.

Then the following two conditions are equivalent. For every integer r ~ r

(1) of Em r

l

, the subquotient-object E~

is the whole object Em , so that we have a canonical isor

l

l

morphism: Both the cycles and the boundaries stabilize in degree

(2)

m past the integer r , in the sense of Definition 1 above. l Proof:

In fact, condition (1) of Corollary 1.1 is clearly

equivalent to:

condition (1) of Part (A) of Proposition I,

and condition (1

0

)

of part (B) of Proposition I, both holding.

Therefore the Corollary follows from Definition 1. Definition 1.1. dm m) (0) (E m r' r,Tr mEO

Q.E.D.

Let D be an additive abelian group and let be a O-graded spectral sequence starting with

r~rO

the integer rO in the abelian category A.

Let m E D be a degree.

Then we say that the spectral sequence (0) converges in degree

m

iff there exists an integer

rl~rO

two equivalent conditions of Corollary 1.1 hold. then given such an integer r

l

such that the If this is so,

we sometimes say that the spectral

Convergence

405

sequence (0) converges in degree m past the integer r

l

(or,

sometimes, we say that the spectral sequence (0) degenerates in degree m past the integer r ). l The spectral sequence (0)

is said to converge if it converg-

es in degree m, for every degree m E o. Definition 1.2.

The spectral sequence (0)

uniformly if there exists an integer r

l

is said to converge

, independent of m E 0,

such that the spectral sequence (0) converges in degree m past the fixed integer r

l

, for all degrees m E

o.

When this is the

case, then given such an integer r l we often say that the spectral sequence (0) converges uniformly (or, sometines, degenerates uniformly) past the integer r

l

.

The notion, Definition 1.1, of what it means for a

Remark:

spectral sequence to converge does not generalize, beyond the case of a D-graded spectral sequence, to that of a spectral sequence in a general D-graded abelian category.

However, the

next corollary shows that the notion, Definition 1.2, of what it means for a spectral sequence to converge uniformly does so generalize. Corollary 1.2.

Let D be an additive abelian group, let B be a

D-graded abelian category, let rO be an integer and let (E ,d r r rOo

)r>r be a spectral sequence starting with the integer - 0 Let r be an integer ~rO. Then the following six condil ,1

r

tions are equivalent. (1) E

r

For every integer r

is the whole object E l

r

~

r

l

, the subquotient-object Er of

, so that we have a canonical isol

morphism of degree zero, E r "" Er

l

Section 9

406

For every

(I' )

of E

r

For every integer r

is a subobject of E l

r

l

r

~

l r l , the subquotient-object E r

l

Fix any integer rO such that rO

one can take >

r

r l , the subquotient-object E r

~

l

(2)

r

r

is a quotient-object of E

r

(1,0)

of E

integer

rb = rO:

or

rb =

r ). l

~

~

rO

rl

(for example,

Then for every integer

, we have that the (r-rO)-fold cycles and the (rl-rO)-fold

cycles of E , coincide, rO Z

(2

0

all r

,(E,)

r-r O

rO

>

r

l

.

Fix any integer rO such that rO ~ rO ~ r l

)

rO: or rO

one can take rO

=

r )· l

(for example,

Then for every integer

r > r l , we have that the (r-rb)-fold boundaries and the (rl-rb)fold boundaries in E , coincide, rO B

( 3)

Note:

d

I

r-r 0

r

(E

o

all r > r

,)

r0

for all integers r

~

r

l

l

.

.

If the D-graded abelian category B is the category AD

of all D-graded objects in an (ordinary, ungraded) abelian category A (see section 3, Example 1), then the above six equivalent conditions are also equivalent to the condition that, "For the fixed integer r l that the condition (1)

~

r

' we have for all degrees mE D, O

(equivalently:

condition (2)) of

Corollary 1.1 holds. Proof:

By the Exact Imbedding Theorem for D-Graded Abelian

Convergence

407

Categories (section 3, Corollary 5.1), it suffices to prove the Corollary in the case in which B is the category of D-graded abelian groups.

(Alternatively, we can use simply Theorem 5 of

section 3, to reduce to the case in which B is the category AD of all D-graded objects in some (ordinary, ungraded) abelian category A.)

But then, the equivalence of conditions (1'),

(2) and (3) follows from part (A) of Proposition 1; and the equivalence of conditions (1,0), (B) of Proposition 1.

0 (2 ) and (3) follows from part

Since also condition (1)

is clearly

equivalent to conditions (1') and 1,0), we have the equivalence of the six conditions of the Corollary. Finally, if the hypotheses are as in the Note to the Corollary, then clearly condition (1) of Corollary 1.1 is identical to condition (1) of this Corollary, which proves the Note. Q.E.D. Definition 1.2.1.

Let D be an additive abelian group, let

B be a D-graded abelian category, let rO be an integer and let

(0) (Er,dr,Tr)r>r be a spectral sequence starting with the in- 0 teger rO in the D-graded abelian category A. Then we say that the spectral sequence (0) converges uniformly iff there exists an integer r

l

~

rO such that the six equivalent conditions of

Corollary 1.2 above hold. such an integer r

l

If this is the case, then given

, we also say that the spectral sequence (0)

converges uniformly (or degenerates uniformly) past the integer

Then, by the Note to Corollary 1.2, we have that Definitions 1.2 and 1.2.1 agree in the only case in which they both make

sensei i.e., in the case that Definition 1.2 makes sense. Therefore,

Definition 1.2.1 is a generalization of Definition

408

Section 9

1.2, to spectral sequences in arbitrary D-graded categories

B,

where D is any abelian group. Considering section 4, Definitions 3' and 4', and section 4, Remark 2' (2

0

),

(1')

following Corollary 3.1', and conditions (1),

and, respectively,

Corollary 1.2.1.

(2),

(l'o), we deduce immediately

The hypotheses being as in Corollary 1.2, we

have that the six equivalent conditions of Corollary 1.2, and therefore also the condition that lithe spectral sequence converges uniformly past the integer r

l

", are also equivalent to

each of the following conditions: (1 00 )

Eoo exists, and the subquotient-object Eoo of Erl is

the whole object E

(2 00 )

r

, l

Fix any integer ro such that rO

~

rO

~

r

l

.

Then

(E ,) exists, and the sub-objects Z (E ,) and Z ,(E,) of rO 00 rO rl-r O rO E , coincide, rO Z

00

Z

(E 00

(2~)

,) r 0

Fix any integer rO such that rO ~ rO ~ r l .

Then

B (E ,) exists, and the

00 rO E , coincide, rO

B

(E 00

(l~)

E 00 exists,

,) r 0

and the subquotient-object Eoo of E

a quotient-object of Er . 1

r

is l

Convergence E a

409

exists, and the subquotient-object E

subobject of E

r

of E

r

is l

l

The proof of Corollary 1.2.1 also shows (using again section 4, Definitions 3' and 4', and section 4, Remark 2' following Corollary 3.1'), that, Corollary 1.1.1.

Let the hypotheses be as in Proposition 1.

Then (A)

The following two conditions are equivalent, (c)

The cycles stabilize in degree m past the integer

(In other words,

"The three equivalent conditions of

Proposition 1, Part (A), hold.") (2)

Fix any integer rO such that rO

Z (Em,) exists, and the '"

r 0

subobjects Z (Em,) and '" rO

Then co-

incide,

If also E: exists, then the following condition is also equivalent: The

(B)

sub quotient E: of Em r

is a quotient-object l

Also, the following two conditions are equivalent. (b)

integer rl'

The boundaries stabilize in degree m past the (In other words,

"The three equivalent conditions

of Proposition 1, part (B), hold.")

(2~)

Fix any integer rO such that rO :5.. rO :5.. r l ,

B (Em,) exists, and the '" rO

,(Em,) subobjects B (Em,) and B rl-r O rO '" rO

Then

Section 9

410

coincide,

If also E: exists, then the following condition is also equivalent: The

subquotient Ern of Ern r

00

Lemma 2.1.

subobject of Ern

is a

r

l

Let A be an abelian category and let t(p+l)

t (p+2) -+

t

vp +l

be an inverse system in the abelian category the integers. object in

Then let V

A, and let t =

(p-l)

A, indexed by all

p

= (V ) pEl' denote the singly graded (t (p» denote the endomorphism of pEl'

degree -1 of the graded object V.

Suppose, for every integer

p, that there exists an integer r(p)~O, depending on

for all integers r all integers i

>

l

p.

~

r(p)

p, such that

,

Then, for every pair of integers i,p with

1, there exists a positive integer ri,p such that ( 2)

for all integers r > r.

~,p

Note:

In equations (1) and (2), t

r , t i , etc., stand for the

respective iterateS of the endomorphism t of the graded object V (not for the components, t(p) :VP Proof:

-+

vp - l

, p E 1", of t).

By the Exact Imbedding Theorem [I.A.C.], it suffices

Convergence

411

to prove the Lemma in the case in which A is the category of abelian groups.

We do this by induction on i

>

1.

For i = 1, take r. r(p). Having proved the assertion l,p r for the integer i > 1, suppose that r 1 ~ 0 and that x E [(t IV) n i+l p . (Ker t ) ] , 1 . e., tha t x E (trlV)p and ti+lx o. Then rl+l -1 i V)p and t (tx) = O. Hence, if r tx E (t > r. 1-1, then l,pl tx E [~ri,p-IV) n (Ker ti)]p-l, and therefore by equation (2) for all positive integers r we have that tx E tr+lv, say tx tr+ly, for some y E Vp - r . r

=

O.

Also x-tryE

are both

~r(p), then

Then t(x-try)

r

t Iv + t V, so that if also rand r

l

x-try E tr(p)v, and therefore x-try E [(tr(p)V)

n (Ker t)]P.

But then by equation (1), we have that x_try E trV, and therefore x E trV.

Therefore, if we let r'+ r = sup(r. 1 l ,p l,p- 1-1, l then whenever x E [(tri+l,PV) n (Ker ti+l)]p, we have

r(p)),

that x E trv for all integers r

>

r(p), completing the induc-

tion.

Q.E.D. The proof of Lemma 2.1 shows that one can take r. l,p

Remark:

sup(r(p), r(p-l)-1,r(p-2)-2, ... ,r(p-i+l)-i+l), all i,pE2" with i

> 1.

The next definition and Lemma appear in [I.L.].

We re-

produce them here for the sake of completeness of exposition. Definition 2.

Let

A be an abelian category and let

t(p+2) (1)

. .. ~ v P +1

t(p+l) Jp) -----.. v P - - - ?

be an inverse system in the abelian category the integers.

A, indexed by all

Then let V = (V P ) P (i7 denote the singly graded

object in A, and let t = (t (p) )pEt" denote the endomorphism of

Section 9

412

degree -1 of the graded object V.

Then we say that the images

stabilize in the inverse system (1), iff, for every integer p, there exists a positive integer r(p) such that

Example.

In terms of Definition 2, the conclusion of Lemma 2

above can be written, "For every integer p the inverse system:

[Vn(Ker t)]p

is such that the images stabilize." Lemma 2.2.

Let A be an abelian category such that denumerable

direct products exist and such that the functor "denumerable direct product" is exact. gory

A is (P.2)

Assume, in addition, that the cate-

such that, Whenever (A., a .. ). . E is an inverse system in 1 1J 1,J 'l' j::i the category

A indexed

by the integers such that

each a .. is an epimorphism, then the map from the 1J inverse limit: [lim Ai]

-+

AO

i~+'"

is an

epimorphism~*)

(E.g., the category of abelian

groups obeys Axiom (P.2).) Let (*)see Introduction, Chapter 1, section 7.

Convergence t(p+2) (1)

•• ,

t(p+l) Vp + l

--;>

--;,

413

t (p)

vP

t (p-l) Vp - l

_ _;>

--;>

•• ,

be an inverse system in the category A such that the images stabilize.

Then (infinitely t-divisible part of V)

(2)

(t-divisible

part of V), and

o.

(3)

Proof:

For every integer p, choose a positive integer r(p)

as in Definition 2.

Then by equation (2) of Definition 2, we

have that

for all integers r,p such that r

~

r(p).

For a given integer p, choose r any integer such that r

>

sup(r(p),r(p+l)+l).

(trV)p

=

t«tr-IV)p+l)

Then (t-divisible part of VIP t«t-divisible part of V)p+l), and

therefore (5)

t maps

(t-divisible part of V)p+l epimorphically

onto (t-divisible part of VIP , for all integers p. We have that lim (t-divisible part of VIP.

pE ?:

414

Section 9

This latter, by equation (5), is an inverse system in which all the maps are epimorphisms.

By Axiom (P.2), we have that the

natural mapping: lim (t-divisible part of VIP

+

(t-divisible part of

+-

p E7'

is an epimorphism, for all integers PO.

These last two equa-

tions imply conclusion (2) of the Lemma. By ea.

(4), in the inv. sys.

V/(t-div. part), the map

from the(p+r(p))'th term to the p'th term is zero. 11ml (V/ (t-div. part)) = O.

By eq.

11ml (t-divisible part of V)

= O.

Therefore

(2) and Chap. 1, sec. 7, Thm. 2

Therefore

Ij,mlv

= 0,

(3).

proving Q.E.D.

Proposition 3.

Let D be an additive abelian group, let B be a

D-graded abelian category, and let

t

v~/ E

be an exact couple in the D-graded abelian category B (see section 5, Definition 3').

be the

Let

spectral sequence of the exact couple (see section 5,

Definition 5').

Let r

l

be a non-negative integer.

following conditions are equivalent.

Then the

Convergence

(1) teger r

l

The spectral sequence converges uniformly past the in. r

(2)

r

415

n

(t IV)

(Ker t), for all integers

>

If the (t-divisible part of V) exists, then it is equivalent to write: (2",)

n

(t-divisible part of V)

r

(Ker t)= (t IV)

n

(Ker t).

r

[Ker(t 1:V7V)] +tV, for all integers r

~

r

l

.

If the (t-torsion part of V) exists, then it is equivalent to write: (t-torsion part of V) Proof:

+ tV

=

r

[Ker(t

l

:V7V)] + tV.

The hypotheses of Corollary 1.2 hold, with rO = O.

Therefore, by Definition 1.2.1 and Corollary 1.2 above, condition (1) of this Proposition is equivalent to condition (2) of Corollary 1.2, with rO

= rO

o.

But then, by section 5,

Corollary 3.1', equation (lr)' and section 5, Corollary 3.2', equation (lr)' it follows that condition (2) of Corollary 1.2 (with rO

=

rO

Proposition.

=

0) can be rewritten as condition (2) of this

Therefore conditions

are equivalent.

(1) and (2) of the Proposition

By definition of " (t-divisible part of V)", (2)~

if this latter exists, then

(2",).

Similarly (using

section 5, Corollary 3.1', equation (2 ), and section 5, r 0 Corollary 3.2', equation (2 ); and condition (2 ) of Corollary r

1.2), we see likewise that (l)~(20), and that, if the (t-

416

Section 9

torsion part of V) exists, then (2)

~(200)'

Q.E.D.

The "degree by degree" form of Proposition 3, is Corollary 3.1.

A be an

Let D be an additive abelian group, let

(ordinary, ungraded) abelian category and let

t

E

be a D-graded exact couple in the abelian category deg (k) + deg (h) E D and

y

= deg (h).

A.

(Then deg (k)

Let

=

p

p-6

E D.)

Let m E D be any fixed degree, and let r

l

be any fixed

non-negative integer. (A.)

Then the following conditions are equivalent. (1)

integer r

l

The cycles stabilize in degree m past the

. r

[(t lV)

(2)

for all integers r

~

r

l

n (Ker t) 1m+p-y,

.

If the (t-divisible part of V) exists, then it is equivalent to write: (2 00 )

(B.)

[(t-divisible part of V)

n (Ker t) lm+ p - y

Also, the following conditions are equivalent. (1°)

The boundaries stabilize in degree m past the

Convergence integer r

l

417

. r

([Ker(t l :V-+V)] + y tV)m- , for all integers r ~ r

l

.

If the (t-torsion part of V) exists, then it is equivalent to write r

[it-torsion part of V) + tV]m- y

(2~)

([Ker(t 1

V-+V)] + tV) m-y . ~:

Of course in the statement of Corollary 2.1, the

mapping of D-graded objects: "tr" stands for "the with itself r-times"

iterate of t

(and not "the r'th component of t as a map

of D-graded objects" -- this latter does not even make sense, in general, since D is not in general contained in the integers, and therefore a positive integer r need not be a degree, i.e., need not be an element of D). The proof is similar to that of Proposition 3. Remarks.

1.

By Lemma 2.1, and the Remark following, condition

(2) of Proposition 3 is equivalent to: r

(t IV)

n

for all integers r Proof:

(Ker ti), ~

r

l

.

By section 3, Corollary 5.1, the Exact Imbedding Theorem

for D-Graded Abelian Categories, it suffices to prove the assertion in the case that B is the category of D-graded abelian groups.

Then, replacing the D-graded abelian group V

=

d

(V )dED

d

with the (ordinary, ungraded) abelian group V = $ V , it sufdED fices to prove the assertion for an abelian group V.

This

Section 9

418

latter follows from Lemma 2.1 and the Remark following. 2.

It follows that if, under the hypotheses of Proposition

3, we have that the (t-torsion part of V) exists, then condition (2) is equivalent to condition r

(trV) n (t-torsion part of V)

(2' )

part of V), for all integers r 3.

~

(t lV) n (t-torsion

rl ·

Under the hypotheses of Proposition 3, if both the

(t-divisible part) and the (t-torsion part) of V exist, then condition (2') of Remark 3 above, and therefore also condition (2) of Proposition 3, are each equivalent to: (t-divisible part of V) n (t-torsion part of V)

(2~:

r

(t lV) n (t-torsion part of V). 4.

The dual of Remark 3 above, is that under the

hypo-

theses of Proposition 3 above, if both the (t-divisible part) and the (t-torsion part) of V both exist, then (t-divisible part of V) + (t-torsion part of V) r

(Ker t 1) + (t-divisible part of V)

;

otherwise stated,

(2~' )

r

The images of Ker t l

and the (t-torsion part of V)

in V/(t-divisible part) are the same. 5.

Another way of writing the condition,

ition 3, is that: in V/tV."

r

"Ker(t) and Ker(t

rl

(2

0

)

of Propos-

) have the same images

Similarly, condition (2~) of Proposition 3, when it r

makes sense, is equivalent to: "the (t-torsion of V) and Ker t l

Convergence

419

have the same image in V/tV." Let A be an abelian category such that denumerable

Theorem 4:

direct products exist, such that the functor "denumerable direct product" is exact, and such that Axiom (P.2) holds (see Introduction, Chapter 1, section 7).

(E.g., the category of

abelian groups, and also its dual category, obey all of these condi tions . ) Let t

E

be a conventional bigraded exact couple in the abelian category Let (Ep,q dP,q ,p,q) r ' r ' r p,q,rE;r r > rO be the associated conventional bigraded spectral sequence start-

A, starting with some integer rOo

ing with the integer rOo

Suppose that, there exists a fixed

integer n, such that (1)

The cycles stabilize in degree (p,n-p) for all inte-

gers p. Then (2)

(left defect)p,n-p = 0 for all integers p, and there-

fore the n~th inverse limit abutment "R n (as defined in Theorem 2 of section 7) is perfect (as defined in Definition 5 of section 7). Proof:

For simplicity of notation, we first replace the given

420

Section 9

exact couple by an exact couple that is isomorphic by means of a map of bidegree (0,0) and such that the map h of the new exact couple is of bidgree (0,0).

Then by Corollary 3.1,

part (A.), the hypothesis (1) is equivalent to asserting that, for every integer p, there exists an integer r p -> sup (rO'O) such that r

[(t PV) n (Kert)jP,n- p ,

(3) whenever r

> r

p

.

Then by Lemma 2.1 applied to the inverse system

-)-

of section 7, Theorem 2, Note gers i

>

... ,

2, it follows that for all inte-

1, and all integers p, that there exists an integer

r, > 0 such that l,p

whenever r

>

r

, p,l

(In fact, one can take r

, = sup(r,r

p,l

p

p-

1-1, ... ,r

'+l-i+l).) P-l

By the Example following Definition 2, in the inverse system: (4 ) ... +[V n Ker ti+ljP+l,n-p-l + [V n Ker tijP,n-p + p + [V n Ker t 2 jP-i+2,n- p +i-2 + [V n Ker tjP-i+l,n-p+i-l ,

(in which the maps are the restrictions of the appropriate components of t), we have that the images stabilize in the sense of Definition 2.

Therefore, by Lemma 2.2, conclusion (2), it

follows that the (t-divisible part) and the (infinitely tdivisible part) of the inverse system (4 ) coincide. p

In terms

Convergence

421

of the inverse system (6 ) of section 7, Theorem 2, Note 2, n this latter observation is equivalent to the statement that (5

.)

p,l

n (Ker ti)]p,n-p

[(divisible part of V)

n

[(infinitely divisible part of V)

for all integers p,i with i

(5

>

1.

Taking i

1 in equation

.), this implies that p,l (6 ) p

n

[(divisible part of V)

(Ker t)]p,n- p

[(infinitely divisible part of V) for all integers p.

By Remark 1

(Ker t)]p,n- p ,

n

following Definition 5 of

section 7, this latter is equivalent to the conclusion, equation (2), of the Theorem.

Q.E.D.

Remarkl. The proof of Theorem 4 shows, in fact,

that under the

other hypotheses of Theorem 4, the condition (1) of Theorem 4 is equivalent to the assertion, that the

(n+l-a-~

'th

inverse

system: ... -+V

p+l,n-a-B-P

+

1 vp,n+ -a-B-P

+

p-l,n+2-a-B-p V

of section 7, Theorem 2, Note 2, Note 1 to that Theorem, so that

......

(where a, sand rO are as in l-a-S = total degree of k),

obeys the hypotheses of Lemma 2.1.

Or, by Lemma 2.1 and the

Example following Definition 2, we have that condition (1) of Theorem 4 is also equivalent to the assertion that, for every integer p, the inverse system:

Section 9

422

... -+[V n (Ker t [V

3

p+2,n-l-a-rl-p )]

+

n (Ker t 2 )] p+l,n-a-~-p

+[V n(Ker t)]p,n+l-a-B-p

is such that the images stabilize in the sense of Definition 2 above. Remark 2. If one replaces the principle hypothesis of Theorem 4, that "the cycles stabilize," by the much more stringent hypotheses, that "conditions (1) and (2) of Proposition 5.1 below hold," then one can eliminate all hypotheses on the abelian category A and still obtain the conclusion of Theorem 4. Proof:

As in the proof of Theorem 4, replace the given exact

couple with one that is isomorphic through an isomorphism of bidegree (0,0), such that the new exact couple is such that the map k is of bidegree (0,0).

Then, as in the proof of Propos-

ition 5.1 below, we see that conditions (1) and (2) of Proposition 5.1 imply that, there exists an integer r

l

, such that for

the rl'th derived couple

of the given exact couple, we have that the mappings tP+l,n-p-l: vP+l,n-p-l r

l

r

l

Convergence are isomorphisms for p

~

PO + 1.

But then clearly ) jP,n- p

[(t-divisible part of V

(1)

r

=

423

l

[(infinitely t-divisible part of V

r

Since (v

)jP,n-p l

,E ,t ,h ,k ) is the r 'th rl r r r 1 r l l l l 1 = t V, derived couple of the given exact couple, we have that V for all integers p.

r

r

l

and therefore (2)

(t-divisible part of V) = (t-divisible part of V

r

), l

(infinitely t-divisible part of V)

= Equations

(infinitely t-divisible part of V

r

) l

(1) and (2) imply that

[(t-divisible part of V)jP,n- p

[(infinitely t-divisible

part of V)jP,n-p

for all integers p.

Intersecting this latter equation with

Ker(t), we obtain that the (p,n-p) 'th object of the left defect is zero. Corollary 4.1.

Q.E.D. Let A be an abelian category such that denumer-

able direct products exist, such that the functor "denumerable direct product": holds.

W

A

'IJV>

A is exact and such that Axiom (P.2)

(See Introduction, Chapter 1, section 7.

E.g., the

category of abelian groups, and also its dual category, obey these conditions.)

Suppose that we have a conventional bi-

graded exact couple in the abelian category A, such that the associated spectral sequence is such that

424

Section 9 For every pair of integers p,q, the cycles in degree

(1)

(No condition of uniformity is assumed.)

(p,q) stabilize.

Then the left defect of the exact couple is zero, and therefore the inverse limit abutments:

n

"H , n E'l' , of the exact couple

are all perfect (in the sense of Definition 5 of section 7). Note:

If one replaces the principle hypothesis (1) of

Corollary 4.1, that "the

cycles stabilize," by the much more

stringent hypothesis, that "For every integer n, condition (1) of Proposition 5.1 below holds," then one can eliminate all hypotheses on the abelian category A, and we still obtain all of the conclusions of Corollary 4.1. Proof:

Follows immediately from Theorem 4.

immediately from Corollary 4.2.

Remark 2

The Note follows

following Theorem 4.

Assume the hypotheses of Corollary 4.1.

Then

for all integers p and n, we have that liml [V

n

(Ker ti)jP-i+l,n-p+i-l

o.

+-

i>l Note:

The proof of Corollary 4.2 shows that, under the weaker

hypotheses of Theorem 4, we have that liml [V

n (Ker ti, jP-i+l,n-
itl where n is as in Theorem 4 and 1 -a-I) is the total degree of k (notation being as in section 7, Theorem 2, Note 1). Proof:

Follows immediately from Remark 1 following Theorem

4 and from Lemma 2.2, conclusion (3). Example 1.

Let A be an abelian category as in Theorem 4, and

Convergence

425

suppose that we have a conventional bigraded spectral sequence (EP,q dP,q ,p,q) that comes from a conventional bigraded r 'r 'r r ~ rO exact couple. Suppose that (1)

We have an integer n, such that there exists an in-

teger PO such that, whenever p ~ PO' we have that E~~n+l-p (or, weaker, such that there exist integers PO,r suc h that EP,n+l-p = 0 wh enever p r

there exist integers PO,r whenever p

~

~

l

Po and r

~

l

with r

r l ).

with r

PO; or, wea k er s tOll 1

l ,

~

0

rO

th a t

~ rO such that d~-r,n-p+r = 0

l

Then, by Theorem 1, we have that

the n' th inverse limit abutment "H perfect.

l

=

n

of .the exact couple is

(Since then obviously for all inegers p, the cycles

stabilize in degree (p,n-p).) Exam~.

Suppose, given a conventional bigraded spectral

sequence starting with the integer rO that comes from a conventional bigraded exact couple in an arbitrary abelian category

A, that the spectral sequence is such that, (1)

For every integer n, there exists an integer Po =

PO(n), and an integer r

l

=

rl(n)

~

r O' such that, whenever

p ~ po(n), we have E~~7~f = 0 (resp.: such that Axiom (P.2) holds and such that whenever p

~

PO(n)

, r

~

rl(n), we have

dP-r,n-p+r = 0) • Then the hypotheses of the Note to Corollary r

4.1 (resp.: of Corollary 4.1) are satisfied and therefore the left defect vanishes; so that the inverse limit abutments, n "H , n E7 , are all perfect. Remarks.

1.

Graphically, a conventional bigraded spectral

sequence obeys the condition of Example 2 above, if and only if along every line of slope -1 in the diagram,

426

Section 9 n=l

n=O

n =-1

' "E -32 r l

' " E r-22 1

"\. E-12 r l

n=2 ' " E02 r l

ceeding in the indicated direction (to the bottom right), we eventually reach a spot (in general varying from line to line) after which all of the terms on that particular line are zero. (The condition in Example 2 reads: "such that all d**'s that r

terminate in a spot on the n'th line, past (i.e., to the bottom right of)

(PO(n),n-PO(n)), and have r

~

rl(n), are zero maps.")

E.g, this is so for an upper half plane spectral sequence (i.e., one confined to the region {(p,q): q

~

a}), or a first

or third quadrant spectral sequence, etc. 2.

An

obvious

case in which the spectral sequence obeys

the condition (1) of Corollary 4.1, is if the spectral sequence converges. 3.

However, it is not difficult to construct a convention-

aI, bigraded exact couple (in fact, even one coming from a double complex, see section 10 below) in the category of abelian

Convergence

427

groups such that the cycles stabilize, but such that the boundaries do not stabilize in degree (p,q) for any pair of integers (p,q).

(Therefore, Corollary 4.1 is "better" than Re-

mark 2 above.) Proposition 5.

Let

A be an abelian category such that denumer-

able direct products and denumerable direct sums exist, such that the functor "denumerable direct product" is exact and such that Axiom (P.2) holds (see Introduction, Chapter 1, section 7). Let C* be a filtered cochain complex in the abelian category and let (EP,q dP,q TP,q) r'r'r r>l n be a fixed integer. (1)

be the spectral sequence.

A,

Let

Suppose that

For infinitely many positive integers p, the filtered

object Hn(co-comp(C*)/F(co-comp C*» p

is co-Hausdorff (this

condition is automatically satisfied if, e.g., the functor "denumerable direct limit" in the abelian category

A is exact),

and that (2)

For every pair of integers p,q such that p+q = n,

there exists a positive integer B(p,q) such that, for every integer i ~ 0, we have that dP-i,q+i r

o

whenever r > B(p,q)+i.

Then the (n+l) ' s t integrated partial abutment, Hn+l, as defined in section 8, is complete. Notes:

1.

If one deletes the hypothesis (1) in the statement

of the Proposition, then the proof shows that there is a canonical isomorphism in the category (3)

A,

G+oo(H n + l ): 11ml (G_oo(Hn«co-comp C*YFp(co-comp C*}»}. p++oo

428

Section 9

If one assumes that the functor "denumerable direct sum" is exact, then G_oo(Hn«co-comp C*)/F (co-comp C*))) p

n (H + l (F _p' C*/

F C*)). p

Therefore, in this case, equation (3) can be rewritten: n liml H + l (F C*/F C*). p' :::+00 _pi p

(3 ')

(of course, if either denumerable inverse limits or denumerable direct limits are exact, then the object (3')

2.

is zero.)

Condition (1) of the Proposition holds, if there exists

n n an integer PO such that F C = F C for all integers p ~ Po p Po n (i.e., if C is "co-discrete but not co-Hausdorff"). Proof:

Replacing C* by the completion of its co-completion if

necessary, we can assume that for all integers n.

cn

is complete and co-complete,

Let

o

E

be the dual exact couple of the filtered cochain complex C*, as defined in section 8, Theorem 1°. vp,q = HP+q(C*/Fp+1C*), for all integers p and q, and the map o

k is of bidegree (0,0). Fix integers nand p.

Then the hypotheses imply that the

Convergence

429

cycles stabilize at the bidegree (p-i,n+i-p) past the integer ro(p-i)

= B(p,n-p)+i.

Therefore, by Remark 1

following Theorem 4 we have, for every integer i

> 1, that the

inverse system:

is such that the images stabilize in the sense of Definition 2. In fact, by the Remark following Lemma 1.2, we have, more ~

precisely, for every integer i

whenever r

>

r. , where r. = sup(rO(p), r (p-l)-l,r (p-2)-2, l,p l,p O O

... ,rO (p-i+l) -i+l). that r. l,p

> 1.

of t

= B (p,n-p) +i,

it follows

1.

>

Let B

=

B(p,n-p)

=

r. , all integers l,p

Then equation (1) can be rewritten, that for all inte-

gers i,r with i r

Since rO (p-i)

B(p,n-p), for all integers i, and is therefore

independent of i i

1, that

>

1 and r

>

B

=

B(p,n-p), that the restriction

is an epimorphism:

o

By definition of nt-torsion part," see section 5, Definition 1', the supremum of the increasing sequence of subobjects, i > 1, of vP+r,n-p-r on the left side of equation (2) is the (t-torsion part of V)p+r,n-p-r.

We have likewise that the

supremum of the increasing sequence of subobjects, i

(tB V)p,n-p

~

1, of

on the right side of equation (2) is the (t-torsion

Section 9

430

t

part of B V). Therefore, by section 2, Lemma preceeding Theorem r 3, part (2) applied to the map t Or

have that the map induced by t , o

(t-torsion part of

( 3)

0

V)

p+r n-p-r ,

~

0 0B (t-torsion part of (t

V) )p,n-p , is an epimorphism, whenever r2:.B=B(p,n-p). Taking r=B in the epimorphism (3), we have therefore that ° ° p+B , n-p-B ) t°B ((t-torsion part of V) (tB V»p,n-p .

(4)

o

(t-torsion part of

Substituting equation (4) into equation (3), we obtain that, whenever r > B = B(p,n-p), then we have that the mapping inducr ed by t is an epimorphism (t-torsion part of V)p+r,n-p-r ~ tB((t-torsion

(5)

part of V)p+B,n-p-B) For the fixed integer n, this being true for all integers p, it follows that the inverse system: V)p,n-p)

° ((t-torsion part of

is such that the images stabilize in the sense of

pEZi"

Definition 2.

Therefore, by Lemma 2.2, conclusion (3), we

have that liml (t-torsion part of V)p,n-p

(6)

..-

o •

p++oo

However, since vp,n-p = Hn(C*/Fp+1C*), and the mapping

t :

vp,n-p

+

vp-l,n-p+l are the maps on cohomology induced by

the natural epimorphismsl from the exact sequence: •••

+

°i Hn(Fp_i+1C*/Fp+1C*) + Hn(C*/Fp+1C*)~ n ~ H (C* /F p-~. +lC*) + •••

Convergence we have that, for every integer i in V)p,n-

p

=

(see section

~ 0, that (precise ti-torsion

Im(Hn(Fp_i+lC*/Fp+lC*)

a,

431

+

Hn(C*/Fp+lC*)).

Proposition 3), this latter is the (p-i+l) 'st

filtered piece of Hn(C*/Fp+lC*).

Therefore

(precise ti-torsion in V)p,n-p

(7)

~

for all integers i

O.

=

F

.

p-~+l

Hn(C*/F

Hn(C*/F Hn(C*/F

p+l p+l

Therefore the supremum of the

C*) on the right of equation (7) C*)

C*) p+l'

By hypothesis (1) of the Proposition,

for infinitely many positive integers p, Hn(C*/Fp+lC*) co-Hausdorff.

But

is

subobjects of is all of

Therefore the supremum of the sub-

objects on the left of equation (7) are all of vp,n-p; or, equivalently, (t-torsion part of V)p,n- p for infinitely many positive integers p.

substituting into

equation (6) yields

o . But by section

a,

Corollary 3.2.1, the group on the left of this

equation is G~n+l(c*). (see section

a,

Therefore Hn+l(C*) is Hausdorff.

Since

Corollary 3.2.1) Hn+l(C*) is also "complete but

not Hausdorff", it follows that Hn + l

= Hn+l(C*)

is complete.

(The fact, observed parenthetically above, that hypothesis (1) of the Proposition holds if the functor "denumerable direct limit" is exact, follows from section Proof of Note 1:

a,

Corollary 3.2.1.3.)

We establish equations (6) and (7) of the

above proof as before.

Passing to the supremum over i

in eq.(7),

Section 9

432

(t-torsion part of V)p,n- p

we have that,

Hn(C*/F

F -00

p+l

C*)

Or; otherwise stated, that we have a short exact sequence: (8)

o

o

-+ (t-torsion part of

0

p n-p

V)'

->-

n H (C*/Fp+lC*)

-+ G Hn(C*/F p +lC*) -+ 0 . -00

is right passing to "liml", and using the fact that "liml" + p-++oo p-++ exact, this implies that we have a natural isomorphism: 00

By section 8, Corollary 3.2.1, the left side of this equation is canonically isomorphic to G+ooHn+l(C*), proving equation (3) of the Note.

The rest of the Note follows from section 8, the

dual of Corollary 3.2.1. Proof of Note 2:

The condition of Note 2 is equivalent to asserting that co-comp(C n ) is co-discrete. Therefore, cocomp (Cn)/Fp (co-compC n ) is co-discrete for all integers p, and therefore Hn(CO-Comp(C*)/Fp(CO-Comp C*)) is co-discrete, and a fortiori co-Hausdorff, for all integers p. Proposition 5.1.

Q.E.D.

Let A be an abelian category such that de-

numerable direct products exist and such that the functor "denumerable direct product":

AW"''''>A is exact.

Let C* be a

filtered cochain complex in A such that co-comp(C*) exists (condition automatically satisfied if denumerable direct sums exist in A), and let n be a fixed integer.

Suppose in addition

that there exists an integer PO and an integer r (1)

o whenever

p

>

PO.

l

~

I such that

Convergence

433

Assume in addition that either Axiom (P.2) holds, or that we can choose the integers PO,r , such that l hypothesis (1) above, and also the following condition, hold: (2)

EP,n-p-l r = 0 whenever p ?.PO . l

n Then the n'th integerated partial abutment H is complete. Proof: As in the proof of Proposition 5, replacing C* with (co-comp C*)' if necessary, we can and do assume that C* is complete and co-complete, and we then once again study the dual exact couple of C*.

n As before, we have that H is "complete but

not Hausdorff" and that

Since also Vp,n-p-l

Hn-1(C*/Fp+1C*), to prove the Proposition

it is necessary and sufficient to prove that, in the dual exact couple of C*, we have that

Let

Section 9

434

be the (r-l) 'st derived couple of a dual exact couple of C*, o

~

for all integers r

1.

V

r

tr-lv, and therefore for r

> 1

and for all integers p we have the short exact sequences of bi-

A,

graded objects in (3 )

o ~

(precise t-torsion part of °p-l,n-p

~ vr+l

-+

Vr )p,n-p-l

0

These sequences, for r fixed and all p, are a short exact sequence of

A.

~-indexed

inverse systems in the abelian category

The leftmost inverse system is such that all of the maps are

zero, and therefore has liml zero. -<-

sequences (3)

Throwing the short exact

through the functor "l1ml", and using the fact

that liml is right exact, it follows that -<-

(4)

A portion of the r'th derived couple is the exact sequence (4.1)

v P +l,n-p-2 __________~) vp,n-p-l r

r

\

I

EP +l,n-p-2

EP+l,n-p-l

r

r

Therefore, by hypothesis (1) of the Proposition, we have that the mappings: (4.2)

are epimorphisms whenever r

> r

l , p

~

PO - 1 .

Therefore, if Axiom (P.2) holds (see Introduction, Chapter 1,

Convergence

435

section 7), liml ~p,n-p-l

(5)

.... p ...+oo

0, whenever r

r

>

r

l

.

(This also follows, of course, from Lemma 2.1.) On the other hand, if hypothesis (2) of the Proposition holds, then from the exact sequence (4.1) we deduce that the °p+l n-p-2 0p n-p-l V' ... V ' are monomorphisms whenever

mappings: r

>

r

~

r , p l (4.3)

r

PO-l.

This, and equations (4.2), imply that,

The mappings:

~p+l,n-p-2 ... ~p,n-p-l are

r

isomorphisms for r

r ~

rl , p

~

PO - 1 ;

and therefore, once again, we obtain equation (5), whenever r > r

l

holds.

Therefore, in all cases, we have that equation (5)

.

, equation (4) for 1 ~ r ~ rl-l l vQp,n-p . imply equatlon (2). Q.E.D.

Equation (5) for r = r

Op,n-p and the fact that V1 Example 3.

Suppose that we have an abelian category A such

that denumerable direct products exist and such that the functor "denumerable direct product" is exact.

Suppose that

we are given a filtered cochain complex C* (such that co-comp C* exists), and such that, for every integer n, there exists po(n) and an integer r

an integer PO

all the terms EP,q r

l

rl(n)

> 1 such that,

in rl'st part of the spectral sequence

l

that lie on the bottom right half line: are zero.

=

p + q

=

n, p

~

PO(n)

Then condition (1) of Proposition 5.1 holds for all

integers n (and therfore also condition (2) of Proposition 5.1 holds for all integers n + 1, i.e., for all integers n). But then by Proposition 5.1, we have that the integrated

Section 9

436

partial abutments Hn are all complete for all nE1.

And also

by Example 2 above, we have that the left defect of (all four of the exact couples) of the spectral sequence is zero. Remarks.

1.

Condition (1) of Proposition 5.1 for the integer

n implies condition (2) of Proposition 5 for both the integers n-l and n.

2.

Condition (2) of Proposition 5, or condition (1) of

Proposition 5.1, for all integers n, implies that the cycles stabilize.

(Neither condition for all integers n in general

implies that the spectral sequence converges.)

Condition (2)

of Proposition 5 for all integers n can be thought of as being a condition that the cycles should stabilize "somewhat uniformly." 3.

Let n be a fixed integer, and let

tegers such that PO + qo

=

n.

(p~qO)

be fixed in-

Then condition (2) of Proposition

5, for the given fixed integer n, is equivalent to asserting that:

"The cycles stabilize

at

all spots on the line p + q

n, and the condition in (2) holds for the specific pair

(po,qO)."

Yet another way of writing condition (2) of Proposition 5 is to say that, conditions (2.1) and (2.2) below both hold:

(2.1)

The cycles stabilize, and

(2.2)

There exists a pair of integers (po,qO) such that Po + qo = n and such that the condition in (2) of Proposition 5 holds for the pair (po,qO)

This latter condition, It states that,

.

(2.2), can be expressed graphically.

437

Convergence

\

A

B

\

p

p + q

=

+

n + 1

q

n

"there exist spots A and B, on the line, p + q = nand p + q = n + 1, respectively, such that, given any point 'to the upper left' of A on the line p + q = n and any point' to the lower right' of B on the line p + q = n + 1, the differential(*) connecting them is zero."

From this point of view, it is

clear that condition (2.2) is self-dual, with nand n + 1 interchanging.

We call a spectral sequence that obeys this condition

(*)If A = (Pl,ql)' then (pi,qi) is on the same line p + q = n as A and is "to the upper left" iff pi + qi = Pl + ql (=n), and pi

<

Pl'

Similarly, if B

line as B and "to the lower right" iff pi + and pi

~

P2'

qi

= P2 +q2 (=n+l),

The condition that "the differential connecting

the spots (pi ,qil and (pi,qi) is zero", of course means that the p' q' pi,qi pi,qi ->- E 2, 2 is zero (if mapping d 'E pi-pi ~ rO; otherwise , p'-p" p'-p' pi-pi 2 1 2 1 ' q'" P l' 1 . the condition of "vanishing of d, ,~s taken to be vacuous.)

P2- P l

Section 9

438

(2.2) for all integers n, one that is semi-stable.

Semi-

stability does not imply, in general, either that the cycles stabilize or that the boundaries stabilize. Thus, condition (2) of Proposition 5 holding for all integers n, is equivalent to the statement that, "The cycles stabilize, and the spectral sequence is semi-stable." 4. category

The hypothesis, in Proposition 5, that "The abelian

A is such that denumerable direct sums exist," can be

replaced by the weaker hypothesis, that "co-comp C* exists." The most important case is when the abelian category the category of abelian groups.

A is

Then the functor "denumerable

direct limit" is exact and Axiom (P.2) holds, and therefore, by Remark 1 above, Proposition 5 is a better theorem than Proposition 5.1.

Perhaps this should be summarized as a Corollary.

Corollary 5.2.

Let

A be

an abelian category such that denumer-

able direct products and denumerable direct sums exist, such that Axiom (P.2) holds and such that the functor "denumerable direct limit" is exact.

Let C* be a filtered cochain complex in

A, and let n be an integer.

Suppose that

(2) for every pair of integers (p,q) such that p + q

=

n,

there exists a positive integer B(p,q) such that, for every integer i ~ 0, we have that dP-i,q+i = 0 whenr

ever r

~

B(p,q)+i.

Then the filtered object, the (n+l) 'st integrated partial abutment Hn + l complete, and

= Hn+l(cO-comp(c*'))

is both complete and co-

Convergence Proof:

439

By Proposition 5 we know that Hn is complete.

Theorem 4, we have that (left defect)p,n-p p.

=

By

0 for all integers

The remainder follows from section 8, Corollary 3.2.1.3.

Remark 1.

Suppose that we have the hypotheses of Example 1

following Corollary 4.2, and that the exact couple in Example 1 comes from a filtered cochain complex C*, such that the cocompletion of C* exists.

Then the hypotheses of Proposition 5

are satisfied, and therefore (by Proposition 5) the (n+l) 'st n l integrated partial abutment H + is complete, and (by Theorem 4) the (p,n-p) 'th object of the left defect is zero. If the abelian category A is such that denumerable direct sums exist and such that the functor "denumerable direct limit" is exact, then by section 8, Corollary 3.2.1.3, these conclusions can be sharpened to read:

That the (n+l) 'st integrated

n l partial abutment H + is both complete and co-complete; and n that the n'th integrated partial abutment H is such that G (H n ) ~ p

EP,n- p , for all integers p. '"

Therefore, in this case,

if the hypotheses of Example 1 hold for all integers n, then the integrated partial abutments Hn, n E~

are an abutment in

the sense of section 7, Definition 1, and are complete and co-complete. 2.

Suppose that we have the hypotheses of Example 3 follow-

ing Proposition 5.1.

Then we have observed in Example 3 that

the integrated partial abutments Hn, n E~, are complete; and also that the left defect of (all four exact couples) of the spectral sequence is zero. If in addition the abelian category A is such that denumerable direct sums exist and such that the functor "denumerable

440

Section 9

direct limit" is exact, then by section 8, Corollary 3.2.1.3, these conclusions can be sharpened to read:

That the integ-

n rated partial abutments H , n E~, are both complete and cocomplete, and are an abutment for the spectral sequence, in the sense of section 7, Definition 1.

Section 10 Some Examples

Example 1.

The Spectral Sequences of a Double Complex

Theorem 1.

Let A be an abelian category and let C**

(Cp,q, 3~i;0)' 3~O;l)) (p,q)E.". Xi?

=

be a (whole plane) double com-

plex in A (see section 3, Example 3).

Then there is induced a

conventional, bigraded spectral sequence (EP,q dP,q TP,q) r'r'r r>O starting with the integer zero, such that EP,q r

cp,q

EP,q

Hq(C P , *) ,

1

and

Proof:

,

HP ( (Hq(C i , * )) i E {')

EP,q 2

Case 1.

First, assume that the abelian category A has

denumerable direct sums.

Then define

Following notations as in [P.P.W.C.), we then define 3n :C n

+

c n + l , for all integers n, by requiring that

nl p,q C

3

=

(' l' ) p,q lnc USlon p+l,q 03(1,0)

for all integers p,q such that p + q

441

n.

Then C*

Section 10

442 is an (ordinary,

~-indexed,

the abelian category

A.

singly graded) cochain complex in

Define

n Then F Cn is a sub-object of C , and

for all integers p,n.

p

n Therefore (F C ) p

a filtered object with decreasing filtration. that dn(FpC n ) (F C*) p

p

E

'l'

C

p

E.". v

is

It is immediate

n l FpC + , for all integers n,p.

Therefore C*

=

is a filtered cochain complex.

We have that (2)

and that these isomorphisms, for a fixed integer p and all integers n, make the n'th coboundary of the cochain complex n p n-p Gp(c*) correspond to (-1 ) a(b,l)

Therefore the isomorphism

(2) induces, by passing to the subquotients, an isomorphism, (3) for all integers nand p. The spectral sequence of the filtered cochain complex (1) (see section 8, Theorem 1) is such that H

n

(G

p

(C*»

Therefore, if we define EPo,q

=

cp,q and dP,q

o

=

a p,q

(0,1) ,

then

the spectral sequence of the filtered cochain complex C*, so extended, has the desired properties. Remark:

Tracing through the construction, it is easy to see that

for r > 1, dP,q:EP,q r r

+

EP+r,q-r+l is induced by the inverser

Some Examples

443

composite relation: (,p+r-l ,q-r+l) -1 (,p+l ,q-l) ( ,p+r-l,q-r+l) 0(1,0) 0 0(0,1) 0· •• 0 0(1,0) p+l,q-l -1 o(d(O,l)

)

p,q

0(3(1,0))

by passing to the subquotients. Case 2. of

General case.

Then let AD be a full exact subcategory

A that is a set and that contains the objects cp,q for all

integers p,q, and let F be an exact imbedding from AD into an abelian category B such that denumerable direct sums of objects exist (e.g., by the Exact Imbedding Theorem [I.A.C.], one can take B

=

the category of abelian groups).

Then by Case 1 we

have the spectral sequence of the double complex F(C**) the category B.

in

By induction on r, I claim that EP,q dP,q are

r

in the image of F, r > O.

'r

and dP,q

Since

°

F(d~O;l))' we have that E~,q and

are in the image of F,

for all p,q E ~

If r _> 1, then since EP,q = Ker(dP,q)/ r+l r Im(dP-r,q+r-l), and by the above Remark dP,q is induced by the r

r

image under F of an additive relation in AD from cp,q into cp+r,q-r+l, it follows by the inductive assumption that E~~i and d~:i are in the image of F, completing the induction. Therefore the spectral sequence of the double complex F(C**) in the abelian category B is the image under F of a uniquely determined spectral sequence in [LA.C.l,

it follows

A.

Using the Introduction to

(again by induction on r

~

0) that the

spectral sequence thus defined in the abelian category

A is

independent, up to a canonical isomorphism, of the abelian category B closed under denumerable direct sums and of the exact imbedding: A'VU> B

chosen.

Q.E.D.

444

Section 10 For the rest of this Example, we will assume for simplicity

of statements that the abelian category

A is

closed under

denumerable direct sums and under denumerable direct products of objects.

The filtered cochain complex C* = (FpC*)pEQ' intro-

duced in the proof of Theorem 1 is the traditional one.

It

is co-complete and Hausdorff, but in general is not complete. Its completion is the filtere (4)

Cn"

=(

ID cP,q) p+q=n p,::O

cochain complex C* , where Ell (

II

C

p+q=n

p,q) ,

all integers

n.

, p >0

By the results of section 8, the "best possible hope" for an abutment for the spectral sequence in Theorem 1 are the integrated partial abutments Hn, n ~ 7

, which are by the definition

the cohomology of the filtered cochain complex C*", (5)

Hn = Hn(C*"), all integers n.

Also, by section 8, the spectral sequence of the double complex has a left defect, which is a quotient of E"" = (E:,q) (p,q~ and a right defect, which is a subobject of E",,'

xi'!

(If "denumer-

able direct limit" is exact, then the right defect is zero; see section 7.) By Corollary 1.1 of section 8, we have a set of partial abutments for the spectral sequence of the double complex C**, which is by definition the partial abutments of the exact couple of the filtered cochain complex (F p C*)PE7' direct limit abutments are

(6)

Explicitly, the

Some Examples

445

and (7)

n Since the filtration on C is co-complete for all integers n, if the functor "denumerable direct limit" is exact; by section 8, Corollary 1.3, equation (6) then simplifies to: (6' )

The objects on the right of equation (6') are called by some people the "total cohomology of the double complex C**." Remark:

The traditional filtered cochain complex used to con-

struct the spectral sequence in Theorem 1 is C*. noted, c*" is actually more natural.

As we have

There is one "worst"

possible choice, worse even than the traditional one; namely, the filtered cochain complex 0*, where (7.0)

-~

n D - p+q=n II cp,q)e ( p+q=n i

P20

n E 'F

p>O

with the analogously defined coboundaries and filtration. in general neither complete nor co-complete. 0* is """

.

C*, co-comp(O* ) = C*.

0* is

co-comp(o*)

yet another filtered cochain

complex that, like the traditional C* (= co-comp 0*), is "intermediately bad"; it is the dual construction of the traditional. (7.1)

Explicitly, 0*" is determined by Dn "::

II Cp , q all integers n. p+q=n '

By section 8 the "worst possible" cochain complex 0** gives

446

Section 10

rise to four exact couples,

(all having the same spectral

sequence,namely the spectral sequence of Theorem 1), o 0 co-comp(2 )---> (2

1'1

)~(2)-7(2

)

By results of section 8, the exact couple (2)

is isomorphic to

the traditional exact couple of the traditional C* Similarly, the exact couple (2 act couple of 0*1'1.

0

= co-comp(O*).

is isomorphic to the dual ex-

)

The partial abutments of this dual exact

couple are

lim -+ p-++oo

(7 )

Hn(O*/F

-p-l

0*)

and (8)

where IT

p'

cP

',n-p'

< p

(As usual with the partial abutments coming from the dual exact couple of a filtered cochain complex, the dual inverse limit abutments are more interesting than the dual direct limit abutments.)

If the functor "denumerable direct limit" is exact,

then since 0* is co-complete,

(7' )

0, all integers n.

Therefore, in this case, by section 8, it follows that the o n filtered objects "H , n

E 7 , of equation (8), the inverse li-

mit abutments of the dual exact couple of 0*, must have the

Some Examples

447

same associated graded as the integrated partial abutments Hn , n E? , of equation (5); and therefore in this case that (8')

n

F_oo("H )

~ Hnt", all integers n.

If "denumerable direct product" is exact, then whether "denumerable direct limit" is exact or not, we have that the cohomology Hn(O*A) of the (product) cochain complex O*A, equation (7.1), for its natural filtration, is "complete but not HausTherefore, if "denumerable direct limit" is exact, then F -00 Hn(O*A)/G +00 Hn(O*A) : co-comp (HnA) , all integers n, as complete, co-complete filtered objects. Let us now assume also that the abelian category A obeys Axiom (P.2)

(see Introduction, Chap. I, sec. 7), or else that the

double complex C** is such that conditions (1) and (2) of section 9, Proposition 5.1, hold. By Corollary 4.1 of section 9, if the spectral sequence is such that the cycles stabilize, then the left defects are all zero.

(Therefore, in this case, if also the functor "de-

numerable direct limit" is exact, then the integrated partial abutments are an "honest" abutment for the spectral sequence.) By Prop. 5.1, sec. 9, this is so--that is, the left defect is zero -- (whether or not (P.2) holds), for example if the double complex C** vanishes outside of a region R, such that, for every integer n, the line:

p+q

n is such that there exists

an integer PO' such that the half line {(p,q) :p+q = n,

p~O}'

to the bottom right of the spot (PO,n-PO)' is entirely outside the region R (or, weaker, if there is such a region R such that,

Section 10

448

for every integer n, there exists an integer r = r(n), such that for every pair (p,q) of integers not in R, with p+q=n, we have EP,q = 0). r

For example, this is so if the double complex

is first quadrant; third quadrant; second quadranc upper halfplane; or, more generally, vanishing in the fourth quadrant. Also, by e.g. section 9, Proposition 5.1, if the double complex C** is confined to such a region R (or, weaker, if there is such a region R such that, for every integer n there exists an integer r(n)

>

0 such that EP,q = 0 for all p,q r(n)

such that p + q = nand (p,q) is not in R), then also the integrated partial abutments are complete.

(Proposition 5 of

section 9 gives a more general such statement when Axiom (P.2) holds).

When this is so, and the functor "denumerable direct

limit" is exact, then the integrated partial abutments Hn, nEll', are therefore a complete and co-complete abutment for the spectral sequence.

And if the spectral sequence is confined

to such a region (resp.:

to such a region that obeys the in-

dicated hypotheses and dual hypotheses) then also the integrated partial abutments are discrete (resp.: finite), since EP,q '" vanishes outside the region. Example 1.1.

If the double complex C** vanishes outside of a

region R such that every line

p + q =' n intersects that region

in only finitely many terms (--or, by section 9, Proposition 5, if say Axiom (P.2) holds and either denumerable direct limit or denumerable inverse limit is exact, and if there is such a region R such that, for every integer

~

there exists an integer

B(n) such that for all (p,q) not in R such that p + q = n we have that EP,q + ( ) = 0 -- or, by section 9, B(n) sup p,q --

Some Examples

449

Proposition 5.1, if there is such a region R such that, for every integer n, there exists an integer rl(n) EP,q rl(n) p + q

= =

>

0 such that

0 for all pairs of integers (p,q) not in R such that

n --) then by the above, the integrated partial abutn E ~ , are an abutment, and each H has a finite fil-

ments Hn, n tration.

Example 1.1.1. If the double complex C** vanishes outside of a region R as in Example 1.1, then the reader will verify that the filtered cochain complexes C*, 0*, C*A, D*A discussed in a Remark above all coincide; and therefore the four exact couples collapse to two, co-comp(2

0

)

=

(2

0

),

and (2)

=

(2 A).

The in-

tegrated partial abutments Hn, equation (5), then also coincide with the direct limit abutments 'Hn, equation (6), and the dual

On

inverse limit abutments "H , equation (8), for all integers n; and each has a finite filtration (and the inverse limit abutments "H n and the dual direct limit abutments dentically zero, n E

~

,~n are then

i-

.)

Two important special cases of Example 1.1.1 are first quadrant double complexes and third quadrant double complexes (the latter can be re-interpreted as "first quadrant homological double complexes" by the indexing convention C

E-P,-q r

p,q

The easy details are left to the reader). Remark.

Suppose that the hypotheses are as in Theorem 1.

Then

we also have the reverse rev(C**) of the double complex C**, such that the (p,q) 'th object in rev(C**) is Cq,p, and such that (rev C**) p,q d(O,l)

C** q,p 11(1,0)'

(rev C**) p,q d(l,O)

C** q,p

d (0, 1)·

The

450

Section 10

spectral sequence of the reverse of C** is called the second spectral sequence of the double complex C**. ('EP,q 'dP,q 'TP,q) r ' r ' r p,q,rE'l'

We will denote it

Thus explicitly,

r>O 'EP,q 0

Cq,p,

'EP,q

Hq(C*'P)

'EP,q 2

q * HP ((H (C , i)) . E 'l') .

1

and ~

.

This second spectral sequence of the double complex C** is in general very different from the one of Theorem 1, which is called the first spectral sequence of the double complex C**. However, the sets of partial abutments and the integrated partial abutments of these two spectral sequences are closely related, but usually with very different filtrations.

For

example, the "best possible" filtered cochain complex C*"', equation (4), for the first spectral sequence (such that the cohomology of C*A is the integrated partial abutments, equation (5), of the first spectral sequence) is the "worst possible" filtered cochain complex for the second spectral sequence -but with a very different filtration. versa.

And, of course, vice

The "traditional" filtered cochain complex C*, equation

(1), for the first spectral sequence is the

~

as the "trad-

itional" filtered cochain complex for the second spectral sequence -- but again with an in general very different filtration.

And similarly for the "dual of the traditional"

filtered cochain complexes D*"', equation (7.1).

Also, if the

Some Examples

451

functor "denumerable direct limit" is exact, then the direct limit abutments of both spectral sequences are both isomorphic as objects to what is traditionally called the "total cohomology" Hn(C**) = Hn(C*), equations (1) and (6'), of C**, but in general have entirely different filtrations.

Explicitly,

the p'th filtered piece of the filtration induced by the first spectral sequence is the image of H*(

1&

p' f-q , =n

cp',q')

n E 6'

+H*(

Sl

p' .fq , =n

cp',q')

n E 6'

p'~

and the p'th filtered piece of the filtration induced by the second spectral sequence is the image of H*(

1&

p' f-q , =n

cp',q')

n E 6'

q'~

which are in general very different. The most important abutment-like invariants, the integrated partial abutments, are in general not isomorphic for the two spectral sequences, not even as objects (forgetting filtration). However, in the very special case of Example 1.1.1 above, they do coincide as objects (but not in general as filtered objects); and then as observed in Example 1.1.1, these then also coincide (as objects) with the direct limit abutments of the exact couple of C*, and with the inverse limit abutments of the dual exact couple of D*

1\

(both of which are then simply the cohomology

of these cochain complexes ignoring filtration). Example 2.

The Spectral Sequence of a Composite Functor.

Theorem 2.

Let A, Band C be abelian categories, such that the

Section 10

452

categories A and B have enough injectives, and let F: A'VV>B n

be additive functors.

and G: B "SV>C

Let F , n

and (Go F) , n > 0, denote the riqht derived GoF respectively. (1)

n

0, G , n

>

0,

functors of F, G and

Suppose that

Q injective in Gn(F(Q))

~

=

A implies that

0 in C, for all integers n > 1.

Then, for every object A in

A, there is induced a conventional,

bigraded spectral sequence with abutment starting with the integer two such that "p,q "'2

( 2)

and such that the n'th object of the abutment is n H = (GoF)n(A), all integers n.

(3) Notes.

1.

Equation (2) implies that the spectral sequence is

first quadrant -- i.e., that E~,q

=

0 unless p,q > 0, all

integers r .:. 2. 2.

n The filtration on H is finite, for all integers n.

The proof of Theorem 2 is very well known, and need not be repeated.

The earliest published reference that I know is

[C.E.H.A.], near the end of the book. Remarks.

1.

The spectral sequence of Theorem 2 is the spectral

sequence of a double complex that vanishes outside the first quadrant.

(That is a special case of Example 1.1.1 above.)

The second spectral sequence of that double complex is also used in the proof of Theorem 2, to compute the abutment. 2.

If all the other hypotheses of Theorem 2 except possibly

Some Examples

453

condition (1) hold, then the conclusions of Theorem 2 hold iff condition (1) of Theorem 2 holds.

3.

Condition (1) (1')

is equivalent to:

A any object in A and n an integer that there exists an object A' in

~l

implies

A and

a mono-

morphism f: A ~ A' such that Gn(F(f»:Gn(F(A»~ Gn(F(A'» Exercise 1.

Let F: A '\N>B

categories, such that tives.

is the zero map in C.

A does

be an additive functor of abelian not necessarily have enough injec-

Then let us define a system of derived functors of F to

be (1)

A non-negative cohomological exact connected sequence of functors from A into B, together with

~ F O, such that

(2)

A map of functors n:F

(3)

For every obj ect A in A and every integer n

~

1, there

exists an object A' in A and a monomorphism f:A

~

A'

such that Fn(f) :Fn(A) ~ Fn(A') is the zero map in B, and (4)

For every object A in A, there exists a directed set D, and an exact sequence

of direct systems indexed by the directed set D in the abelian category A, where A denotes the "constant direct system A", such that the direct system indexed by D in the abelian category A,

454

Section 10

trivializes in the sense of section 6, Exercise 1 following Lemrna 7, and such that, the mapping of direct systems: (n

f l.

°

° ° °

1 1 )·ED:(Ker(F(Q.) .... F(Q·)))·Eo-,.(Ker(F (Q.) .... F (Q')))'Eo l l l l l l l

induces an isomorphism on the direct limit.

(Notice that by

section 6, Exercise 1, the direct limit of the leftmost direct system exists in rightmost

°

A; and, by left-exactness of F , the

direct system is isomorphic to the constant direct

system FO(A), and therefore has direct limit FO(A). Exercise 2.

It is not difficult to show that, if F: A'vu> B

is

a left-exact functor of abelian categories, then if a system of derived functors of F exists as in Exercise 1 above, then they are unique up to canonical isomorphism.

(Also, if the category

A has enough injectives, then the derived functors of F as in Exercise 1 above exist, and are canonically isomorphic to the usual derived functors as defined for example in [C.E.H.A.]). One can give examples of additive functors such that the derived tunctors as defined in Exercise 1 above do not exist. (For example, the functor F(A) = Hom(

n~l

(7./116':) ,A),

from the

category of finitely generated abelian groups into the category of abelian groups.) Exercise 3. F:

It can be shown that, given an additive functor

A~> B of abelian categories, if there exists a left-exact

functor FO: A ~> B

and a map of functors n:F .... FO such that

condition (3) of Exercise 1 holds, then FO is a universal leftexact functor into which F maps. false. )

(The converse is in general

Some Examples Exercise 4.

455

Theorem 2 can be generalized to a version that does

not involve injectives. Theorem 2'. and

G:B~~>C

Let A, Band C be abelian categories, let

F:A~~>B

be additive functors, such that the derived functors:

F* and G* of F and G exist.

Suppose also that condition (1')

of Remark 3 following Theorem 2 holds.

Then the derived functors

(GoF)* of GoF exist, and for every object A in A, we have a conventional bigraded spectral sequence starting with the integer two, such that

and such that the abutment is Hn

=

(GoF)n, all integers n.

And

Notes 1 and 2 following Theorem 2 above hold. Remarks.

1.

It is easy to show that, if all of the hypotheses

of Theorem 2' except possibly condition (1') hold, and if the conclusion of Theorem 2' holds, then condition (I') must hold. 2.

The proof of Exercise 2 is not difficult.

and 4 are far less obvious as stated.

Exercises 3

However, they can be

proved by an elegant method, using an (as yet unpublished) construction of the author, which he calls the category of direct limits . Example 3. The Adams Spectral Sequence We describe this construction only in very general terms, concerning only the spectral-sequence-theoretic aspects.

The

deep algebraic-topological aspects are handled well in the original reference [F.A.]. I am indebted to Professor John Harper of the University of Rochester for familiarizing me with this material.

456

Section 10

Definition 1.

A continuous function f: (X,x o )

+

(Y'Yo) of base-

pointed topological spaces is a stable homotopy fibration iff for every integer n, there exists a positive integer i

>

3

such that the induced mapping on the relative homotopy groups, i i i -1 '!Tn (S f): '!Tn (S X,S (f (Yo)))

i '!Tn (S Y)

+

(Where SZ is the suspension of Z, for all

is an isomorphism.

base-pointed topological spaces Z.

We define the suspension

of Z to be the double cone on Z, modulo the equivalence relation that identifies the double cone of the base-point to a point. ) Definition 2.

A spectrum Z is a sequence

(Zn'Tn)nE~'

indexed

by the integers, where Zn is a base-pointed topological space and Tn:SZn

+

Zn+l is a continuous, base-point preserving func-

tion, for all integers n.

If Z

=

(Zn,T ) and Y n

are spectra, then a map of spectra f: Z f = (fn) nE

~'

where f n : Zn

+

+

=

(Yn,on)

Y is a sequence

Yn is a continuous, base-point

preserving function such that the diagram of topological spaces and continuous functions:

.,

zn+l

'n

r

SZ

Yn + l

r

') SY

n Sf

°n

n

n

is commutative, for all integers n.

I f f:Z + Y is a map of

spectra, and i f Yn is the base-point of Y for all integers n, n'

Some Examples

457

- 1 (y))) E'7I (f -1 (y),T 1 S(f n n n n n n v. spectrum, which we call the fiber of the map f.

then th e sequence of fibers:

is a

If X is a base-pointed topological space, then define

X n

>

0,

<

O.

Then, since the functors Sand n are (X ) n E'Z' n

adjoint, the sequence

is in a natural way a spectrum, the associated

spectrum of X. Definition 3. the fiber.

Let f:Z

Y be a map of spectra and let F be

+

We say that f is a stable homotopy fibration iff

there exists an integer nO such that, whenever n continuous function: Zn

+

~

nO' the

Y is a stable homotopy fibration n

of base-pointed topological spaces in the sense of Definition 1. Then if f:X

+

Y is a continuous, base-point preserving function

of base-pointed topological spaces, and if (f ): (X ) n n

+

(Y ) is n

the induced map on the associated spectra, we have that f is a stable homotopy fibration in the sense of Definition 1 iff (fn) is a stable homotopy fibration in the sense of Definition 3. If Z and X are spectra, then for every integer n, define {Z,X}n = lim {homotopy classes of continuous, basek++oo

point preserving functions from Zn+k into XnH.} . Then {Z,X}n is an additive abelian group, for all integers n. If A is a pointed topological space with associated spectrum (A ) , and X is any spectrum, then define likewise k

458

Section 10

If B is any pointed topological space, then define likewise

for any spectrum or pointed topological space A, all integers

n. Also, for every spectrum Z, define

TI~lZ)

=

{SO,Zln' all integers n, TIslZ) is called the n'th

where sO is the zero sphere.

n

stable homotopy group of Z, for all integers n. Remark:

One traditionally assumes more hypotheses in the defin-

ition of a spectrum Z

=

(Z ,T) n

n n

E

7'

than we have in Definition

2 above, e.g., connectedness conditions on the maps Tn' n E 7'. For a spectrum Z obeying suitable such sets of conditions, one can show that, for any finite polyhedron A, that {A,Zln n < 0.

=

° for

(For example, this is so if Z is the spectrum of a

topological space.)

In particular, taking A to be the zero

sphere, it follows that TIst(Z) n

=

0 for n <

° and

for such a

spectrum Z. Lemma.

Let p:Z

~

X be a map of spectra that is a stable homo-

topy fibration, and let F be the fiber.

Then for every finite

polyhedron A there is induced a long exact sequence of abelian groups:

Proof:

The proof immediately reduces to the case in which Z

and X are spectra of topological spaces; and then replacing Z and X by the geometric realizations of their sing. cxs. and

Some Examples

459

replacing Z with an appropriate homotopy equivalent CW-complex Z' and the mapping Z'

~

X with an appropriate homotopic mapping,

we can assume that the continuous function homotopy lifting property.

f: Z

~

has the

But then the corresponding theorem

even for ordinary homotopy classes of maps: and A

-+ X

A

~

F, A

~

Z

X (rather than stable homotopy classes of maps) is

well-known, and implies the Lemma. Now, suppose that we have a sequence Xp' p E together with maps tp+l:Xp+l

~

~

, of spectra,

Xp that are stable homotopy

fibrations. p.

Let Yp + l be the fibers of these maps, all integers Then, by the above Lemma, for every finite polyhedron A,

we have a long exact sequence (1 )

()

Jl

{A,Y} p n- l~'"

where k is the inclusion from Y into X. Therefore, if we define ( 1.1)

vp,q

(1. 2)

EP,q

and

then V bigraded abelian groups, and we have the conventional, bigraded exact couple starting with the integer +1, (2)

Section 10

460

where k is of bidegree (0,0) and is induced by the inclusion k:Y

~

X, t is of bidegree (-1,+1) and is induced by the projec-

tions t:X

p+l

~ X

p'

so that h has bidegree (1,0). tral

a-p-q

and hP,q =

in the exact sequence (1),

By sec. 5 and 7, we have the spec-

sequence of the exact couple (2), which one can call a

generalized Adams spectral seguence.

It is a conventional, bi-

graded cohomological spectral sequence starting with the integer +1, such that

(3)

=

EP,q

1

{A,Y } P -p-q

The inverse limit abutments are

(4)

and the direct limit abutments are (5)

'H

n

=

lim {A,X

pt+a>

} +1 -p-n

As always, the inverse limit abutments are complete and the direct limit abutments are co-complete for their natural filtrations. As stated above, this construction is of course much too general to prove very much.

In the

"true" Adams spectral

sequence [F.A.j, for example, one assumes enough conditions on the spectra Y such that the groups {A,Y} are zero if n is p p n negative.

Then the spectral sequence (3) is zero outside of

the slanty half plane

n < O.

In addition, in the original

spectral sequence we have that Xp

=

therefore also Yp = (a point) for

p~

(a point) for p -1, whence

~

-1, and

Some Examples

o

for p

<

461

-1 ,

so that the spectral sequence is confined to the region: n

<

p~O,

0, i.e., to the shaded area below.

Figure 1 Also, we then have, by equation (1.1), that vp,q = 0 unless p

~

O.

Then from equation (5) it follows that the direct

limit abutments are all zero.

And, by section 7, Proposition

4, we have that the inverse limit abutments "H

n

are complete

and co-discrete. Therefore the inverse limit abutments will be an "honest" abutment iff the left defect is zero.

The (p,n-p)'th group of

the left defect is zero iff every element of {A,X}

p -n

---

that

comes from an element of {A,Xp+k}_n for all integers k > 0, comes from an element of the inverse limit. By section 9, Corollary 4.1, this will be so if the spectral sequence is such that the cycles stabilize.

However, this is

not trivially clear from the region pictured in Figure 1. However, for the special situation studied in the "true" Adams spectral sequence, one has a direct algebraic-topological proof that the cycles stabilize (see [F.A.]).

Therefore, by section

9, Corollary 4.1, it follows that the left defect is zero, and therefore: n The inverse limit abutments "H , n E., , are complete and

462

Section 10 co-discrete, and are an abutment in the sense of section 7, Definition 1.

Remarks:

1.

In the actual Adams spectral sequence, each Y p

is of the homotopy type of a finite direct product of K(n,n) 's. Therefore, for each p, {A,Y} is zero for all but finitely p n many n.

In terms of the spectral sequence, this means that in

Figure 1 for EP,q , each vertical line has only finitely many 1

non-zero terms.

(However, that in itself is not enough to

imply that the cycles stabilize.) 2.

Also, in the actual Adams' spectral sequence, Adams

proves that, for every integer n, there exists an integer rO ro(n), such that for all integers p the cycles in degree (p,n-p) stabilize past the integer rO

=

rO(n).

By section 9, Proposi-

tion 1, it is equivalent to say that r

[(t °V)n(Ker t))p,n-p=[(t-divisible part of V)n(Ker t))p,n-~ for all integers p.

But then, by Lemma 2.1 and the Remark

following, we have that

[(troV)n(t-torsion part of V))p,n-p=[(t-divisible part of V)n (t-torsion part of V))p,n-p i but since vp,n-p

=

0 if P < 0,

(t-torsion part of V)

V.

Therefore (t-divisible part of V)p,n- p , for all integers p. Of course, the above condition implies that the cycles

Some Examples

463

stabilize, and therefore (section 9, Corollary 4.1) that the left defect is zero.

Therefore

(t-divisible part of V)n(t-torsion part of V)

=

(infinitely

t-divisible part of V)n(t-torsion part of V), i.e., every t-divisible element in vp,q is infinitely t-divisible, for all p,q Therefore, we have that

(infinitely t-divisible part of V)p,n-p ,

for all integers p.

That is:

exists an integer rO

=

u

E {A,X}

p n

For every fixed integer TI, there

rO(-n), such that if p

o=

for all integers i

>

0

u and such that {A,projectioni+l}n maps u i + l

into u ' for all integers i i 3.

0 and if

is the image of an element in {A,X + } , then p rO n

there exist elements u1.' E {A,X +' } P 1. n such that U

~

>

0

Spectral sequence-theoretically, to obtain the results

indicated in Remark 2, all that is needed (see sec. 9, Rk. 1 after Prop. 5) is: That the cycles stabilize, and that the spectral sequence is semi-stable (sec. 9, Rk.3 after Prop.5.1). a lower (or upper) half plane spec. seq. is automatically 4.

(And

sem~s~le).

Since the spectral sequence is confined to the lower

half plane, we have trivially that the boundaries stabilize. Therefore the fact that the cycles stabilize implies that the spectral sequence converges. 5.

In a later construction I understand that Adams

produced an alternate construction of an "Adams' spectral

464

Section 10

sequence," that also comes from an exact couple, but in this case one such that vp,q

=

0 whenever p > O.

Therefore the

spectral sequence in that case is confined to the left half plane, and the inverse limit abutments vanish. 7, Proposition 4, part (2), we have that:

And by section

The direct limit

abutments 'Hn, n E 7 , are discrete and co-complete, and are an abutment for the spectral sequence.

(Notice that for such

a spectral sequence, we do not need to know that the stable homotopy { }n vanishes in negative dimensions.)

CHAPTER 1 THE GENERALIZED BOCKSTEIN SPECTRAL SEQUENCE

Let

A

be a ring with identity and let

element in the center of the ring A. Let

A1\t

pletion of

Then

A

for the t-adic topology. n

n

c* = (C , d ) nE.?!'

Let

tE

A

be a fixed

denote the comAl'lt

is a ring.

be a cochain complex of left A-modules

indexed by all the integers, such that multiplication by t:C

n

~ c

n

is injective (i.e., such that

t-torsion), all integers

o

n.

n C

has no non-zero

Then from the short exact sequence:

~ C* ~ C* ~ C*/tC* ~ 0

of cochain complexes of left A-modules, we obtain the long exact sequence of cohomology: (1) If we let

V =(Hn(C*»

nEZ'

and

E = (H

n

(C* /tC*) ) nE.?!' then the long

exact sequence (1) can be "wrapped around" to give the singly graded exact couple (see Introduction Chapter 2, section 5) (2)

where ~

t

is degree-preserving and is multiplication by

is degree-preserving and induced by the natural maps: 465

tEA,

Chapter 1

466

n Hn (C*) .... H (C* /tC*) n E 'l' , and

d

increases degrees by one and

is induced by the coboundaries in the long exact sequence (1). Therefore this exact couple is a conventional, singly graded exact couple as defined in Introduction, Chapter 2, section 5. Then, see the Introduction, Chapter

2, section 5, Definitio

5', we have the associated spectral sequence of the exact couple (2), which is a conventional, singly graded spectral sequence starting with the integer zero, where n En = H (C* /tC*) , all integers

( 3)

n E'Z ,

o

(i.e.

EO

=

the graded group E).

This spectral sequence is

called the Bockstein spectral sequence of the cochain complex C*

with respect to the element

t

in the center of

A.

As usual, the r'th derived couple of the exact couple (2) is an exact couple, of the form: (4) d

where

trV

is the graded subgroup of

subquotient of

o

r

V, Enr

E~, and the other two maps

and +1) are determined from

1T

at

-r

and

is of course a 1T

d,

r

,

d

r

respectively,

the exact couple (2) by passing to the subquotient. is the cohomology of the cochain complex boundary the n'th coordinate of

1T

r

En r

(of degree in

And

,with n'th co-

nd • r

Studying the exact couples (2) and (4), we have seen in the Introduction, Chapter 2, section 5, Corollary 3.1' equations (lr) and (2 r ), and Corollary 3.2', equations

(lrl and (2 ), that r

Bockstein Spectral Sequence Lemma 1. (1)

Let

n

be an integer.

Then

uE En = Hn(C*/tc*)

An element

is an r-fold cycle

o

dn:Hn(C*/tc*) ~ Hn+l(C*)

iff its image under the coboundary of (2)

The restriction of

d

467

n

Hn+l(C*),

induces an epimorphism:

(One might call this latter group: "precise t-torsion in Hn+l(C*) (3) subgroup in

that is r-times t-divisible"), The r-fold boundaries in n Ker (t r :H (C*)

-+

are the image of the r (this is the "precise t -torsion

Hn (C*))

Hn(C*)") under the natural map:

and (4)

n u E H (C*)

An element

is such that the image under

the natural map: boundary iff there exist v

v,wE Hn(C*)

with v E Kert

a precise tr-torsion element) such that

Definition.

uE M,

we say that

t-torsion element iff t·u = 0; if tr-torsion element iff

tr·u

iff there exists an integer tr-torsion element.

n

tr·M.

=

(Le.,

v + two

Let us introduce some terminology formally.

M is a left A-module and

u E

u

r

u

r u

0;

r

~

We say that

~

0

then

u

is a 12 reci se u is a 12 reci se

is a t-torsion element such that

0

u

If

u

is a precise

is t-divisible iff

is infinitely t-divisible iff there exists a

r~O

sequence of elements u

= u o'

ur,r

~

0, of elements of

M, such that

all integers r ~ O. t·u + l = u ' r r Then, by Introduction, Chapter 2, section 5, Corollaries and such that

3.1' and 3.2', equations (1) and (2) of both Corollaries, we

468

Chapter 1

have that Corollary 1.1. (1)

Under the hypotheses of Lemma 1

An element

E E~

u

= Hn(C*/tc*)

iff its image under the coboundary

is a permanent cycle

dn:Hn(C*/tc*)

+

Hn+l(C*)

is

t-divisible. (2)

The restriction of the n'th coboundary in the long

exact sequence (1) induces an epimorphism of left A-modules (in fact of left (A/tA)-modules) ZOO(E~)

it-divisible, precise t-torsion elements in

+

Hn+l(C*) } The permanent boundaries in

(3)

the natural map: it-torsion in

Hn(C*)

E~

Hn(C*/tc*)

of the subgroup

Hn(C*)}.

An element

(4)

+

are the image under

the natural map:

u E Hn(C*)

n H (c*)

boundary iff there exist

+

is such that the image under

n H (c*/tC*)

E~

v,w E Hn(C*)

is a permanent

with

vat-torsion

u = v + two

element such that

Then, by Introduction, Chapter 2, section 5, Theorem 4', we have Theorem 2.

Let

A

be a ring with identity, let

element in the center of the ring complex of A-modules.

A, and let

denotes the

be an

be any cochain

Then consider the singly graded, cohomo-

logical spectral sequence defined above.

If

~

t

It starts with

Eoo-term, then for each integer

the short exact sequence of A/tA-modules:

n

we have

Bockstein Spectral Sequence (1)

o

+

469

(A/tA) 8~n(C*)/(t-torsionD A

t-divisible, precise t-torsiOn) ( elements in Hn + l (C*)

+

O.

The monomorphism in the short exact sequence (1) by the natural map:

Hn(C*)

+

Hn(C*/tc*), while the epimorphism

in the short exact sequence (1) dn:Hn(C*/tc*)

+

is induced

is induced by the n'th coboundary

Hn+l(C*), in the long exact sequence of cohomol-

ogy. Remarks 1.

I was first made aware of the Bockstein spectral

sequence by Birger Iversen, in the case that in fact a discrete valuation ring, and erated A-module, all

n E 7

cn

A

is Noetherian,

is a finitely gen-

(Then the third group in the

short exact sequence of Theorem 2 vanishes, so that the monomorphism in that sequence is an isomorphism).

However, we need

the greater generality. 2.

Much of the preceding depends

only on the long exact

sequence of cohomology (1) at the beginning of the chapter. More precisely,

(and more generally), let:

dn - l n ->V be any long exact sequence of abelian groups and homomorphisms, indexed by all the integers,

(such that the groups in spots

congruent to zero mod 3 are equal to those in spots congruent to one mod 3, as indicated in the display of the sequence). Then we can define a singly graded exact couple by defining

n

V = (V ) nEzr '

Chapter 1

470

where

t

is deduced from the

is deduced from the deduced from the

d

TIn n

tn

and is of degree zero,

and is of degree zero and

and is of degree plus one.

a singly graded cohomological spectral sequence

d

IT

is

Then we obtain E~,

r ~ 0,

such that

and all the results of Lemma 1, Corollary 1.1 and Theorem 2 hold (with

"~,,

replacing

"Hn(C*/tC*}" throughout.} A-module where

A

the endomorphism

=

"Hn(C*)" (Here,

and

~

n "H "

replacing

can be regarded as an

P[Tj, and where the action of

tn

of the abelian group

T

on

n V

is

~, all integers

Taking this point of view allows us to use terminology

n EP

such as lit-torsion", "t-divisible", etc., in Definition 1 above). (A practical application of Remark 2 is that, if one is working with a cohomology theory, such as e.g. cohomology of sheaves, one might work directly with a long exact sequence of cohomology, without worrying whether it comes from cochains) . Remark 3.

Do results generalize to other abelian categories?

Of course the answer is "yes." Assume, for simplicity, that

Let

A be any abelian category.

A has the property that denumenble

infima and suprema of subobjects exist.

(E.g., this is the case

if denumerable direct sums and denumerable direct products of objects exist).

Then we pose a definition.

object, and

+

t:A

A

If

A

is any

is any map, then the precise t-torsion

471

Bockstein Spectral Sequence in

A

is Ker(t); the precise {-torsion in

integers r

~

O.

A

is Ker(t r ), all

Also we define the t-torsion in

supremum of the subobjects of

L

A:

A

to be the

(precise {-torsion in

A) .

r>O We also define the infinite t-torsion in Ker(A

+

lim (A

t

+

A

t

+

A

t

+

A

t

+

... )) ,

A

to be

whenever the direct limit

+

exists.

Notice that the infinite t-torsion contains the t-

torsion - sometimes properly, if denumerable direct limit is not an exact functor.

We define the t-divisible part of

be the infimum of the subobjects:

n (Im(tr:A

+

A)).

A

to

The

r~O

infinitely t-divisible part of [lim( ... ~ A ~ A ~ A)l exists.

+

A

is the image of the map:

A, whenever the indicated inverse limit

Notice that the infinitely t-divisible part of

A

is

contained in the t-divisible part, sometimes properly, when denumerable inverse limits are not exact.

(Note:

As we have

observed in Introduction, Chapter 2, section 2, Example 2, the concepts of "t-torsion part" and "t-divisible part" are selfdual, in the sense that one flips into the other (modulo the usual one-to-one correspondence between subobjects and quotient objects of a fixed object

A

passing to the dual category).

in a fixed abelian category after Similarly for "infinite t-torsion

part" and "infinitely t-divisible part".)

Then, the generaliza-

tion of Ie. g., Theorem 2 to abelian categories

(which

follows from Introduction, Chapter 2, section 5. Theorem 4') is: Theorem 2'.

Let

A be an abelian category such that denumer-

able suprema and infima of subobjects exist.(*)

Let

(*) If one deletes this hypothesis, there would be some difficulty in defining "permanent cycles", "permanent boundaries" or "E~" ,.•hen attempting to construct the indicated spectral sequence.

472

Chapter 1

be any long exact sequence, indexed by all the integers, in the abelian category

A (such that the objects in any spot con-

gruent to zero mod 3 is the same as that in the next spot, congruent toone mod 3, as indicated in the display).

Then there is

induced a singly graded, cohomological spectral sequence, such that

E~

n H , all integers

and such that, for each integer

n,

n,

we have a short exact

sequence: (2)

o ~ [Vn/(t n -torsion part of Vn »)/ t n . [vP/(t n -torsion part of Vn ») ~ En ~ (t-divisible, precise t-torsion 00 part of ~+l) ~ 0,

where the monomorphism, respectively epimorphism, in this sequence is induced by

nn' respectively

n d ,

where

lit • [~/(t -torsion part of V»)" denotes the image of the n

n

n

endomorphism induced by

of the object:

part of V »), and where

is the n'th group of the

n

Eoo-term

of the indicated spectral sequence. Theorem 2' is of course a special case of Introduction, Chapter 2, section 5, Theorem 4'.

And of course, Lemma 1, and

Corollary 1.1 also generalize to the hypotheses of Theorem Remark.

2~

As we have observed in Introduction, Chap. 2, section 5,

Remark 2' after Theorem 4', notice that the short exact sequence (2) of Theorem 2' is self-dual--that is, if we pass to the dual

Bockstein Spectral Sequence abelian category

O

A

,

473

and re-index the sequence (1)

so as to

obtain an analogous long exact sequence in the category

AO ,

and therefore also a Bockstein spectral sequence (cohomological, singly graded)

in

O

A , and a short exact sequence

(2°)

in

AO

analogous to (2), then the sequence (2°) is simply the sequence (2) thrown into the dual category (and "written backwards") , with the first and third terms interchanged.

Thus, at this

level, the "t-divisible" third term in the sequence (2) becomes symmetrical to the more "ordinary looking" first term in that sequence. Also, the short exact sequence of Theorem 2' has an analogue for

instead of

Namely, if

abelian category,

(whether or not

A has the property that

A

is any

denumerable sups and infs of subobjects exist), if

C*

cochain complex (indexed by all the integers) and if is any endomorphism of all integers

C*

such that

t

n, then for every integer

is a t*:C*

~

C*

is a monomorphism,

n

r

~

we have the

0

short exact sequence (2' ) r

o

~ (A/tA)0{precise tr-torsion part of Hn(C*)) ~ En A r

t·(precise tr+l-torsion part of Hn+l(C*)) ~ (Here we have used notations as if where element

A

is a ring, t*:C*

+

C*

tEA in the center of

-+

o.

A = category of left A-modules,

is multiplication by some A.

In the general case, if

is any object in an abelian category and

t:M

~

M a map, then

"(A/tA) 0M" must of course be replaced by "Coker(t:M ~ M) ". A (Of course, Also tlt·M" is understood to be "Im(t:M -+ M) "). also, all exact sequences in this Chapter involving

En r

(as

M

474

Chapter 1 En) hold for any abelian category, even if denumer00

opposed to

able suprema and/or infima of subobjects do not exist).

The

proof is Introduction, Chapter 2, section 5, Corollary 2.2' Remark 4.

One might wonder, can one generalize Theorem 2'

(and

Lemma 1, Corollary 1.1, and the exact sequences in the preceding Remark,) to cochain complexes not injective?

C*

such that

t*:C*

~

C*

is

The answer is "yes", even at the abelian category

level.

A be an abelian category (such that infima and suprema

Let

of denumerable sets of subobjects of objects exist), let be a cochain complex in let

t*:C*

~

C*

C*

A ( indexed by all the integers) and

be any map of cochain complexes (not necessarily

a monomorphism).

Then we define the (generalized) Bockstein n

n

r

r

spectral seguence, a simply graded spectral sequence, (E ,d ) ElI n

.

r.::O Namely, for each integer a coboundary [LA. C.]) that

n

define

dn:O n ~ on+l n

n D

=

Cn x e n + l , and define

by requiring (in terms of elements n

n n+l d (u, v) = (d (u) +t (v), -d (v)) , Then

0*

=

n n (0 ,d )nE;?!,

11 -indexed cochain complex in the abelian category

e*(+l )

be the cochain complex such that

n'th coboundary is

n _d + l ,

first injection, and d*:O* (Thus,

n E ~. ~

Let

is a A.

Let

1 e n (+ 1) -_ en + , and the 'IT*: C* .. D*

denote the

C*(+l) the second projection.

')[n(u) = (u,O) ,dn(u,v) = v, all

u EC n , vE

c n + l , nE

11 ).

Then we have a short exact sequence of (1/ -indexed) cochain complexes in the abelian category

A:

(0) 0 ~ C* ~*O* §* C*(+l) ~ O.

Noting that

Hn(C*(+l)) =

n+l H (C*), the long exact cohomology sequence of the short exact

Bockstein Spectral Sequence

475

sequence (0) is of the form: T

(1)

where

and

T

n+l

n+l

is the n'th coboundary

of the cohomology sequence of the short exact sequence (0), all is an endomorphism of n H (C*), all

n E 7.

An explicit computation,

uSing the

usual construction of the coboundary, shows that where

t*:C*

~

C*

the cochain complex

n Tn = H (t*) ,

is the orginally specified endomorphism of C*.

But then the long exact sequence (1)

is such that Lemma 1, Corollary 1.1, Theorem 2 (or more precisely, Theorem 2') and all the exact sequences of the last Remark hold (with "Hn(D*)"

replacing

"Hn(C*/tC*) ",

n E

;1).

There-

fore, we have the generalized Bockstein spectral seguence with

obeying Theorem 2'

and

(and Lemma 1, Corollary 1.1,

and the last Remark all hold). Theorem 2'.

This generalizes

(Because in the special case that the hypotheses

of Theorem 2' hold (i.e., is easy to see that the

t*:C*

~

C*

is a monomorphism)

Hn(D*) ~ Hn(C*/tC*)

it

canonically, and

similarly for the Bockstein spectral sequence of Theorem 2' and the one just constructed. below) .

(*)

For a proof of this, see the footnote

(*)

The construction of this last Remark can be motivated as

fOllows.

Consider the special case in which

of A-modules, where

A is the category

A is a commutative ring, and

tE A

is an

element that is not a divisor of zero, and such that the given

Chapter 1

476

endomorphism

t*:C*

~

C*

(which we are not assuming to be

injective) of the given cochain complex by

t".

There exists a

-indexed cochain complex

7

of A-modules and map of cochain complexes of A-modules ~

a*:C*

C*

such that (1)

induces an isomorphism on cohon n Multiplication by t:'C ~ 'C is injective,

mology, and (2) all integers acyclic

n E

'?

[C.E.H.A., pg. 363] choose an

~uch

n E 'l' ,

then

'

P~ ~ p~+l

~**

that if we define

Then

Bockstein spectral seguence of ~

'C*.

(2)

resolution of

-m

Let

AltA

p*

t:C*

~

as right A-module. F*.

'C

all

Define 'C* =

n

is projective,

n

a*:'C*

,

~ 'C +

C*

n

is inthat in-

Then define the generalized

be any flat

obtaining a cochain complex

'C

t":

n E 'l' ; and we have a map

duces an isomorphism on cohomology). t:'c*

c-n,m=pn

is a double cochain complex.

n E 'l' , so that "multiplication by

jective, all

c n ~ c n+l ,

over the coboundary:

the associated singly graded complex. all

P~,

proiective homological resolution of en, call i t

all integers

E

a*

(E.g.,

,?

and choose a map:

n,m

is "multiplication

Then we give three alternative constructions of the

spectral sequence.

'C*

C*

C*

to be that of

(e.g., projective) Define Fn = P-n' "Hn(F*® C*)"

Then if one uses

A

"Hn(c*/tc*)"

in lieu of

of Theorem 2 one obtains a long exact

sequence (see "percohomology", chapter 5 below, for details), n-l dn +l _d__ >H n (C*)---1>H n (C*)->H n (F* ® C*) - - > A

which gives rise to a Bockstein spectral sequence.

'C*

and

F*

as in (1) and (2).

Then use

(3)

Choose

"Hn(F* ® 'C*)"

in

A

lieu of

"Hn(c*/tC*) "

of Theorem 2, giving rise to a long exact

sequence: (1)

which yields a Bockstein spectral sequence.

(The long exact

Bockstein Spectral Sequence

477

sequences constructed in (3) map into that constructed in (1) (using the map of cochain complexes: (A/tA)

F* .... (A/tA) , where

is regarded as a cochain complex concentrated in dimen-

sion zero)

and the long exact sequence in (3) also maps into

I

that in (2)

a:' C* .... C*) .

(using the map

three constructions (1),

(2),

(3),

This shows that the

yield canonically isomorphic

long exact sequences, and therefore also generalized Bockstein spectral sequences - and also, that the long exact sequence, and also generalized Bockstein spectral sequence, constructed in (1) 'C*

is independent of the choice of such a cochain complex

and map

a*).

A special case of the construction (2) is as follows. specific projective resolution of the A-module

o .... A P

by

t).

0,

n

F- l

=

FO

=

A,

is

.... A/tA .... 0

n"l

0,1.

d

l

(multiplication

=

The corresponding cochain complex

o .... i.e. ,

1A

A/tA

A

F*

is

° . . A ....t A .... ° . . ° . . F

n

=

0,

n "I -1,0,

d- l

=

"multiplication

F* ® 'C* F;j D* , the cochain complex constructed A in the Remark above (which was constructed in any abelian

by

til.

category

But then

A).

Thus, the long exact sequence and the generalized

Bockstein spectral sequence constructed in that Remark, special case by

t,

tEA

A = category of A-modules,

t*

=

in the

multiplication

a non-zero divisor, coincides with the one con-

structed in this footnote. This shows that: 1.

In the construction in Remark 4 above,

T

n

H

n

(t*),

as previously asserted. (Proof:

By the Exact Imbedding Theorem

[IAC~,

to prove such an

Chapter 1

478

A = category of

assertion, it suffices to prove the case: abelian groups. acts on

en

Then regard

as a

7 [Tl-module,

where

T

Therefore we are reduced to

E ;[.

n

as

e*

proving the assertion under the hypotheses of this footnote, in the special case

A

=

7 [Tl,

observed in this footnote,

t

=

T.

But then, as we have

Hn(D*) = construction (2) of this

footnote, which we have demonstrated to have the desired properQ.E.D. )

ties. 2.

Under the hypotheses of Remark 4 above, in the special

case that

t*:C*

~

C*

is a monomorphism, then the long exact

sequence and generalized Bockstein spectral sequence constructed in Remark 4 above are canonically isomorphic to those in Theorem (Proof:

2'.

Again, using the Exact Imbedding Theorem [I.ACJ,

A = category

it suffices to prove the assertion in the case that of

7 [Tl-modules, and

where

t

=

TE 7 [Tl

=

t*:e* A.

->-

C*

is "multiplication by

t",

But then, the construction of the

above Remark is the special case construction (2) of this footnote when ~

0

=

P* ~

A

t

~

A

(the projective resolution: ->-

A/tA

->-

0)

of the A-module,

A/tA.

And the

construction of Theorem 2' in this case is the special case construction (1) of this footnote, with' 'e* = e*,

0.*

=

identity.

Therefore, the proof of observation 2 follows from the fact, observed above, that the three constructions (1),

(2),

(3) of

this footnote all yield canonically isomorphic long exact sequencE and generalized Bockstein spectral sequences.)

Remark 5.

It should be noted that every singly graded exact

couple (in any abelian category) :

479

Bockstein Spectral Sequence

V -->V

\1 E

~

that is such that the map:

V

such that the maps:

and

V

E

+

V

is of the degree zero, and E

+

V

are maps of singly

graded objects, of whatever (possibly different)

degrees,

can

be re-interpreted as a long exact sequence as in the hypotheses of Theorem 2' of Remark 3 above graded objects E

and

maps:

E

V

E

~

and

is positive and

+

V. V

n

to

-n

is not

-r,

to make it

E

If the sum of the degrees of the +1, then:

+r, then we obtain

sequences; negative and E

(possibly after reindexing the

r

e.g.,

such long exact

then we re-index

+r,

if this integer

n V

to

V-n ,

and proceed as in the last case.

(The case in which this integer is zero is trivial, and reduces to denumerably many ungraded exact couples, another special case)).

Therefore essentially every singly graded exact couple

in every abelian category,

such that the map

t:V

+

V

of the

exact couple is of degree zero, is covered by Theorem 2' of Remark 3. The observations of this last Remark are related to those made in Intro. Chap.2, section 4, Remark following Example 3. We conclude this section with a corollary to Theorem 2: Proposition 3.

Under the hypotheses of Theorem 2, suppose that

the ring

is left Noetherian.

A/tA

suppose that

Hn(C*/tc*)

Let

n

be an integer and

is finitely generated as a left A-

module (equivalently, as left (A/tA)-modula. an integer

(1)

r,

Then there exists

depending on the fixed integer n-l d r +l

..• =dn s

l

= 0,

all

n,

s ~ r

such that

480

Chapter 1

(2)

u E Hn(c*/tc*)

If

element in

Hn(C*), then

element in

Hn(C*)

then

dn-l(u) If

(4)

that

=

u

is the image of a precise tr-torsion

and

u E Hn(C*)

tr·v

n n d - l (u) E t r .H (C*),

is such that

is t-divisible in

Hn(C*).

is a precise t-torsion element such v E Hn(C*),

for some

Let

(5)

u

u E Hn - l (C*/tC*)

If

(3)

is the image of at-torsion

u E Hn(C*)

then

is t-divisible.

u

be any t-torsion element of

Hn(C*).

Then the following conditions are equivalent:

u

t-divisible;

v E Hn(C*)

that Note:

u

=

u

is t-divisible; there exists

is infinitely

tr·v.

Whether or not

A, AAt

of

AltA

is left Noetherian,

under the other hypotheses of Prop. 3, whether or not is finitely generated, for every pair of integers r~O,

such

conditions (1),

(2),

(3),

(4)

Hn(C*)

n,r

with

and (5), in the statement

of the Proposition are equivalent. Proof.

First, let us prove the Note.

In fact, by definition

of a spectral sequence, n-l d r +l

0,

But by Lemma 1, part (3), the image of a precise Corollary 1.1, part (3),

u E

E~

all integers

is in

image of a t-torsion element in

is in Hn(C*).

~

r

u

iff

tr-torsion element in u E E~

s

Boo(E~)

is

Hn(C*); and by iff

u

is the

This proves (1)<=> (2).

Bockstein Spectral Sequence

481 u ~ E~-l

On the other hand, by part (1) of Lemma 1, if u E Zr(E~-l)

iff

d

n-1

(u) E t

Corollary 1. 1, i f

u E E~-l

is t-divisible in

Hn(C*).

Next,

(3)

<=>

of Corollary 1.1. element, and

·H (C*); and by part (1) of u E Z'" (E~-l)

This proves (1)

d n - l (u)

iff

<=>

(3).

(4) by using part (2) of Lemma 1 and part (2) {u E Hn(C*):

such that

=

u

u

is a precise t-torsion

try in Hn(C*)}

=

{d n (w):w E Zr (EOn-l ) } , by part (2) of Lemma 1; and u

then

n

then

(Since

3v

r

n {u E H (C*) :

is a precise t-torsion, t-divisible element} =

{d n (w):w E Z'" (EOn-l )}, by part (2) of Corollary 1.1). Finally, obviously

(5)

remains to show that (4)

=>

=>

element such that there exists u

To prove the Note, it

(5). That is, we must show that,

under the hypotheses of (4), if

then

(4).

u E Hn(C*) v E Hn(C*)

is infinitely t-divisible.

is at-torsion such that

u = tr·v,

We first show that

u

is

t-divisib1e. Since

u

is a t-torsion element of

h ~ 0

an integer

such that

th.u

=

O.

Hn(C*), there exists The proof that

u

is

t-divisib1e now precedes by induction on the non-negative integer

h.

If

h

=

0, or resp. 1, then

assertion follows from condition (4). We have that (4),

th·u

u = tr·v

=

O.

Therefore

for some

u

=

0, or resp. the

So assume that

th-l. (tu)

=

O.

that

t·u

In fact, if

is t-divisible. N

We now show that

is any integer

divisible, there exists

By condition

v E Hn(C*), and therefore

Therefore, by the inductive assumption applied to

~

u

r+1, then since

w E Hn(C*)

such that

h > 1.

(tu) = tr. (tv). t·u,

we have

is t-divisible. t·u

is t-

Chapter 1

482 But then

u = tN-I·w+o:, t·o: = O.

where

We also have, since

t r . (tN-I-r) ·w,

t·o: = 0,

so that

and

t-divisible.

=

u-t

~

r+l, that

t

N-I

w = Thus,

w

is

0:

0:

In particular, we can write

S E Hn(C*).

exists

0:

N-I

N

0:

=

tN-I·S,

there

But u

t

N-l

w+a,

so that u

= ~

N

being an arbitrary positive integer

u

is t-divisible, proving the induction.

r+l,

it follows that

Thus, under the hypotheses of condition (4), we have shown that,

if

u

is any t-torsion element in u

In fact,

since

u u

is t-divisible.

is infinitely t-divisible. is t-divisible, we can write Then

Let Since u

l

=

u

is a t-torsion element, so is

2

=

u . l

But since

tr.ui' it follows from the last proved assertion that

is t-divisible. u

To

(4) => (5), we must show, under this

complete the proof that same hypothesis, that

Hn(C*), and if

t

r

·u

2,

etc.

u

l

Define

But then we can write

Proceeding by induction, we obtain a sequence (EXplicitly,

having defined us,s

~

1, and establised that

u

s

is at-torsion,

Bockstein Spectral Sequence t-divisible element, choose

483

such that

u~+l

u •

s tr·u' ) Thus, u is an infinitely u s +l = s+l . t-divisible element of Hn(C*), as asserted. This completes

Then define

the proof of Note 1.

It remains to prove the Proposition.

In fact, consider the (A/tA)-module hypothesis, this is Boo(E~)

Hn(C*/tc*)

is a submodule.

Since by hypothesis the ring

as a left (A/tA)-module.

Boo(E~)

But

Boo(E~)

is the increasing union

is finitely generated as

follows that there exists an integer Br(E~)

=

Boo(E~).

r n B (EO)' r

r

s

~

~

O.

(A/tAl-module, it ~

0

such that

But then, by the general theory of spectral

sequences, this is equivalent to integers

(A/tA)

is finitely generated

of the sequence of left (A/tA)-submodules:

Boo(E~)

E~. By

a finitely generated left (A/tA)-module.

is left Noetherian, likewise

Since

=

=

••• = d

n

s

=

0, all

1, and this is condition (1) of the Proposition.

Q.E.D. Corollary 3.1.

Let

A

t ~ A

be a ring with identity and let

be an element of the center of the ring

A.

Let

be a cochain complex of left A-modules indexed by all the m integers, such that multiplication by t:c m ~ c is injective, all integers

m.

Let

n

be any fixed integer and let

any fixed non-negative integer.

Let

M

=

r

be

{t-torsion inHn(c*)}.

Then the five equivalent conditions of Proposition 3 are all equivalent to each of the following conditions: (5' )

M

as left submodules of elements of

M} and

M, T

=

D+T

where

D = {infinitely t-divisible

{precise tr-torsion elements in

M}

484

Chapter 1

(5" )

D+T

M

as left submodules of

M,

where

D

is some left submodule of

M every element of which is t-divisible, and submodule of

M such that

tr'T

T

is a left

= {oJ.

There exists a short sequence of left A-modules

(5 "')

o

D

~

~

M

~

T

~

0,

where D is a left A-module every element of which is t-divisible, and

tr'T

= {oJ.

In which case, the sequence (5'") is uniquely determined up to canonical isomorphism preserving image of

D

M -- that is, the

is necessarily {x E M: x is t-divisible} and

is necessarily isomorphic to the quotient A-module. this is the case, if

x E T

x' E M such that

th·x'

Proof.

(5')

Obviously,

=

and

°

=>

in (5'") to coincide with

=

Also, when

0, then there exists

and the image of

(5").

D

th·x

T

x'

in

Also (5") => (5"')

T

is

(Take

x.

D

in (5"), and complete to a short

exact sequence). Also,

(5'") => condition (5) of Proposition 3:

suppose that (5 "') holds, and let element in

n H (C*)

v E Hn(C*l

such that

(i.e., u E M) u = tr·v.

In fact,

n u E H (C*) be at-torsion be such that there exists Then

v E M.

Throwing through

the epimorphism of the short exact sequence in (5"'), we see that the image of But

tr'T

Therefore

= {oJ. u E D.

u

in

T

is in the subgroup

Therefore the image of

u

in

tr'T T

of

T.

is zero.

Since every element of the left A-module

D

is t-divisible, every element of D is also infinitely t-divisible.

Bockstein Spectral Sequence Therefore

u

485

is infinitely t-divisible, proving condition (5).

To complete the proof of the equivalence of conditions (5), (5'),

(5"),

and (5"') i t suffices to show that (5) implies (5').

In fact, assume condition (5) and let v ~ tr·u

tion (5), M.

t-divisible element of

v ~ tr·w,

of

But then if

M.

cise tr-torsion elt. of u = w+x,

where

w

x

u-w,

tr. x ~ t

M (since

~

r

it follows that

M

M}

elements of

is an infinitely

• u_t

x r

is a pre-

• w~v-v=O), so

the sum of an infinitely t-divisible element

and a precise tr-torsion elt. of M.

M

Then by condi-

is an infinitely t-divisible element of

Therefore we can write

that

u E M.

=

D+T,

and

T

=

where

D

=

uEM

being arbitrary,

{infinitely t-divisible

{precise tr-torsion elements in M}.

This proves (5'), completing the proof of the equivalence of (5),

(5'),

(5"),

and (5"').

Assume now that we have a short exact sequence as in (5"'). To show that the image of

D

is necessarily

{x E M: x is t-

divisible}. We can assume that the monomorphism: inclusion. in

T

Let

x E M

be t-divisible.

of

is zero. D

Then the image of

Therefore

xED.

Therefore the image of

x

Therefore

D D

~

x

in

Conversely, since every element

is t-divisible considered as an element of

element of

of

is an

is t-divisible, and in particular lies in the submodule

{a}. T

D ~ M

D,

every

is t-divisible considered as an element of

M.

{x E M:x is t-divisible considered as an element

M}. Finally, to complete the proof of the Corollary, let x E T

and let T.

h

be a non-negative integer such that

We must find an

x' E M

representing

x

th·x = 0 such that

in

486

Chapter 1

o

in

M.

In fact,

let

the image of fore

th·u

u E M be any element representing

th·u

in

T

is zero.

exists an infinitely t-divisible element v E D)

element

=

th.x' u

=

0

(since

x'+v.

Since

T

is

th·x'

u

in

=

tho (u-v)

v E D,

But the image of in

th.v = th.u.

such that

u

and

T

is

x' x.

E M (i.e., an

Let

=

x'

th.u_th·v

=

u-v.

0),

Then

and

have the same image in Therefore the image of

T. x'

Q.E.D.

X.

Remarks: 1.

There-

Therefore there

M.

v

Then

th.u E D.

Therefore

is infinitely t-divisible in

x.

If the ring

A

should be a principle ideal domain,

then the short exact sequence of A-modules (5"') of Corollary 3.1 splits as sequence of A-modules (but not,

in general,

canonically), whenever the equivalent conditions (1)-(5"') of Proposition 3 and Corollary 3.1 hold (for any pair of integers n, r

with

r

~

0

such that (1)-(5"') hold, whether or not

is finitely generated) . Proof. trivial.

In this case, if Otherwise,

AAt

t = 0

or a unit, the assertion is

is the direct product of finitely

many discrete valuation rings, and the image of

t

in each of

these is an element of each of the respective maximal ideals. Therefore, an

AAt_module

divisible iff

D

D

is such that every element is t-

is injective as

AAt_module.

sequence (5"'), which is a sequence of 2.

If the ring

Therefore the

AAt_modules, splits.

AAt, the t-adic completion of

A,

is a

finite direct product of discrete valuation rings, then again the conclusion of Remark 1 above holds (whether or not principle ideal domain).

A

is a

487

Bockstein Spectral Sequence 3.

Proposition 3, Corollary 3.1 and Remarks 1 and 2 above

all generalize to the situation of Remark 2 following Theorem 2.

I.e., to the case in which:

A

is a ring with identity,

and we have a long exact sequence of left A-modules: (1) indexed by all the integers (such that the left A-module in each spot indexed by an integer congruent to zero modulo three coincides with the left A-module in the next spot (indexed by an integer congruent to one modulo three), as indicated in the display of the sequence (1».

Then if

is the generalized Bockstein spectral sequence, as defined before Lemma 1 (and as generalized in Remark 2 following Theorem 2), so that e.g., n,r

En, n E 1',

then for every pair of integers

such that r 2:. 0, the conditions (1),

Proposition 3 and (5'),

to

"Em"

"Hm(C*)" to

all integers

if the ring

A

(4),

"~",

And for any integer

is left Noetherian and if

En

is finitely

generated as left A-module, then there exists an integer depending on Example 1.

n Let

(5) of

m); and the last two

observations of Corollary 3.1 then hold. n,

(3),

(5") and (5"') of Corollary 3.1 are all

equivalent (with the notation changes: "Hm(C*/tC*)"

(2),

r

such that all of these conditions hold. A

be a ring with identity and let

element in the center of the ring

A.

Let

t

be an

Tn, n 2:. 0, be an

exact connected sequence of functors on the category of left A-modules and let

M be a left A-module such that multiplication

488 by

Chapter 1 t:M

~

M is injective.

Then we have the long exact sequence

to which Remark 3 above, and Remark 2 following Theorem 2, can be applied to the corresponding generalized Bockstein spectral sequence (as defined in Remark 4 following Theorem 2). Example 2.

Let

X

be a topological space,

set of

let

A

be a ring with identity, and let

X,

an element of the center of the ring

A.

U

Let

be an open sub-

F*

tEA

be

be a cochain

complex of sheaves of left A-modules (indexed by all the integers), such that multiplication by

t:F

n

~ F

n

morphism of sheaves of abelian groups, all integers

is a monon.

Then

we have the long exact sequence of cohomology: n-l t n d --->H n ( X,U,F *) -->H n ( X,U,F *) ->H n ( X,U,F */ tF *) --> d ... , a long exact sequence of A-modules, obeying the "parenthetical modulo three condition", of Remark 3 above, and therefore Remark 2 following Theorem 2, applies. The point of Remark 3 above is that the results of this chapter apply to any appropriate (that is, obeying the "parenthetical modulo three condition") long exact sequence, indexed by all the integers, of left A-modules, whether or not it comes ~

priori from some cochain complex, or whether or not from some

kind of cohomology theory (which is how most applications arise - although not always obviously from a cochain complex

C*

of

left A-modules (although, e.g., Example 2 above can be interpreted as coming from such a cochain complex)). Remark 4.

Suppose, under the hypotheses of Theorem 2, that the

489

Bockstein Spectral Sequence ring

A

is commutative and the cochain complex

is a differential graded A-algebra (i.e., that graded A-algebra, with a unit in d(u)

u v + (_l)deg Uu U d(v),

H*(e*)

whenever

products.

(d ) r

(u)

~

"has cup products";

preserves cup

0 the coboundaries

are an "anti-derivation"; Le.,d(uUv)=

r nE;r

d

r

d (uU v) =

H*(e*/te*), and

Hn(e*) ~ Hn(e*/te*), n E Z,

And, for each integer

n

is also a

u E en, v E em, n,m Ea' ).

is a graded A-algebra, as is

the natural map:

e*

of A-modules

and such that

Then the long exact sequence of cohomology (1) i.e.,

e*

r

(_l)de g Uu U d (v) u E En, v EE m, n,m E a'. r' r r

U v +

fore, by induction on

r,

n n (E r + l , d r + l ) nE1

There-

is a graded (A/tA)-

algebra, and is the cohomology of the differential graded (A/tA)-algebra directly.

This is all easily verified

(I've gone through the simple details myself).

There-

fore we say that the exact couple (2) has cup products, and that the associated generalized Bockstein spectral sequence: also has cup products.

In the more general circumstances of Remark 2 following Theorem 2, one must require that the long exact sequence (1) have cup products in the same sense (i.e., ring,

V*

is a graded A-algebra,

the natural map: V* tion on r, algebra (Er+l)nE;r n

(Er)nE;r ).

E*

n

is a commutative

is a graded A/tA-algebra;

preserves cup products; and by induc-

that the coboundaries

(E )nE7. r

n

~

E *

A

make the graded

into a differential graded algebra, so that

becomes a graded algebra (as the cohomology of In which case, we again say that the long exact

Chapter 1

490

sequence (1) has cup products, that the singly graded exact couple associated to the long exact sequence (1)

(with cup

products) has cup products and that the associated generalized Bockstein spectral sequence:

has cup products. (Note:

In practice, most long exact sequences (1)

usually

can be interpreted as coming from an appropriate cochain complex of left A-modules

C*

is injective, all

n E

such that multiplication by ~.

t:c

n

+

c

n

And when the long exact sequence (1)

"has cup products" in the sense just defined above, then it usually comes from such a cochain complex ential graded A-algebra.

C*

that is a differ-

E.g., this is the case in the situation

of Example 2 of Remark 3, see [RRWG1, Chapter I, last section. Considering the construction of an appropriate cochain complex in that reference, we see in fact that, under the hypotheses of Example 2 of Remark 3 above, if the cochain complex of sheaves of A-modules

F*

over the topological space

X

has the

structure of (0' -indexed) sheaf of differential graded A-algebras then both the singly graded exact couple associated to the long exact sequence of cohomology: dn- l

(1)

t

dn

--->Hn(X,U,F*)-->Hn(X,U,F*)~Hn(X,U,F*/tF*) --> •••

has cup products in the sense of this Remark, and therefore so also does the generalized Bockstein spectral sequence

Bockstein Spectral Sequence

491

associated to this exact couple, as described in this Remark (and defined just preceding Lemma 1). that, in Chapter I, last section of Hn(X,U,F*) = Hn(C*), where of

[~P.WCJ,

C*

see Chapter I, is a

(The reason for this is [~~WC.J,

we note that (3)

C*(X,U,F*), in the notations ~-indexed

differential graded

A-algebra, and the isomorphism (3) preserves cup products (and in fact is the way, in [p.p.we], Chapter I, that we define cup products in

Hn(X,U,F*)

whenever

ential graded A-algebras) and where as graded A-algebras. C*(=C*(X,U,F*»

F*

is a sheaf of differ-

Hn(X,U,F*/tF*) ~ Hn(C*/tC*)

(Note also that the definition of

is such that, if multiplication by

t:F

n

~ F

n

is a monomorphism of abelian groups, then so is multiplication by

t:C

n

~ cn, all integers

definition of C*(X,U,F*)

in

n.

This is immediate from the

[P'p'~CJ,

Chapter I.)

We have shown: Example 2, continued. Remark 3 above, if also

Under the hypotheses of Example 2 of F*

is a sheaf of differential graded

A-algebras, then the singly graded exact couple associated

to

the long exact sequence (1) of Example 2 of Remark 3 has cup products as defined in this Remark (Remark 4), and therefore so does its associated generalized Bockstein spectral sequence (as defined preceding Lemma 1),

("having cup products" as

defined in this Remark, Remark 4). [P.P.W~~,

(The reason being that, by

Chapter I, last section, this reduces to the assertion

about cochain complexes

C*

of A-modules that are differential

graded A-algebras, as discussed at the beginning of this Remark) .

This Page Intentionally Left Blank

CHAPTER 2 THE SHORT EXACT SEQUENCE 1.8.

In this chapter, I recall an exact sequence which I established in an earlier paper, Theorem 1.

Let

A

[P.P.w.cJ, 1.8.1, pgs. 159-160.

be a ring with identity, and

ment in the center of

A.

Let

tEA

an ele-

be a cochain

complex of left A-modules (indexed by all the integers). that (multiplication by n.

n t): C

Then for every integer

->-

n C

Suppose

is injective, all integers

n,

(1)

We have a short exact sequence of

(*)

O-+Hn(C*)J\t-+lim[Hn(C*/tic*)]-+ i>O.

lim (precise -t~-torsion in

~t-modules

1 Hn + (C*»

->-

O.

i>O

(Here M,

"ri't"

e •g • ,

M

denotes the t-adic completion of

= Hn (C *) ) •

in

Hn+l(C*».

~>O

"

(In the inverse system

t.")

there is induced an isomorphism of liml[Hn(c*/tic*)] i~O

dn:Hn(C*/tic*)-+ "(proecise ti-torsion

the maps of the inverse system are all in-

duced by "multiplication by

(2)

i~O,

Hn(C*) -+Hn(C*/tic*),

and the epimorphism by the coboundaries i ~ 0).

all A-modules

(The monomorphism in the exact sequence

(*) is induced by the natural maps:

Hn+l(C*),

M,

Also, for every integer A"t-modules

~~iml(precise ~~O

4093

n,

ti-torsion in

Hn+1(C*».

494

Chapter 2

(Where the isomorphism is induced by the coboundary mappings . +1 d n :H n (C*/t1. c *) ~Hn (C*), i,:,O). For want of a better name, we shall refer to the exact sequence (*) of Theorem 1 and the isomorphism (2) of Theorem 1 as "the exact sequence 1.8". Proof:

The proof is exactly as given in [P.P.W£J, 1.8.1; but we

repeat it. For each integer (~-indexed)

i,:, 0,

the short exact sequence of

cochain complexes of A-modules

gives rise to the long exact cohomology sequence:

n l H + (C*)

-;.

...

from which, for each integer

n,

we extract the short exact

sequences (1)

o~ [Hn(C*)/ti(Hn(C*)) ~Hn(C*/tic*) .... (precise ti-torsion in

The functor

"lim" i>O

Hn + l (C*)) ~ 0,

i > 0.

is a left-exact functor from the cate-

gory of inverse systems of A-modules indexed by the non-negative integers into the category of A-modules. functor is

"liml",

The first derived

and the higher derived functors are zero.

bO If we fix variable

i >

°

n> 0,

then the short exact sequences (1) for

can be thought of as a short sequence of in-

verse systems of A-modules indexed by the non-negative integers.

The Short Exact Sequence 1.8

495

Applying the system of derived functors

lim, liml, 0, i>O i>O to this short exact sequence yIelds an exact sequence

0, ... ,0 .•.

of A-modules with six terms: (2)

O-+Hn(c*)l\t-+lim[Hn(c*/tic*)]-+ i>O lim({precise tl-torsion elements in i>O

Hn+l(C*)})-+

l n Ij,m [H (C*) /ti

i>O i liml ({precise t -torsion elements in i>O all integers system:

n.

n 1 H + (C*) }) -+ 0,

But, in the exact sequence (2), the inverse

(Hn(C*)/t. Hn(C*)) i>O

is an inverse system of A-modules

and epimorphisms, so that (3)

But equation (3) tells us that the fourth group in the exact sequence of six terms (2) is zero.

Therefore, the exact sequence

(2) becomes an exact sequence of three terms (conclusion (1) of this Theorem) and an isomorphism (conclusion (2) of the Theorem). Q.E.D. Corollary 1.1.

Under the hypotheses of Theorem 1, let

arbitrary fixed integer.

n

be an

Then the following conditions are

equivalent:

(1)

The natural map of

Al\t-modules:

Hn (C*)l\t .... lim Hn(C*/tic*)

i"to

(I') The natural monomorphism of

is an isomorphism. (A/tA)-modules:

[(A/tA) ~ Hn(C*)] .... (A/tA) ~ [lim Hn(C*/tic*) 1 A

isomorphism.

A

ito

is an

Chapter 2

496

(2)

The A-module

Hn+l(C*)

has no non-zero infinitely

t-divisible, t-torsion elements. (2') Same as condition (2), with "precise t-torsion" replacing "t-torsion". (3)

Let

u E Hn(C*/t C*)

u, E Hn(C*/tic*),

i ~ 0,

1

that the image of integers

be such that there exist

u i +l

such that in

that the image of

v

is

is

all

ui '

vE:Hn(C*)

H (C*/t C*)

in

and such

= u

l Hn(C*/tic*)

Then there exists

i > l.

u

such

u.

Before proving Corollary 1.1, we state and prove two related Lemmas. Lemma 1.1.1. ger

n

Under the hypotheses of Theorem 1, for every inte-

we have a short exact sequence of

(* .1)

0 .... [(A/tA)

(A/tA)-modules

~ Hn (C*)] - - - -...., (A/tA) ~ [lim Hn (c*/tic*)] A

i~O

A

.... (precise t-torsion in

Hn+l(C*)

that is in-

finitely t-divisible) .... O. More generally, for every integer

n E'2

we have a short exact sequence of

(A/tjA)-modules

(*.j)

and every integer

j

~

0 .... [(A/tjA) ®Hn(C*)] .... (A/2lV ® [lim Hn(C*/tic*)] A

A

.... (precise tj-torsion in

i>O

Hn+l(C*)

that is in-

finitely t-divisible) .... O. Remark:

The inverse limit of the short exact sequences (*.j)

of Lemma 1.1.1 for

j

~

0

is the short exact sequence (*) of

Theorem 1. Lemma 1.1.2.

Under the hypotheses of Theorem 1, for each inte-

0,

The Short Exact Sequence 1.8 ger

n E 'Z,

the natural map of (A/tA)

(A/tA) -modules:

[lim Hn(C*/tiC*)]->-Hn(C*/tC*) A i>O

@

is a monomorphism. every integer

497

More generally, for each integer

j~O,

the map of

n E'Z

and

(A/tjA)-modules:

(A/tjA) ® [lj,m Hn(C*/tiC*)]->-Hn(C*/tjc*) A i>O is a monomorphism. Proof of Lemma 1.1.1:

Regard every left A-module as a module

over the polynomial ring ring 'Z

as

'Z

'Z[T]

in one variable

of integers, by letting 'Z[T]-module by letting

left A-module

M,

"T"

"T"

act as

T "tn.

act as zero.

over the Also regard

Then for any

we have that

M ® 'Z "" M/tM"" (A/tA) ® M 'Z[T] A and

TorI [T] (M,.?) "" (precise t-torsion in

Notice also that if

D

M).

is the third group in the short exact

sequence (*) of Theorem 1, then

D

has no non-zero t-torsion.

Otherwise stated, (1)

TorI [T] (D,'Z) = O.

Therefore if we throw the short exact sequence (*) of Theorem 1 through the system of derived functors: on the category of

TOr{ [T] ( ,'Z),

k ~ 0,

7 [T]-modules, then the last three terms of

the resulting long exact sequence become the short exact sequence (*.1) claimed in the Lemma.

(The fourth-from-the-last

term in the long exact sequence is zero by equation (1).) establishes the short exact sequence (*.1).

This

Applying (*.1),

Chapter 2

498

with

t

j

replacing

t,

we see that (*.j) follows from (*.1). Q.E.D.

Proof of Lemma 1.1.2.

Consider the commutative diagram with

exact rows: 0->- [(AltA)

~Hn (C*) 1 --.;;,

Ii

n H (C*/tC*) ->- (precise t-torsion in

r

n

n

Hn+l (C*)) ->- 0

r

i

0->- [(AltA) @H (C*) 1 -+ (AltA) @ [lj:m H (C* It C*) 1 ->- (precise t-torsion in A A PO Hn+l (C*)- that is infinitely t-divisible) ->- O.

The top short exact sequence is the short exact sequence occurring in equation (1) of the proof of Theorem 1, in the special case

i = 1.

The bottom short exact sequence in this diagram is

the short exact sequence (*.1) of Lemma 1.1.1.

The first and

third vertical maps in the above diagram are, respectively, an identity map and an inclusion map, and are therefore monomorphisms.

But then, by the Five Lemma, it follows that the middle

vertical map in the diagram is a monomorphism. first assertion of Lemma 1.1.2; replacing the second assertion, for each integer Proof of Corollary 1.1:

t

This proves the by

t

j

then proves

j ,:,,1.

Clearly, the last group in the short

exact sequence (*) of Theorem 1, is zero iff condition (2') holds.

Therefore condition (1) of Corollary 1.1 is equivalent

to condition (2') of Corollary 1.1.

Similarly, the last group

in the short exact sequence (*.1) of Lemma 1.1.1 is zero iff condition (2') holds.

Therefore condition (I') of the Corollary

is equivalent to condition (2') of the Corollary.

Also it is

obvious that conditions (2) and (2') of the Corollary are equivalent (since if there exists a non-zero t-torsion element

u

The Short Exact Sequence 1.8

499

that is infinitely t-divisible, then choose i > 0 such that i i+l i t ·u=O, t·u;iO. Then t • u is a non-zero precise ttorsion element that is infinitely t-divisible).

Therefore

conditions (1), (1'), (2) and (2') of the Corollary are equivalent. On the other hand, condition (3) asserts that if

I

is

the image of the natural map: n Hn (C*) .... [lim H (c*/tic*)], i< 0

(1)

I,

then the image of in

Hn(C*/tC*),

and of the right side of equation (1),

are the same.

But since

(A/tA)-module, the natural homomorphism: Hn(C*/tC*) (2)

Hn(C*/tC*)

is an

[lim Hn(C*/tiC*)] .... i>O

can be factored: n [lim Hn (C*/tic*)] .... (A/tA) ® [lim H (C*/tic*) 1 ....

ito

A

i>O

But by Lemma 1.1.2, the second map in equation (2) is a monomorphism.

Therefore, condition (3) of Corollary 1.1 is equivalent

to the assertion that, the images of

I

and of the right side

of equation (1) under the first map of equation (2), coincide. But since the image of the right side of equation (1)

under the

first map of equation (2) is the whole of (A/tA) ® [lim Hn(C*/tic*)] A

we see that condition (3) of the Corol-

i>O

lary is equivalent to the assertion that the natural mapping of (A/tA) -modules: (3)

[(A/tA) ® Hn (C*)] .... (A/tA) A

is an epimorphism.

@

A

n [lim H (C*/tiC*)] itO

The mapping (3) is the first mapping in the

500

Chapter 2

short exact sequence (*.1) of Lemma 1.1.1, and therefore is always a monomorphism.

Therefore, condition (3) of the Corol-

lary is equivalent to the assertion that the above displayed mapping (3) is an isomorphism.

But this latter assertion is

condition (1') of the Corollary.

Therefore condition (3) of the

Corollary is equivalent to condition (1') of the Corollary, completing the proof. Remarks.

1.

The reader should notice that, in both conclusions

(1) and (2) of Theorem 1, the inverse system: sion elements in , by t~on

Hn+l(C*)}),

~>O

has for its maps, "multiplica-

, -, 1 ements ~n ' t"{ : prec~se t i+l -tors~on-e

{precise ti-torsion elements in 2.

In

({precise ti-tor-

Hn + l (C*) },

Hn+l(C*)}->each integer

i > O.

[P.P.WCJ, 1.8.1, we left the statement of the

Theorem as the exact sequence of six terms, equation (2) in the proof of Theorem 1.

It is, of course, trivial to go from this

sequence to the slightly improved conclusion (equations (1) and (2) of the conclusion of Theorem 1 above). 3.

The reader will notice that the proof of Theorem

1 would work with any suitable "cohomology theory" replacing the cohomology of cochain complexes. Example 1.

Let

A

obeys Axiom (P.l) Let from

Tn, A

n E 71, into

and

S

be abelian categories, such that

(see Introduction, Chapter 1, section 7). be any exact connected sequence of functors

S.

an endomorphism of

Let M

MEA in

A

be an object and let

o ->- Tn (M) At ->- lim i~O

Tn (M/tiM) ->-

lim (precise ti-torsion in

i~O

t:M ->- M

that is a monomorphism.

obtain a short exact sequence: (1)

a

Tn + l (M»

->- 0

be

Then we

The Short Exact Sequence 1.8

501

and an isomorphism (2)

· 1 Tn (M/ t i M) "" 1 ~m .... l '~m 1

i!O

all integers

( prec~se . ti

(Here,

n.

bl

"Tn (M) At"

Let

center of

A

and "(precise ti-torsion in

A

be a ring, let

and let

F*

t

til: Fn .... F n

groups, all integers

be an element of the

be a cochain complex of sheaves of

x.

A-modules over a topological space cation by

denotes as usual

"Ker(Tn+l(t i »".)

denotes

Example 2.

T n + l (M»,

. ~n

i>O

"l~m(eoker(Tn(ti):Tn(M) .... Tn(M»)" Tn~l(M»"

. -tors~on

Suppose that "multipli-

is a monomorphism of sheaves of abelian

n.

Let

U

be any open subset of

x.

Then we have a short exact sequence (1)

O .... Hn(X,U,F*)At+lj,m Hn(X,U,F*/tiF*) .... i> 0 lim {precise ti-torsion elements in i>O

Hn+l(X,U,F*)} .... 0,

and an isomorphism (2)

liml Hn(X,U,F*/tiF*)~ liml (precise ti-torsion in i>O i!O Hn+l(X,U,F*»,

all integers

n.

Does Theorem 1 generalize to abelian categories? Theorem I'.

Let

A

Yes.

be an abelian category such that denumerable

direct products exist, and such that the direct product of a denumerable set of epimorphisms is an epimorphism (i.e., such that the fUnctor "denumerable direct product" preserves epimorphisms). Then let A

e*

be any cochain complex of objects and maps in

(indexed by all the integers), and let

endomorphism of the cochain complex

e*

t*: e* .... e* such that

t.e any

tn: en .... en

Chapter 2

502

is a monomorphism, all integers

n.

Then for every integer

(1)

We have a short exact sequence

(*)

O-+Hn(c*)At-+lj,m [Hn(C*/(t*)i C*)]-+ i>O IJ,m (precise ti-torsion in i>O

(where by

"C*/(t*)ic *"

"Coker [(t*)i: C*-+C*]",

n

n l H + (C*)) ... 0,

we mean the cochain complex: we mean

and by

"lim [Coker(Hn((t*)i): Hn(C*) -+Hn(C*))]", i>O

and where "precise

ti-torsion" is defined as in the Definition in Chapter 1, Remark 4 following Theorem 2, and also in the Introduction), and for every integer

n

we have an isomorphism

liml[Hn(C*/(t*)ic*)],! i>O

(2)

lim (precise ti-torsion in i>0 Proof:

Hn+l(C*)).

The hypotheses on the abelian category

A are necessary

and sufficient for

A

"lim" to exist for all inverse systems in i>0 indexed by a denumerable directed set not having a maximal

element, for cally zero.

l "lim "

2

"1j,m " to be identii>O Therefore the proof of Theorem-l generalizes to

{to

to exist, and for

prove Theorem I'. (*) 4.

After a suitable modification in the statement of

Theorem 1 (or of Theorem I'), can the hypothesis that "multiplication by

t"

(or that the map

tn)

be a monomorphism: Cn -+ cn,

(*) For a description of the elementary properties of an abelian category A obeying the indicated conditions, see the Introduction, Chapter l. se,ction 7.

The Short Exact Sequence 1.8 all integers question in

n,

be removed?

Theore~2

503

The answer (as in the analogous

and 2' of Chapter 1) is "yes".

We state

the more general theorem, first at the abelian-category theoretic level (as it is no harder in this case) and then, in Remark 5 below, in the more concrete situation of a cochain complex of A-modules. Theorem 2'.

Let

A be an abelian category such that denumerable

direct products of objects exist, and such that the denumerable direct product of epimorphisms is an epimorphism (i.e., such that the functor "denumerable direct product" preserves epimorphisms). gory

A

Let

C*

and let

cochain complex

be an arbitrary cochain complex in the catet*: C* .... C* C*.

For each integer plex

D~ l

l

1 , = C~l x C~+ l

d~:D~ .... D~+l d~(u,v)

i> 0,

in the category

D~

be an arbitrary endomorphism of the

define a

A

(;r-indexed) cochain com-

as follows:

and the coboundary:

is the map such that, in terms of elements [I.A.C.]

= (dn(u) +"(t*)i,,(V), _dn+l(v)).

l

chain complex

t*

of the

phism

"t*"

of

D~ l

is the co-

constructed in the proof of Theorem 2' of

0*

Chapter I, but with the i'th iterate phism

(Thus,

~ochain

C*

complex

C*

"(t*)i"

of the endomor-

replacing the endomor-

in that construction).

Then, the notations

being as in Theorems 1 and I', (1)

For each integer

n

we have the short exact sequence

in the abelian category (*)

A:

504

Chapter 2 lim (precise ti-torsion in i>O

Hn + l (C*)) + 0,

and the isomorphism

Proof: i

~

0,

Hn

lim i~O

(2)

(D~) ~ lim ~

n H + l ((C*).

(precise ti-torsion in

i>O

As in the proof of Theorem 2' of Chapter 1, for each we have a short exact sequence of cochain complexes: O+C*+Di+C*(+l) +0,

(where

. ~s

C*( + 1)

t he

. cocha~n

and with n I th coboundary

comp 1 ex such t h at

- d!l+ 1 , n E 7! ,

Cn(+l) __

cn +l

,

which as in the proof

of Theorem 2' of Chapter 1 gives rise to a long exact cohomology sequence of the form:

Hn + l (C*) -;:.

(In this long exact sequence, as in the proof of Theorem 2' of Chapter 1, the map labelled a&

"Hn((t*)i)"

is the (n-l) 'st

coboundary for the indicated short exact sequence of cochain complexes, and is explicitly computed to be the map n H - l (C* (+1»

after we identify other maps).

n

= H (C*).

Hn ( (t*) i) ,

Similarly for the

The proof of Theorem 2' then closely folIDWS that

of Theorem 1', with the long exact sequences,

i

~

0,

just con-

structed taking the place of the corresponding long exact sequences, Remark 5. Theorem 2.

i

~

0,

used in the proof of Theorem 1'.

An important special case of Theorem 2' is Let

A

be a ring with identity and let

an element in the center of the ring

A

t E: A

be

that is not a divisor

The Short Exact Sequence 1.8 of zero.

Let

C*

be an arbitrary cochain complex of A-modules

(indexed by all the integers).

01

505

For each integer

i

~

0,

let

be anyone of the following cochain complexes: (1)

01 =

('c*)/ti(,c*),

where

'C*

is any (;z'-indexed)

cochain complex of A-modules such that "multiplication by 'C

n

-+ 'C

n

is inj ecti ve, all integers

n,

and where we have a

fixed map of cochain complexes of A-modules Hn {¢*): Hn(,C*) -+Hn(C*)

that n.

(Such

a

'C*,

¢*

ttl:

¢*:

'C* -+ C*

such

is an isomorphism, all integers

exists, as is easy to see - we have

made this construction in the footnote to Remark 4 following Theorem 2 of Chapter 1), or (2)

O~

~

=

F~

~

@C*,

where

A

F~

is a

~

(non-positively indexed)

cochain complex constructed as follows: be any acyclic, flat (e.g., projective), homolo-

Let

gical resolution of the right A-module Ft

~

A/tiA,

and then let

be the corresponding (non-po£itively indexed) cochain com-

plex such that

F~ = pi ~

-n'

n

~;z'.

pi = 0

(E.g., one can take

n

to be "multiplication by (3)

where

'C*

o~ ~

=

F~ @ ( ' ~A

and

¢*:

till),

or

C*) , 'C* -+C*

are as in (I) above, and the non-

positively indexed cochain complex Then for each integer (I)

and

'

F~

~

is as in (2) above.

n,

We have a short exact sequence of

~t-modules

506

Chapter 2 n

O .... H (C*)

(* )

At

. n .... It-m [H i>O

(D~)l l

....

Hn + l (C*)) .... 0

lim (precise ti-torsion in i>O and an isomorphism (2)

Proof:

The proof is similar to that of the footnote to Remark

4 following Theorem 2 of Chapter 1. fine

Indeed, if we were to de-

as in Remark 4 above of this chapter,

"D~" l

(let's call

this Definition (0)), instead of using Definition (I), (2) or (3), then Theorem 2 becomes Theorem 2' of Remark 4 of this chapter.

It remains to show that Definitions (I), (2) or (3)

above yield cochain complexes

D~ l

having the same cohomology

as that using Definition (0).

But the

nD~" l

of Definition (3)

maps into that of Definition (2), and by the Kunneth relations (since each

F

n

Similarly, the

is flat over "D~n

A)

has the same cohomology.

of Definition (3) maps into that of Defi-

l

(F* ® 'C* .... (A/tiA) l~ 'C*), and since "multiplication A A n n 'C .... 'C is injective (or equivalently, since

nition (1) by

tin:

Tor~(A/tiA, 'C n ) J

i

A

Tor. (A/t A, ) J

_

= 0,

= 0,

all integers j

~

2,

since

j t

~1

(of course,

is a non-zero divisor)),

again the usual Kunneth relations spectral sequence implies that this map induces as isomorphism on cohomology. nitions (1), (2) and (3) oE the same cohomology groups. tion (1)

D~ l

Thus, Defi-

give cochain complexes having

Also, this shows that in Defini-

(or (2), or (3)), the cohomology of

Di'

is indepen-

dent of the arbitrary choices (in Definition (I), of a suitable 'C*

and

~*i

in Definition (2), of

P*i

in Definition (3),

The Short Exact Sequence 1.8 of both of these).

507

To complete the proof that [Theorem 2' of

Remark 4 above of this chapter implies Theorem 2 above], it suffices to show that

using Definition (0)

(i. e.,

as de-

fined in Theorem 2' of Remark 4 of this chapter) has the same cohomology as

Di

using Definition (2).

is a non-zero divisor,

But in fact, since

t

(as in the footnote to Remark 4 follow-

ing Theorem 2 of Chapter 1 with

t

i

replacing

t)

in Defini-

i p*,

tion (2) we can take for flat resolution the projective i t i resolution -+ 0 -+ 0 -+ A---'»A -r A/t A .... 0 of the right A-module A/tiA.

But then (as observed in the footnote to Remark 4 fol-

lowing Theorem 2 of Chapter 1, with

t

i

replacing

Definition (0). Q.E.D. (Remark:

as

D'!'

1.

as defined in

D~ 1.

defined in Definition (2) coincides with

t),

(For details, see "percohomoloqy", Chap. 5).

In the proof of Theorem 2' of Remark 4, one has

to compute explicitly the map: that this map is

Hn«t*)i).

Hn-l(C*(+l)) -+Hn(C*)

and show

This is not difficult, going back

to the explicit construction of the coboundary in the cohomology sequence of a short exact sequence of cochain complexes.

But

there is an alternative way, using Theorem 2 and [I.A.C.), analogously

to our last observation in the footnote to Remark 5 fol-

lowing Theorem 2 of Chapter 1. Namely, first suppose that the hypotheses are as in Theorem 2 of Remark 5 of this chapter; to prove that the long exact cohomology sequence of Theorem 2' has the indicated form.

In

fact, in the proof of Theorem 2 of thi& chapter, we have shown that Definitions (I), (2) and (3) of isomorphic cohomology.

all have canonically

And (as in the proof of Theorem 2 of

Remark 5 of this chapter) the construction

D~ 1.

of Theorem 2'

Chapter 2

508

in the terminology of the proof of D*" i ' Theorem 2 of Remark 5 of this chapter) is a special case of i ti i Definition (2) (when P*= ... O+O+A ->A+A/t A-+O). There("Definition (0) of

fore, to prove the assertion under the hypotheses of Theorem 2, it suffices to prove the analogous assertion with, e.g., Definition (1) of

D'i!'. ~

the exact

But under Definition (1) of

sequence (1) and isomorphism (2) in the conclusion of Theorem 2 follows from Theorem 1 applied to the cochain complex of A-modules

'C*.

This proves the assertion under the hypotheses of

Theorem 2 of Remark 5 of this chapter.

Finally, knowing Theo-

rem 2' of Remark 4 of this chapter is true in the case A = the category of A-modules,

A

phism "multiplication by the center of

A

a ring, and t",

t

t*: C*-+C* = the endomor-

a non-zero divisor in

A

in

(i.e., under the hypotheses of Theorem 2 of

this chapter), implies Theorem 2' of Remark 4 of this chapter in general, by using the Exact Imbedding Theorem,

[I.A.C.]

(by the

same argument as in the last portion of the footnote to Remark 4 following Theorem 2 of Chapter 1». Remark 6. chains? Theorem.

Can Theorem 2' of Remark 4 be generalized beyond coYes. Let

A

and

B

be abelian categories.

denumerable direct products exist in the category

Suppose that B,

and that

the denumerable direct product of epimorphisms in the category B

is an epimorphism (that is, that the functor "denumerable

direct product" is an exact functor in

B).

Let

ColA)

be the

abelian category having for objects all cochain complexes (indexed by all the integers) of objects and maps in the abelian category

A,

and all maps of such cochain comlexes.

If

The Short Exact Sequence 1.B F* ECo(A)

and if

O*=O*(t*,F*)

t*:F* ->F*

is an endomorphism, then define

to be the object of

CoCA)

cochain complex of objects and maps of

d all

n

(i.e., the

is

(in terms of ele-

the map

n n+l (u,v) = (d (u) + t* (v) , - d (v»

UE:Fn,

VE:Fn+l,

nE:7.

Also, if

,

F*ECo(A)

F* (+1) E: Co (A) to be the cochain complex such that all integers Fn+l(+l)

n,

F*(+l)

n n F + l ->F + 2

(n+l) ' s t coboundary:

(I.e., roughly speaking,

is the negative of the

in the cochain complex

F*(+l)

is

F*

n E 7,

abelian category

C,

F* E C,

1

C into the

and let i

t* :F* -> F*

~

n

E~)

is a map in

"F* C

shifted by of

in the category (+1) = tn+l,

C,

is an object in the subcategory F* ->

(the maps in

o~ 1

F n -> F n x Fn+l,

such that the cochain complex

+1", is also an object in the sub-

Co (A), such that the map, Co(A)

be any

the cochain com-

0,

whose n' th coordinate is the inclusion:

category

n

Let

C, such that, for each integer

F*(+l),

t

B.

and such that the inclusion map:

all

CoCA)

be any cohomological exact connected se-

O'ii=O*((t*)i,p*) E:Co(A)

Co (A)

F*.

"shifted by +1").

quence of functors from the abelian category

plex

Fn(+l)->

C be any exact, abelian subcategory of

and let

map in

then define F n (+1) = pn+l,

and such that the n'th coboundary:

in the cochain complex

Now let

7-indexed

A) such that

dn:O n -> on+l

and such that the coboundary ments, see [I.A.C.])

509

t* (+1) :F* (+1) .... F* (+1),

lies in the subcategory

all integers

n),

C

(where

such that the map in

Co (A) ,

Chapter 2

1510

"t* xt*(+1)":05 ->-05 a map in in

C,

Co(A),

such that for each integer call it

is the map: F n x F n+l

(whose n'th coordinate is tnx(tn(+l»)

"(1 xt*)i",

(identity F

->-

i.:: 1;

x «n+l) I st coordinate of

n

Co(A)

projection:

F n x F n + l ->- F n + l ,

category

of

C

"(1 x t *) i

gers

n.

nE;??,

t*):

• 1S a map 1n

II'

n E;r)

c,

is a map in the sub-

Tn (05) = 0,

(**)

n H (05) = 0,

all integers (1)

so this last condition will hold if, whenever Hn (G*) = 0,

0t-l

whose n'th coordinate is the

all

It is easy to see that

that

->-

Co(A).

Suppose also that mark:

ot

and such that the proj ection map:

(the map in

°t->-F*(+l)

the map

whose n'th coordinate,

Fn x F n + , l .1S SUC h t h at

all integers

i >1

is

all integers

n, then

n.

(Re-

all integers G* E C

Tn (G*) = 0,

Most exact connected sequences

n,

is such all intethat one

comes across in practice have this latter property, and therefore a fortiori are such that condition (**) holds for any such

F* E C ). Then for every integer

sequence in the category (1)

o ->-T n (F*)l\t*

n

B:

->-lim Tn(O*)

i>O

there is induced a short exact

->-

1

lim (precise (t*) i-torsion part of

Tn + l (F*)) ... 0

i>O and an isomorphism in

B:

(1) To show this, by the Exact Imbedding Theorem [I.A.C.] it suffices to prove in the case that A = category of abelian groups, C = all of Co (A). But then, we have seen in the footnote to Remark 4 following Theorem 2 of Chapter 1 (in the special case t = 1) that Hn (OB*) '" Hn (C* /1 • C*) = 0, all integers n. (Use Definition (1) of *(=0 of that footnote, with 'C*=C*, <jJ*=identityof C*).

0)

The Short Exact Sequence 1.8

511

(2)

(Here, as usual, eg., "Tn(F*)At*"

Proof:

"lim (Coker(Tn«t*)i):Tn(F*) ->-Tn(F*)))". i>O

First, for each integer

sequence in (* i)

denotes

0 ->- F*

Co(A),

-7

and in

Dt ->- F* (+1) ->- 0,

i = 0,

by hypotheses

we have the short exact

C, which yields the long exact

sequence in the category

For

i >0

(**),

B:

Tn(D

O)

=

all integers

0,

Therefore, from the long exact sequence (1 ), 0 the

n.

we deduce that

(n-l) 'st coboundary is a canonical isomorphism:

(2)

Tn-1(F*(+1)) :;Tn(F*)

in the category

all integers

B,

Next, observe that we have a commutative diagram with exact rows in the category

Co(A),

1 ft

and also in

C,

0 +F* ->-Dt ->- F*(+l)->-O

(3)

(identity)

0->- F* ... where

(t*(+l))i

t*(+l)

of

D~

~

l(t*(+l))i ->- F* (+1) ->- 0

denotes the i'th iterate of the endomorphism

F*(+l)

in

C,

and where

P!:D! +D(l

is the com-

posite of the mappings: "(lxt*)," Di

~>D!_l

"(lxt*)i_l"

>D!_2~···

" (1 x t*) "

1) D*

o

n.

Chapter 2

512

in the category

C.

The diagram (3) yields a map from the long

exact sequence (li) into the long exact sequence (1 ), all in0 tegers

i > O.

The portion about the (n-l) 'st coboundaries is

B:

the commutative square in the category

(4)

where the bottom isomorphism is the isomorphism (2), and where is the (n-l)'st coboundary of

the top map, labeled the long exact sequence (li)

(of the short exact sequence (*i».

Therefore, by the diagram (4) and the isomorphism (2), if for every integer

n

we identify the objects

B

Tn(F*) in the category

Tn-l(F*(+l»

and

by means of the canonical isomorphism

(2), then the (n-l)'st coboundary in the long exact sequence (li) is identified to the endomorphism (5)

Tn-l«t*(+l»i)

of

Tn-l(F*(+l»,

all integers

n.

But we also have a commutative diagram with exact rows in the category

Co (A),

1

and in the category

C,

O--l>F*-:>DCi ---------':>F* (+1) (6)

t*

l't* x t* (+1)"

0-> F*--l>D*

o

:>0

}* (+1) :> F* (+1) - > 0

where each of the rows is the short exact sequence (*0).

This

gives rise to an endomorphism of the long exact sequence (1 ), 0 and considering the portion of (1 ) around the (n-1) 'st co0

The Short Exact Sequence 1.8

513

boundary, to the commutative square: Tn-l(F*(+l))

,., i>Tn(F*)

Tn-l(t*(+l))l

(7)

Tn - l (F* (+1))

lTn(t*) ,., i> Tn (F*)

in which the horizontal isomorphisms are the isomorphisms (2). Since we made the isomorphisms (2) into identifications, it follows that, under the identification Tn-l(t*(+l)) Tn(t*)

of

of

Tn-l(F*(+l))

Tn(F*).

(2~,

the endomorphism:

corresponds to the endomorphism

Hence similarly for the i'th iterates.

Considering observation (5), it follows that, if we identify Tn-l(F*(+l))

and

all integers

n,

Tn(F*)

by means of the isomorphism (2) for

then the (n-l) 'st coboundary in the long

exact sequence (Ii) is identified to the endomorphism of the obj ect all integers

Tn (F*) n.

in the category

B,

Tn«t*)i)

all integers

i.:: 0,

But then the long exact sequence (Ii) is

identified with a long exact sequence

all integers

i > O.

Using the map

"(1 x t*) i+l",

map from the short exact sequence (*i+l) sequence (*i)

in the category

exact sequence (li+l) category

B.

it is the map map

C,

we obtain a

into the short exact

and therefore from the long

into the long exact sequence (Ii) in the

(This map is such that, "along the first "Tn(t*)";

"identity"· Tn (F*) ,

"Tn("(1xt*)i+l")",

"along the second

Tn(F*)",

Tn(F*)", it is the

and "along

all integers

Then using the long exact sequences

n) ,

all integers

(1i)'

i .:. 0,

i > O.

and the map

514

Chapter 2

of long exact sequences from (li+l) ly constructed, all integers

i

(i~O)

long exact sequences

~

into

(li)

just explicit-

0, in lieu of the corresponding

and maps of long exact sequences

used in the proof of Theorem 1, we now obtain similarly (and just as easily) the two conclusions,

(1) and (2), of the TheoQ.E.D.

rem. Example 1.

Let

subset of

X let

X,

be a topological space, let A

U

be an open

be a ring with identity and let

be an element in the center of the ring

A

sarily be a non-zero divisor).

be any cochain complex

of left A-modules over G!

F*

need not neces-

and for each integer

X,

i >0

let

be the cochain complex of sheaves of left A-modules over

G~

such that

0=

the coboundary Ex, x

n u E Fx

the image of ment

Let

(t

tEA

F n )( Fn+l,

all integers

dn:Gr: .... Gr:+ l

and

l

l

n+l v E Fx

(u,v)

Then for every integer sequence of

and such that the

is the map such that, whenever (the stalks at

in the stalk at

(d n (u) + tiv, _d n + l (v) ),

n,

X

x

of

all integers n,

we have that

x) ,

Gn + l

is the ele-

n.

there is induced a short exact

Al\t = (=t-adic completion of

A) -modules:

(1)

Hn + l

->-lim {precise t -torsion elements in i>O

(X,U,F*)}

.... 0,

and for each integer

n

we have an isomorphism of

Al\t-modules

(2)

liml {precise ti-torsion elements in i;:O

Hn+l(X,U,F*)}.

The Short Exact Sequence 1.B Proof:

Take

515

A = category of all sheaves of A-modules over the

topological space whole category

X,

B = category of all A-modules, and

CoCA)

n E 6"',

nected sequence of functors from hypercohomology":

to be the exact con-

CoCA)

Tn(F*) =Hn(X,U,F*),

C = the

into

B,

"relative

all integers

n.

Then

the hypotheses of the Theorem in Remark 6 above are all obviously satisfied.

(The only one requiring verification is (**);

and this follows for the parenthetical reason given following the statement of (**) - that is, if the cohomology sheaves

Hn(K*)

K* E CoCA)

are all zero, all integers

then the relative hypercohomology groups: are zero, all integers made directly.

is such that

Hn(X,U,K*)

n,

(=Tn(K*»

n-and this latter observation is easily

(Notice that if the cochain complex

bounded below, i.e., if

n K = 0, all

n < N,

3 N E6"',

K*

is

then this

also follows from the second spectral sequence of relative hypercohomology.}) mology

(Note that in Example 1 the relative hypercoho-

Hn(X,U,F*),

C* (X,U,F*)

nEl',

does come from a cochain complex

(e.g., use "Godement cochains" as I define them in

[P.RWCJ, Chapter I,---or, if our "punctual cochains").

X

is punctual ([P.P.WCJ, Chapter I),

Therefore, Theorem 2' of Remark 4 (or

even Theorem 2 of Remark 5) can be used, if one wishes, in lieu of the Theorem of Remark 6, to cover the case of relative hypercohomology and an endomorphism of a cochain complex of sheaves that is not a monomorphism.

I prefer the Theorem in Remark 6

since it does not necessitate making use of cochains). Remark 7.

Of course, under the hypohtheses of Example 1, or of

Theorem 1', of Remark 3; or of Theorem 2' of Remark 4: or of Theorem 2 of Remark 5; or, respectively, of the Theorem of Re-

516

Chapter 2

mark 6, then of course we have a Corollary with three equivalent conditions analogous to Corollary 1.1.

(The formulation

is obvious and left to the reader in each case, and of course the proof that 1.1.

(1)~(2)

exactly duplicates that of Corollary

Similarly for the statement of condition (3) and the

proof that

(l)~

(3), in all cases except in "abelian category

theoretic" ones, that is, in all cases except Theorem I' of Remark 3, Theorem 2' of Remark 4, and the Theorem of Remark 6. However, in these cases condition (3) can also be stated (e.g.,

in the Corollary 1.1' to Theorem 1', it reads "the images of the natural mappings:

n n H (C*) -+ H (C* /tC*),

and

coincide"), and it is not

[lim (Hn(C*/tiC*))]-+Hn(C*/tC*),

i>O too difficult to prove once again that essentially the same as in

(1)~(3)

(the proof is

Corollary 1.1; again, the objects on

both sides of the map of condition (1) are "complete", in the sense that N "" lim N/tiN

i>l for each of these objects duced by condition

t*i

N,

were h

i.::. O.

all integers

uti

II

denotes the map in-

One proves that, under

(3), the map in condition (1) induces an isomorphism

"after taking modulo

till

(i.e., after, e.g., for the first

object in the map of condition (1) forming "Coker (H n «t*) i): n H (C*) -+ Hn (C*) ) ") ) • Remark 8.

i >0

Under the hypotheses of Theorem 1 for each integer

the hypotheses of Theorem 2 of Chapter 1

non-negative integer

i,

let

(En (t i ) d n )

r

' r n(:;;?1

r>O

hold. and

For each

The Short Exact Sequence 1.8 (E n

( i)) t nE&,,'

CL

517

denote the generalized Bockstein spectral se-

quenc~

and the Ero-term as described in Theorem 2 of Chapter 1, for i the cochain complex C* and for the element t E A (which is

in the center of ti:C

n

-+ en

A,

and is such that multiplication by

in injective, all integers

n).

Then by the conclu-

sion of Theorem 2 of Chapter 1, for each integer

i::: 0,

we

have the short exact sequence (1)

i 0-+ (A/tiA) ® [Hn(C*)/(t-torsion))-+En(t )-+ 00

A

{precise ti-torsion elements in t-divisible} -+

Hn+l(C*)

that are

o.

The long exact cohomology sequence, the exact couple and the generalized

Bockstein spectral sequence described in Theo-

rem 2 of Chapter 1 for the cochain complex t i +l E A

and the element

each admits a map into the corresponding datum for

cochain complex i::: 0).

C*

C*

and the element

ti EA

Therefore, if we fix the integer

n

(all integers and let

i

vary

through the non-negative integers, then the short exact sequences

(1) define a single short exact sequence of inverse

systems, indexed by the non-negative integers, of A-modules fact, of

At A -modules).

(in

As in the proof of Theorem 1, if we

throw this short exact sequence through the system of derived functors,

lim,

i>O

, 1 I -tm ,

0,0, ••• ,0, ••• ,

then we obtain a short

i>O

exact sequence and an isomorphism, for each integer

n.

We

have proved CorollarLl:..:1..

(Inverse limit of the generalized Bockstein

"abutments" ). Under the hypotheses of Theorem 1, let n E Z,

denote the "abutment" (i.e.,

E~ (t i

),

Eoo-term) of the generalized

Chapter 2

518

Bockstein spectral sequence, as defined in Chapter 1, Theorem 2, corresponding to the cochain complex of A-modules:

the element:

t

i

E A,

non-negative integer

i > O.

all integers En(ti)

i,

00

is the

C*

and

(That is, for each E -term 00

of the gen-

eralized Bockstein spectral sequence corresponding to the long exact cohomology sequence: n-l i n i d_ _ >H n (C*) _t_> Hn (C*) -+ Hn (C* /tic*) ...£-H n + 1 (C*) i..;. ... ) . Then for every integer (1)

n,

there is induced

A short exact sequence of

AAt-modules

(*)

lim{precise ti-torsion elements in i>O and for every integer

n

Hn+l(C*)} -+0,

there is induced an isomorphism of

A" t-modules (2)

that are t-divisible}. Proof:

The observation immediately preceding the statement of

the Corollary

prove equation (2) of the conclusion of the Corol-

lary, and also establish a short exact sequence similar to (*), except that the third group in the sequence is l!m{precise ti-torsion elements in i>O t-divisible},

Hn+l(C*)

that are

rather than the one occurring in conclusion (*) of the Corollary.

However, obviously

The Short Exact Sequence 1.8

519

Hn+l(C*)} ~

lim{precise ti-torsion elements in

i>O

l~m{precise ti-torsion elements in

Hn+l(C*)

that are

i>O t-di visible} , and in fact these groups are also isomorphic to lim{precise ti-torsion elements in ij;"O finitely t-divisible}.

Hn+l(C*)

that are in-

(Since, in e.g., the inverse system ({precise ti-torsion elements in

Hn+l(C*)})i>O'

ments in

Hn + l (C*) l .... {precise ti-torsion elements in

is "multiplication by

the map:

{precise ti+l-torsion ele-

~

i

t", all integers

0).

Hn + l (C*)}

This proves

the Corollary. Notes 1.

Let

tity and let

M be an A-module, where tEA

be any element.

M

for the t-adic topology in the A-module M}.

is a ring with iden-

Then one can define the

topological t-torsion in the A-module

elements in

A

to be the closure, M,

of

{t-torsion

(This definition can be generalized to abelian

categories such that denumerable direct products exist:

M

If

is an object of an abelian category such that denumerable

direct products exist, and if M in the abelian category

A,

t:M .... M is any endomorphism of then define

and define topological t-torsion in of (t-torsion part) of

i M't = lim (Coker t ), i>O

M = pre-image of the image

M under the natural mapping:

M .... M't) •

Then, in the short exact sequence (*) of the conclusion (1) of Corollary 1.2, notice that the first can be written equally well as Hn(C*)At/(topological t-torsion)

AAt-module in the sequence

Chapter 2

520

- i.e., the first

AAt-module of the short exact sequence (*)

of conclusion (1) of Corollary 1.2 is canonically isomorphic to the first

~t-module of the short exact sequence (*) of

the conclusion (1) of Theorem 1, taken modulo topological

t-

torsion. 2.

Notice also that the third

AAt-module in the short

exact sequence (*) of the conclusion (1) of Corollary 1.2 coincides with the third

AAt-module in the short exact sequence

(*) of conclusion (1) of Theorem 1.

Thus, the short exact

sequence (*) of conclusion (1) of Corollary 1.2 is a quotient (obtained by modding out by the topological t-torsion in Hn(C*)At

in the first

AAt-module, and its image in the second

AAt-module) of the corresponding short exact sequence,

(*) of

conclusion (1), of Theorem 1. 3.

Th e group

' 1 { prec~se , ' ' l im ti - t ors~on e 1 ements ln

i>O Hn+l(C*)

that are t-divisible}

1.2, is often zero.

in conclusion (2) of Corollary

E.g., if every t-divisible, precise t-

torsion element in

Hn+l(C*)

is infinitely t-divisible (which

is often the case.

See e.g., Proposition 3 of Chapter 1, and

Chapter 4 below), then by an induction on used in a

p~rtion

i >0

similar to that

of the proof of Proposition 3 of Chapter 1,

it is easy to see that every t-divisible,' precise ti-torsion element in gers

i > O.

Hn+l(C*)

is also infinitely t-divisible, all inte-

But, when that is the case, the inverse system of

which

liml is taken on the right side of equation (2) of Coroli~O lary 1.2; is an inverse system in which all the maps are epi-

morphisms.

Therefore, when this is the case( that is, when

every t-divisible, precise t-torsion element in

Hn+l(C*)

is

521

The Short Exact Sequence 1.8

infinitely t-divisible - a condition that often (although not always) holds -

) then both

, (2) of the AAt -mo d u l es 'ln equatlon

conclusions of Corollary 1.2 are zero.

(Thus, roughly speaking,

the two groups in equation (2) of Corollary 1.2 "more often are zero" than are the two groups in equation (2) of Theorem 1.

In

fact, more precisely, it is easy to show that the natural monomorphism of inverse systems:

({precise ti-torsion elements in

Hn+l(C*)

that are t-divisible})i>O +({precise ti-torsion ele-

ments in

Hn+l(C*)}), 1>0

induces a monomorphism after taking

liml (reason: the quotient inverse system is isomorphic to: i>O 'l (Tprecise t -torsion elements in Hn+l(C*)/(t-divisible elements)})i>O'

and this latter clearly has inverse limit zero.

Considering the exact sequence of six terms deduced from the short exact sequence of inverse systems by throwing through the

' ' 1 0 , ••• , 0 , ••• glves t he lim, l 1m, ito ito We record this latter parenthetical observa-

system of derived functors: desired result».

tion as a Corollary: Corollary 1.2.1. any fixed integer.

Under the hypotheses of Theorem 1, let

n

be

Then the two canonically isomorphic AAt_

modules appearing in equation (2) of the conclusions of Corollary 1.2 are canonically isomorphic to an AAt-submodule of either of the two canonically isomorphic

AAt-modules appearing

in equation (2) of the conclusions of Theorem I--the monomorphism:

liml({preciSe ti-torsion elements in i>O

Hn+l(C*)

t-divisible}) + liml({precise ti-torsion elements in ito being induced by the inclusion of inverse systems, ti-torsion elements in

Hn+l(C*)

({precise ti-torsion elements in

that are

Hn+l(C*)}) ({precise

that are t-divisible}) i>O + Hn+l(C*)}), O· 1>

522

Chapter 2 4.

We conclude Remark 8 with another corollary to Corol-

lary 1.2 to Theorem 1: Corollary 1.2.2.

Under the hypotheses of Theorem 1, the nota-

tions being as in Corollary 1.2, we have that for any fixed integer

n,

each of the equivalent conditions of Corollary 1.1

is equivalent to each one of the following two conditions: (1')

The natural map of

AAt-modules:

n [H (C*) I (t-torsion) ]A t ... lim [E~ (ti) ] i>O is an isomorphism. (3' ) i

~

uEE~(t)

Let

I,

such that

En(ti) 00

is

vEHn(C*)

u

be such that there exist l

= u

and such that the image of

all integers

ui '

n i ui E Eoo (t ),

i> 1-

such that the image of

v

ui +l

in

Then there exists (under the monomorphism in

the short exact sequence (*) of conclusion (1) of Corollary 1.2) in

E~(t)

Proof:

is

u.

Let condition (2') be identical to condition (2) of

corollary 1.1.

Then the proof of Corollary 1.1 (that conditions

(1), (2) and (3) of Corollary 1.1 are equivalent) shows equally well that conditions (I'), (2') and (3') above are equivalent (where, in the proof, the short exact sequence (*) of conclusian (1) of Corollary 1.2 takes the place of the short exact sequence (*) of conclusion (1) of Theorem 1). (3) and (1')-=(2')-=(3').

Thus (1)

~(2)~>

Since conditions (2) and (2') are

identical, the Corollary follows. Notes 1.

As noted above, the proof of Corollary 1.2.2 proceeds

by analogy to that of Corollary 1.1.

In particular , one also

The Short Exact Sequence 1.B

523

proves analogues of Lemma 1.1.1 and Lemma 1.1.2, namely Lemma 2.1.1. integer

n

Under the hypotheses of Corollary 1.2.2, for each and every integer

j

~

0,

we have the short exact

sequence 0->- (A/tjA)

(* , )

J

@

[Hn(C*)/(t-torsion)]

A

->- (A/tjA)

@

[I'm En (ti»)

A

Jo

00

->-(precise tj-torsion in

Hn + 1 (C*)

that is infinitely t-divisible) ->- O.

Lemma 2.1.2. every integer

Under the hypotheses of Corollary 1.2.2, for n,

and for every integer

j

~

0, the natural

mapping: (A/tjA)

@

A

[~im E~(ti)] ->-E~(tj) 1>0

is a monomorphism. The proofs of Lemmas 2.1.1 and 2.1.2 are exactly analogous to those of Lemmas 1.1.1 and 1.1.2, and need not be repeated. (And again, the proof of the equivalence of conditions (2') and (3')

of corollary 1. 2.2 makes use of Lemma 2.1. 2,

(in the same

way that the proof of Corollary 1.1 makes use of Lemma 1.1.2). Note 2.

The proof of Lemma 2.1.2, as the proof of Lemma 1.1.2,

makes use of the Five Lemma, and involves a commutative diagram in which the top row is the exact sequence (*,) of Lemma 2.1.1, J and the bottom row is the short exact sequence: (1)

j) 0->- (A/tjA) 0 [Hn(C*)/(t-torsion») ->-En(t co A

->- (precise tj-torsion in

Hn + 1 (C*)

that is t-divisible)

Chapter 2

524 +0,

established in Chapter 1, and is exactly as in Lemma 1.1.2. Notice also that, by the Five Lemma applied to that diagram, the mapping described in Lemma 2.1.2 will be an epimorphism (and therefore an isomorphism, since by the conclusion of Lemma 2.1.2 said mapping is always a monomorphism) iff the mapping from the rightmost group in the short exact sequence (*.) into the rightmost term of the short exact sequence (1) J

above is an epimorphism.

But clearly, that latter condition

will be so iff every precise tj-torsion element in that is t-divisible, is infinitely t-divisible. corollary 1.2.2.1. let

n

Hn+l{C*)

We now prove:

Under the hypotheses of Corollary 1.2.2,

be any fixed integer.

Then the following six conditions

are equivalent: (1)

For all integers [lim

j ,::,1,

the natural mappings:

E~(ti)l +E~(tj)

i>O are epimorphisms. (1' )

There exists an integer

j.::.l

such that the mapping:

is an epimorphism. (2)

For all integers

i,j

with

i.::.j.::.l,

the natural

mapping:

is an epimorphism. (2')

There exists an integer

j.:.l

such that for every

The Short Exact Sequence 1.8 integer

i.:. j

525

the natural mapping:

is an epimorphism. (3)

Every t-torsion element in

Hn+l(C*)

that is t-divis-

ible, is infinitely t-divisible. (4)

Every precise t-torsion element in

Hn+l(C*)

that

is t-divisible, is infinitely t-divisible. Proof:

(3) implies

fact, if we let

(4) is obvious.

Proof that

(4)~(3):

In

M=Hn+l(C*)/(infinitely t-divisible elements),

then throwing the short exact sequence: 0 .... (infinitely t-divisible elements in .... H

n+l

n H +l (C*))

(C*) +M .... O,

through the exact connected sequence of functors: 1

HO~ [T 1 (7 [T 1/T • it [T l,

),

letting

"t"--therefore the first of these func-

Ext [Tl (7 [Tl/T·7,1 [Tl, ) ,0, ••• ,0, ••• 7 (where every left A-module is regarded as a 7 [Tl-module by "T"

act as

tors, regarded as a functor on the category of left A-modules, is the functor:

N~>

(precise t-torsion in

of these functors is the functor:

N),

N~>N/tN),

and the second we obtain an

exact sequence of six terms, a portion of which implies that every precise t-torsion element in precise t-torsion element in (4)~(3),

that

~.

M in

Let

A

M

is the image of some

Hn+l(C*).

Therefore, to prove

it suffices to prove the following be a ring, let

t

be an element of

A

and let

be a left A-module such that every precise t-torsion element M

that is t-divisible is zero.

Then

M

has no non-zero

Chapter 2

526

t-divisible, t-torsion elements.

(Proof of Lemma:

be at-divisible t-torsion element. fact, since J. _> 1

If

Lemma.

To show that

t

j = 1,

j

By induction on

• u = O.

t-divisible.

By inductive assumption,

and

of the Lemma,

u

v

= tu.

Then let

= v = 0,

Since

Then

is t-divisible.

u == 0,

In

we show that

j.:::. 1,

t

j

• v

to prove it

and that

=0

and

v == O.

u

is

v

is

But then

Therefore by the hypotheses

completing the induction and therefore

the proof of the Lemma.) Therefore

j,

t j +l • u = 0

I.e., suppose that

t-divisible.

tu

u = O.

then this follows from the hypotheses of the

Suppose the assertion is true for

j + 1.

u EM

is a t-torsion element, there exists an integer

such that

u = O.

for

u

Let

This completesthe proof that

(4)~(3).

(3)~(4).

(3)

<~

(4), it follows that, for each integer

j

~

1,

that (3) and (4) are both equivalent to the condition. (4 ) j

Every precise tj-torsion element in

Hn+l(C*)

that

is t-divisible, is infinitely t-divisible. But by the observation immediately preceding the statement of Corollary 1.2.2.1, the condition (4.) is equivalent to the J

mapping of Lemma 2.1.2 being an epimorphism.

And this latter

condition is clearly equivalent to (lj)

The natural mapping: [ 1'*"m En (t i) 1 + En (t j ) i>O 00

00

is an epimorphism. Thus, for every integer Since also

(4j)~(4)~(3),

tion (1.) is independent of J

j

~

1,

we have that

it follows that, for j.

(1.)-=(4.).

J

j '::1,

J

condi-

Therefore conditions (1) and

The Short Exact Sequence 1.8

527

and (1') of the Corollary, and condition (1.), for any fixed J

~

integer

j

1,

are all equivalent.

Therefore (1)

¢=>(4).

It is obvious that (1)<=>(2) and (1')

~(l') ~(3)

<~(2').

proves Corollary 1.2.2.1. Remark 9.

This Q.E.D.

Under the hypotheses of Example 1 of Remark 3; or

of Theorem l' of Remark 3; or of Theorem 2' of Remark 4; or of Theorem 2 of Remark 5; or of the Theorem of Remark 6; in lieu of the hypotheses of Theorem 1, we obtain a Corollary exactly analogo us to Corollary 1.2 of Remark 8 - with observations analogous (and in no way weaker) to Note 1 of Remark 8, to Note 2 of Remark 8, to Note 3 of Remark 8, incl. Cor. 1.2.1 of Note 3 of Remark 8, and to Note 4 of Remark 8, including Corollary 1.1.2 of Note 4 to Remark 8, all holding under these modified hypotheses. (Of course, under the hypotheses of Theorem 2' of Remark 4, for each integer

i >1

the Bockstein abutment

n E 'l',

(and indeed the whole generalized Bockstein spectral sequence n

(Er(t

i

n

),dr)nE~)

is taken to be that corresponding to the long

r>O exact cohomology sequence displayed in the first paragraph of the proof of Theorem 2' of Remark 4--i.e., the long exact sequence of cohomology of the short exact sequence of cochain complexes: O+C*+Dt+C*(+l) +0,

where 4).

D~ 1

is defined in the statement of Theorem 2' of Remark

(And under the hypotheses of Theorem 2 of Remark 5, for

each integer

i

~

0,

the Bockstein abutments

i ) En(t 00

deed the whole generalized Bockstein spectral sequence

(and in-

528

Chapter 2

(E~(ti),d~)n~)

is taken to be that defined by the cochain

r>O

complex of A-modules the center of

A

C*

t i EA

and the element

and is a non-zero divisor in

(that is in

A),

as defined,

in any of three different ways, in the footnote to Remark 4 fol(And, of course, under the

lowing Theorem 2 of Chapter 1).

hypotheses of the Theorem in Remark 6, for each non-negative integer

i,

the Bockstein abutments

i) En(t 00

(and indeed the

whole generalized Bockstein spectral sequence

n

i

n

(Er(t ) ,d r )nE7) r>O

is taken to be that corresponding to the long exact sequence (I' )

(the labeling of the sequence being as in proof of the Theorem of Remark 6)--recall that this long exact sequence is obtained from the long exact sequence (Ii)'

(as labeled in the proof of

the Theorem of Remark 6) by making the identifications (2)

(as

labeled in the proof of the 'rheorem of Remark 6); and that the long exact sequence (Ii) of the proof of the Theorem of Remark 6 is the long exact sequence obtained by applying the exact connected sequence of functors

Tn,

n E :l ,

to the short exact

sequence

in the category

C,

all integers

i >0

(again, the labeling

being as in the proof of the Theorem of Remark 6». under these various

Similarly,

sets of hypotheses, the analogues of Lemmas

2.1.1 and 2.1.2 also hold.

And under these modified sets of

hypotheses, the analogue of Corollary 1.2.2.1 also holds-with

The Short Exact Sequence 1.8

529

the exception that, in the abelian-category theoretic cases (that is, in Example 1 and Theorem I' of Remark 3; Theorem 2' of Remark 4; or in the Theorem of Remark 6) one should delete the conditions (2) and (2') in the statement of Corollary 1.2.2.1, unless one also assumes the Axiom (P.2), see Introduction, Chapter 1, section 7 (in the category

A when under the hypo-

theses of Theorem I' or Theorem 2'; and in the category

6

when

under the hypotheses of Example 1 of Remark 3 or of the Theorem of Remark 6).

This Page Intentionally Left Blank

CHAPTER 3 COHOMOLOGY OF AN INVERSE LIMIT OF COCHAIN COMPLEXES

In this chapter we recall a classical exact sequence, corning from a well-known spectral sequence.

The material of

this brief chapter is indeed very well known (all that is used is among the simplest, best-known cohomological spectral sequences), and Chapter

is included merely for internal com-

3

pleteness of this book.

In consequence, proofs are not given

in extreme details (and theorems are labeled as propositions). Proposition 1.

Let

(C~,a~.) . . 0 1. 1. J 1., J'::'

be an inverse system indexed

by the non-negative integers of cochain complexes (indexed by all the integers) of left A-modules, where identity. (I)

A

is a ring with

Then there are induced A sequence , integer

n

An,

E; J! ,

n E J! ,

of A-modules, and for each

a short exact sequence

(2)

and also there is induced a long exact sequence (extending to

+ '" on the right):

on the left, and

{3}

n-l ~> Hn (lim c~) i~O

.... An .... Hn - l (l1ml

i>O

1.

n

n

H + l (lim Cp .... A + l ....... i~O

531

n

c~) J!..., 1.

532

Chapter 3

Note:

The statement and proof of this Proposition go through

A such that denumerable direct products

to any abelian category

of objects exist; and such that the direct product of denumerably many epimorphisms is an epimorphism (i.e., such that the functor "denumerable direct product" preserves epimorphisms). Proof:

We define a double complex of left A-modules

(O p,q , dP,q (1,0)' dP,q (0,1) ) p,qE.?· C~ ....

C~

J]

1

i>O

1

F"lrs, t f or eac h ln " t egerI n, t e

be the map such that

all integers

i

~

0,

where

C~ .... C~

1\":

J

1

is the i I th projec-

1

and where

tion, all non-negative integers is the map in the inverse system Then, define on,O = On, 1 =

c~,

J]

all integers

n,

i>O on,m = 0,

all integers

d70~l): on,O .... on,1 d70~1) = 0, n

the map

all integers

m=O J]

i>O

d~: 1

such that

to be the map

m,. 0,

and for

n,m

1,

or J]

i>O

C~ .... 1

n.

define J]

i>O

c~+l, 1


d7i~ 0) = 0, Then

i,j,

all integers

dP,q ) (O p,q , dP,q (1,0)' (0,1)

Define

and (necessarily) define

Finally, for each integer

dn,m . on,m on+l,m (1,0) . " .... where

n'th coboundary in the cochain complex all non-negative integers

m,. 0,1.

d~: C~ .... C~+l 1

Ci,

1

1

to be is the

all integers

n,

and (necessarily) define n,m

such that

m,. 0,1.

is a (whole plane) double complex

Inverse Limit of Cochain Complexes of left A-modules such that

op,q

=0

unless

q

533

=0

or

1.

Therefore we have the two spectral sequences of this double complex, both abutting at the total cohomology of the double complex (4)

Let

0**.

An = Hn (0**),

the n'th total cohomology group, all integers

n.

Then the two

spectral sequences are respectively of the form: (5)

EP,q 2

=

HP(limq (C*

*)

'>0 1" a 1J" ,1,J_

+' i>O

)~ An

'

and (6)

(of course with, in general, different filtrations on

An).

The

EP,q-term of the spectral sequence (5) is confined to the hori2

zontal strip

q

=0

or

I,

and the

EP,q-term of the spectral 2

sequence (6) is confined to the vertical strip

p

=0

or

1.

Therefore, the spectral sequence (5) simplifies to the long exact sequence: n C"!,)....Q......,,

(5' )

1

where the map labeled boundary

n "d "

in the sequence (5') is the co-

d~-l,l: E~-l,l -.. E~+l, 0

in the

E~' q-term of the spec-

tral sequence (5), and where the other maps in the long exact sequence (5') are the edge homomorphisms in the spectral sequence (5).

Similarly, the spectral sequence (6) simplifies to

the short exact sequences

Chapter 3

534

(6' )

all integers

n.

Equations (4),

(5') and (6') imply the Propo-

sition. Remark 1.

Notice that, if

(C~,U~

'>0

is an inverse system

of cochain complexes as in Proposition I,

(or as in the Note to

Proposition I), then

1

,),

1J 1, J_

An=An[(Ci'Uij)i,j~Ol

the proof of Proposition 1 for each integer

as constructed in n

depends only on

the inverse system:

h

(a ij )n-2::,h < n+l

i, j

':.0

- i.e., only the portion of the cochain complexes, from

n-2 n+l n-2 n ' Ci the coboundaries: d i , ... ,d , and the maps: ' ... 'C i i n-2 n+l n ui,i-l,···,ui,i-l are required to define A [(ct,utj)i,j~Ol, for each integer Remark 2.

n.

Notice that the groups

structed in the proof of Proposition I, all integers

conn, form an

exact connected sequence of functors from the abelian category having for objects all inverse systems (indexed by the non-negative integers) of cochain complexes (indexed by all the integers) of left A-modules (or, under the more general hypotheses of the

Inverse Limit of Cochain Complexes

535

Note to the Proposition, of objects in the abelian category

A

obeying the hypotheses of the Note), into the category of left A-modules (or into Remark 3.

A).

If we restrict Proposition 1 to inverse systems (in-

dexed by the non-negative integers) of non-negatively indexed cochain complexes of left A-modules, then it is easy to see that the restriction of the exact connected sequence of functors An, n E 'l',

to this subcategory, for

n >

a

form a system of de-

rived functors of the functor: AO[(C'!",CI.'!',),

lim HO(C'!'). And, for such a special inverse i>O ~ system (of non-negatIvely indexed cochain complexes), ~

(C~, CI.~ ~

~J

,),

~J

'>OJ

~,J_

'>0'

=

it is easy to interpret the spectral sequences

~,J_

(5) and (6) of Proposition 1 as being the spectral sequences of the composite functors: O (H )

0

(lim), i>O

and

respectively.

(This observation even makes sense at the abelian

category-theoretic level,

i.e., under the hypotheses of the

Note to Proposition 1, with no additional hypotheses, once one observes that all the indicated derived functors exist in the reasonable sense, and that the spectral sequences of the indicated composite functors also exist). Corollary 1.1.

Let

(C'!',CI.'!",) , 1

~J

'>0

~,J_

be an inverse system (indexed

by the non-negative integers) of cochain complexes of A-modules (indexed by all the integers) and let

C* = lim C!. i>O

Suppose

Chapter 3

536

that the natural mappings: Cn

-r

all integers

C~

are epimorphisms, all integers

n.

Then there are induced short exact sequences:

~

0+ Ij,m l [H n - l (C"«) 1 ~

i>O

all integers Note:

+

Hn(C*)

+

i ~ 0,

Ij,m[Hn(C~) 1 + 0,

i>O

~

n.

The hypothesis in this Corollary "of A-modules", can be

replaced by:

"of objects and maps in the abelian category

A,

A is any abelian category such that denumerable direct

where

products exist and such that the denumerable direct product of epimorphisms is an epimorphism". Proof:

The proof is the same.

The hypotheses of Proposition 1 are obeyed.

have (1) a sequence

n E ;r ,

Hence we

of A-modules, and the exact se-

quences (2) and(3) of the conclusions of Proposition 1. hypothesis, for each integer epimorphisms, all integers all integers Ij,m

i>O

n, l

n, i :> O.

the mappings: Therefore

C

n

+

i>O

are

Cr: ~

liml[C~l ~

But by

=

0,

so that

C"« = O. ~

But then the long exact sequence (3) of Proposition 1 becomes a sequence of isomorphisms: (3' )

substituting these isomorphisms into the short exact sequences (2) gives the conclusions of Corollary 1.1. Remark:

In the short exact sequence, for each integer

the conclusion of Corollary 1.1, the epimorphism:

n,

in

Inverse Limit of Cochain Complexes Hn(C*) -+ lim[Hn(C~)]

ito

537

is induced by the natural mappings:

1

Hn(C*) -+ Hn (C~) ,

all integers

1

i > 0

Hn(natural map:

means, explicitly,

(where "natural mapping" C*-+C!».

This observa-

tion follows easily by tracing the construction. Corollary 1.2.

Let

C*

be a cochain complex, indexed by all

the integers, of left A-modules where be a left ideal in

A

A

is a ring, and let

such that the left A-module

cally complete, all integers

n.

n C

I

is I-adi-

Then there are induced short

exact sequences O-+lj,ml[Hn-l(C*/IiC*)]-+Hn(C*) -+~!m[Hn(C*/Iic*)]-+o,

i>O

1>0

all integers Proof: n C

n.

C~=C*/Iic*, 1

Define

all integers

i > O.

Then since

is I-adically complete as left A-module, all integers

n,

we have a canonical isomorphism of cochain complexes of left A-modules hold.

C* "" lim

i>O

C~;

1

and the rypotheses of Corollary 1.1

Therefore-we have the conclusion of Corollary 1.1, which

implies the conclusion of Corollary 1.2.

Q.E.D.

Remark:

Another special case of Corollary 1.1 is as follows.

Let

be a

C*

(~-indexed)

abelian category to Proposition 1. cochain complex

A, Let C*.

cochain complex of objects in the

where

A obeys the hypotheses of the Note

t*:C* -+C*

be any endomorphism of the

Then we obtain short exact sequences:

0-+ liml [H n - l (C*/ (t*) i c *) 1 -+ Hn [(C*)At*] i.::O n -+ lim [H (C*/ (t*) i c *)] -+ 0,

i.::O where

"C*/(t*)ic *" = "Cokernel of the i'th interate of

t*",

538

Chapter 3

"C*l\t*" = "lim c*/(t*i)c*". (Proof: Let c~=c*/(t*)ic*. 1 i>O n Then the natural map: C .... C~ is an epimorphism, therefore so and

1

is the natural map:

(Cn)l\t* .... C~ 1

Therefore Corollary 1.1 applies).

all integers

i, n,

i> O.

CHAPTER 4 COHOMOLOGY OF COCHAIN COMPLEXES OF t-ADICALLY COMPLETE LEFT A-MODULES

In this chapter, we discuss some consequences of the exact sequences (I.8) of (P.P.WCJ,

(i.e. of Theorems 1 and 2 of Chapter

2), and of Corollary 1.1 of Chapter 3. Theorem 1.

Let

element of

A.

A Let

be a ring with identity and let C*

t) :C n

(multiplication by tegers

be an

be a cochain complex (indexed by all

the integers) of left A-modules, such that n (1) C is t-adically complete, all integers (2)

t

-+

Cn

n,

and

is injective, all in-

n.

Then (1)

For every integer

n,

Hn(C*)/(t-divisible elements)

is t-adically complete. (2)

For every integer

n,

Also, Hn(C*)

has no

non-zero in-

finitely t-divisible, t-torsion elements. every integer

n,

we have the canonical isomorphism

of abelian groups (or of A-modules if center of (3)

Also, for

t

is in the

A):

Hn(C*)/(t-divisible elements) ~ Hn(c*)At~ lim(Hn(C*/tic*»). Also, for every integer n, there i>O are induced canonical isomorphisms of abelian groups

(4)

(or of A-modules, if

t

(t-divisible part of

n H (C*) ) ~ 539

is in the center of

A):

Chapter 4

540

~ liml[precise ti-torsion in itO

Hn(C*)]

~ limlHn-l (c*/tic*). i>O Remark:

We will see later (Theorem l' below), that under the

hypotheses of Theorem 1, a stronger conclusion than (2) holds; namely, (2')

For every integer

n,

Hn(C*)

has no non-zero in-

finitely t-divisible elements. An amusing, and useful, corollary of Theorem 1 is Lemma 1.1.1.

Let

A

element of the ring MAt

be a ring with identity and let A.

Let

be the completion of

Let

?l. [T]

-+

A

identity that sends

M

for the t-adic topology.

~t

replacing

"A" by

If

M

has no non-zero t-torsion.

be the unique homomorphism of rings with T

ring in one variable

be an

M be a left A-module and let

has no non-zero t-torsion, then Proof.

t

into T

t,

where

over the ring and

"?l.[T]"

"T"

prove the Lemma in the case that

by

?l.[T]

is the polynomial

of integers.

?l.

"tn,

A =?l. [T],

t

Then

it suffices to

= T.

We assume

this for the rest of the proof. Let

P*

be a free resolution of the

?l. [T]-module

M.

Then define

. IP

C~ =

i

. ,

-~

i > 0.

0,

Then c

n

C*

2. 0,

is a cochain complex (that is non-positive) such that

is a free

(1)

?l. [T]-module, all integers

H" (C<) - /

M,

n = 0,

0,

n

i

0.

n,

and such that

Cohomology of Cochain Complexes Then

c

M

(=

c

M

T),

541

being the T-adic completion of a free n E J7..

J7. [Tj-module, is with no non-zero T-torsion, all integers

Therefore the cochain complex of the hypotheses of Theorem 1. of Theorem 1,

z[Tj-modules:

limlH-l(C*AT/Tic*AT).

ito

A

obeys all

In particular, by conclusion (4)

(T-divisible part of

(2)

C*

But (3)

HO(C*AT)) ~

(C*AT /TiC*AT) ~ C*/TiC*.

A por-

tlon of the long exact sequence of cohomology of the short exact sequence of cochain complexes of i . 0 ... C*~ C* ... C*/T~C* ...

z[Tj-modules

°

is the exact sequence: i

(4)

••• ..!.....;.H- l (C*) ... H- l

But by equation (1), Ti: HO (C*) ... HO (C*)

.

i

-1

(C*/T~C* )~HO (C*~HO (C*)-

HO(C*) =M,

so that the endomorphism

is injective, since by hypothesis

non-zero T-torsion; and also by equation (1),

H-

l

M

has no

(C*) = 0.

Therefore from equation (4) we deduce that (4')

H- l (C*/TiC*) = 0,

all integers

i > 0.

substituting this into equations (2) and (3) yields (5)

(T-divisible part of

HO (C*AT)) = 0.

Equation (5) and conclusion (3) of Theorem 1 (for the cochain complex

(C*)AT

and the integer

n = 0)

imply that

substituting equation (3) into this latter result yields (6)

HO (C*AT)

~ lim HO (C*/TiC*). i>O

542

Chapter 4

But, from the cohomology sequence of the short exact sequence of cochain complexes:

°

i -+

.

c*....!...- C*

-+

C* ITlC*

-+

0

and from equation (1) and the fact that

M

has no non-zero T-

torsion we deduce that i

(7)

. MIT M, n H (C* ITlC*) = 0,

j

all integers

n,i

with

i > 0.

n=O n

~

0,

Substituting equation (7) into

equation (6) yields (8 )

A portion of the cohomology sequence of the short exact sequence of cochain complexes:

is the exact sequence (9)

~

But equation (3), and equation (4') with

i=l,

Substituting into equation (9), it follows that no non-zero T-torsion. non-zero T-torsion. Corollary 1.1. i ) r>O En(t r I _

'

Therefore by equation (8)

imply that

HO(C*AT)

~T

Q.E.D.

Under the hypotheses of Theorem 1, let

i) En(t 00

...

be the generalized Bockstein spec-

has

has no

Cohomology of Cochain Complexes

543

tral sequence, as defined in Chapter 1, corresponding to the cochain complex t

i

":

C*

->-

C*.

C*

and the endomorphism "multiplication by

Then for every integer

n,

there is induced a

canonical isomorphism of abelian groups (or of A-modules if is in the center of

(1)

t

A):

Hn(C*)/(topological t-torsion)~ lim E~(ti),

i>O and an isomorphism: (2)

that is t-divisible in

Hn+l(C*)}).

(In equation (1) of Corollary 1.1, the phrase "topological ttorsion" means: left A-module Note:

Hn(C*)

of the subgroup

{t-torsion elements}".)

Theorem 1 and Corollary 1.1 both generalize to abelian

categories. "Let

"the closure for the t-adic topology of the

Namely, replace the first sentence of Theorem 1 by

A be an abelian category such that denumerable direct

products of objects exist, and such that the denumerable direct product of epimorphisms is an epimorphism (i.e., and such that the functor "denumerable direct product" preserves epimorphisms). And, in the second sentence of Theorem 1, replace the phrase "of left A-modules" by "of objects of endomorphism of the cochain complex

A,

and let C*".

t* :C*

->-

C*

be an

The rest of Theorem 1

and Corollary 1.1 then remain the same, with

"to

changing to

"t*", and with the obvious understanding (as discussed in Chapter 2) of what "t*-adically complete", or "t*-divisible" means. (E.g. ,

"Cn

is t*-adically complete" means that the natural map

544

Chapter 4

Cn-+lim [Coker«t*)i:cn-+cn)] i"t"O Proof of Theorem 1.

is an isomorphism).

Since (multiplication by

injective, all integers

n n C -+ C

t):

is

n, the hypotheses for the exact sequence

(1.8) of [ERWCJ, i.e., of Theorem 1 of Chapter 2 hold, and therefore the conclusions of Theorem 1 of Chapter 2, hold: For each integer (1)

n,

we have a short exact sequence

S; lim [H n (C * / tiC *) ] -+ i"t"O n lim (precise ti-torsion in H + l (C*) ) -+ 0,

o -+ Hn (C * ) At i>O

and for each integer

n

we have an isomorphism

(2)

cn

On the other hand, since by hypothesis plete, all integers integer (3)

n

n,

is t-adically com-

by Corollary 1.1 of Chapter 3 for each

we have the short exact sequence

O-+lj:ml[Hn-l(c*/tic*)] .... Hn(C*) f}li m [Hn(C*/tic*)] .... O. i>O i>O

As we have observed in Chapter 2, respectively: monomorphism

a

(respectively:

short exact sequence (1)

the epimorphism

(respectively:

Chapter 3, the

S)

in the

in the short exact se-

quence (3)) is induced by the natural mappings:

But since the group natural map: n H (C*)

~ lim [H n (C* /tic*) ] i>O

can be factored:

is t-adically complete, the

545

Cohomology of Cochain Complexes

Hn(C*) .... Hn(C*)At ~ Ij,m[Hn(c*/tic*) J.

(4)

i>O

8

Since

is an epimorphism (by exactness of the sequence (3»,

it follows from the factorization

(4) that

But from the exact sequence (1),

a

fore the natural map

a

a

is an epimorphism.

is a monomorphism.

There-

is an isomorphism:

( 5)

substituting the isomorphism (5) into the short exact sequence n l lim (precise ti-torsion in H + (C*» = 0, or i~O equivalently, that- Hn+l(C*) has no non-zero infinitely t-divi-

(1) we see that

sible, t-torsion.

n

being an arbitrary integer, this proves

conclusion (2) of the Theorem. Equation (3)

implies that the composite map

8

of the maps

in (4), is an epimorphism, and equation (5) that the map (4)

is an isomorphism.

a

in

Therefore, it follows that the first

mapping in (4), the natural mapping:

It is equivalent to say that Hn(C*)/(t-divisible elements) is t-adically complete. Theorem.

That proves conclusion (1) of the

Next, note that conclusion (1) of the Theorem and

equation (5) above imply conclusion (3) of the Theorem. The short exact sequence (3) the map that

(in which the epimorphism is

e), the factorization (4) and the fact, equation (5), a is an isomorphism imply that the kernel of the natural

Chapter 4

546 mapping: y:Hn{C*) -+Hn{C*)At

is

canonically isomorphic to

liml[Hn-l{C*/tiC*)] . i>O But clearly the kernel of

y

(the natural mapping into the

completion) is {t-divisible elements in

Hn{C*)}.

Thus, we

have established the canonical isomorphism (6)

Hn (C*) ) :';' l~ml [H n - l (C* /tic*) ] , i> 0

(t-divisible part of

all integers

n.

Equation (2), with n-l replacing

n,

and equation (6) above

imply the conclusion (4) of Theorem 1. Proof of Corollary 1.1.

Q.E.D.

By Corollary 1.2 in the terminal Re-

marks (Remark 8) of Chapter 2, for each integer

n

we have the

short exact sequence (1)

0-+ Hn(c*)At/{topological t-torsion) -+llm[En{t i )]-+ i>O n l lim{precise ti-torsion in H + (C*» -+ 0,

i>O and for each integer (2)

n,

we have a canonical isomorphism:

lim En {til ~ liml ({precise ti-torsion in i>O ito Hn+l(C*)

that is t-divisible in

Hn+l{C*)}).

Equation (2) is conclusion (2) of the Corollary. (2)of the Theorem,

Hn+l{C*)

By conclusion

has no infinitely t-divisible,

t-torsion elements - equivalently, lim (precise ti-torsion in i>O

Hn + l (C*»

=

o.

Cohomology of Cochain Complexes

547

substituting this last equation into the short exact sequence (1) gives conclusion (1) of this Corollary. Proposition 2.

Q.E.D.

Under the hypotheses of Theorem 1, let

any fixed integer.

n

be

Then the following conditions are equiva-

lent:

(1)

Hn(C*)

is t-adically complete.

(2)

Hn(C*)

has no non-zero t-divisible elements.

(3)

The natural mapping:

is an isomorphism of abelian groups, ter of (4)

A,

(a)

(b)

t

is in the cen-

of left A-modules). If u

and

(or, if

ufHn(C*),

t·u=O,

and

u

ist-divisible,then

is infinitely t-divisible. n u f H (C*)

If

and

t

i

u'l

0,

all integers

i,

then

u

is not t-divisible.

(5) (6)

, i , , , 1( I 1m preC1se t -tors10n 1n

ito

limlHn-l (C*/tiC*) = O. i>O

Corollary 2.1.

Under the hypotheses of Theorem 1, let

n

be an

integer such that the six equivalent conditions of Proposition 2 above hold.

Then, the notations being as in Corollary 1.1, we

have that (1)

lir'llE~(ti) = O. i>O

Proof of Proposition 2.

(1)1=* (2)

follows from conclusion (1) of

Chapter 4

548

Theorem 1.

Conclusion (2) of Theorem 1 is that

Hn(C*)

non-zero infinitely t-divisible t-torsion elements.

has no

Therefore

condition (4a) of this Proposition is equivalent to saying that Hn(C*)

has no non-zero t-divisible, precise t-torsion elements,

which is in turn equivalent to asserting that

Hn(C*)

has no

non-zero, t-divisible elements that are t-torsion elements. condition (4b) says that that are not

Hn(C*)

But

has no t-divisible elements

t-torsion elements.

Therefore (because of conclu-

sion (2) of Theorem 1), condition

(4)<~

condition (2).

The iso-

morphism of the first and third groups in conclusion (3) of Theorem 1, implies that condition (2) of this Proposition is equivalent to condition (3) of this Proposition. (l~ (2)~3~(4).

Therefore

Finally, conclusion (4) of Theorem 1 implies

the equivalence of conditions (2),

(5) and (6) of this Proposi-

tion.

Q.E.D.

Proof of Corollary 2.1.

Consider (for the fixed integer

n) the

short exact sequence of inverse systems, indexed by the nonnegative integers, of abelian groups: (1)

0 ->- ({precise ti-torsion in

Hn(C*)

that is t-divisible in

Hn(C*)})i>O ->- ({precise ti-torsion in ->-

Hn(C*)})i~O

({preCiSe t~-torsion in Hn(C*)}) {precise tl-torsion in Hn(C*) that is t-divisible in Hn(C*)} i>O

->- 0, (where the maps, e.g. {precise ti+l-torsion in {precise ti-torsion in

Hn(C*)}

Hn(C*)}->-

in the middle inverse system,

Cohomology of Cochain Complexes is induced by "multiplication by Let

(ui)i>O

549

t", all integers

~

i

0).

be an element of the inverse limit of the

third of the inverse systems in the short exact sequence of inverse systems (1). represent

u ' i

we have that

Let

u. E {precise ti-torsion in 1.

all integers

Then for all integers

i, j

~

0,

j

• u.+. -u.) E{t-divisible elements in Hn+l(C*)}. 1. J 1. j In particular, we can write t • u . . - u . = tjv. for some ele1.+J '1. J

n l v. E H + (C*).

ment u. E t 1.

(t

i.

Hn(C*)}

j

J • Hn(C*),

sible in

But then

all integers

Hn(C*).

But then (Ui)i>O

u. = t j • (u. +' - v . ), so tha t 1.

U = 0, i

J

1.J

j ~ O.

Therefore

u.

1.

all integers

is t-divii> O.

There-

being an arbitrary element of the

inverse limit of the third inverse system in the short exact sequence (1), it follows that the inverse limit of that inverse system is zero:

(2)

lim

{precise

itO

{precise ti-torsion in that is t-divisible in

Therefore, throwing the short exact sequence (1) through the system of derived functors

lj,m, lj,ml, i>O i>O

a portion of the re-

suIting exact sequence of six terms is the short exact sequence

(3)

1 Grecise ti-torsion in~ I (precise Hn (C*) that is t-divi- ->- lj,m sion in i~O ible in Hn(C*) i~O

o ->-lj,m

->-

liml~precise i>O -

)-+

n ti-torsion in H (C*) } {precise ti-torsion in HnAC*) that is t-divisible in H (C*)}

O.

But by condition (5) of Proposition 1 we know that the middle term in the short exact sequence (3) is zero.

Therefore so is

Chapter 4

550

the first term in this sequence.

But this is the second group

in the isomorphism in conclusion (2) of Corollary 1.1.

There-

fore the first group in conclusion (2) of Corollary 1.1 is zero, i.e., we have verified conclusion (1) of Corollary 2.1. Remarks:

1.

The full strength of the hypotheses of Theorem 1

were not used in the proof of Corollary 1.2 above.

Thus, the

proof of Corollary 1.2 above can be used to show that: Proposition 3.

("Corollary

1.2.1, Chapter

2." )

Under the

hypotheses of Corollary 1.2 in the terminal Remarks (Remark 8) of Chapter

2, for any fixed integer

n, if

then

l!mlE~(ti)

=

o.

i>O Proof:

We prove the assertion for the integer

n - 1

instead of

n. The proof of Corollary 2.1 of this chapter establishes the short exact sequence (3) of the proof of Corollary 2.1 above, under the hypotheses of Corollary 1.2 of Chapter the isomorphism (2) of Theorem 1 of Chapter phism (2) of Corollary 1.2 of Chapter of "Corollary 1.2.1, Chapter

2.

Then use

2, and the isomor-

2, to complete the proof

2", for the fixed integer

n-l.

(This observation could have, and possibly should have, been included in a Remark to Corollary 1.2 of Chapter "Corollary 1.2.1, Chapter

2.

Thus

2" generalizes to all of the many

situations (including abelian categories, restricted cochain complexes in abelian categories, etc.) under which Corollary 1.2

551

Cohomology of Cochain Complexes of Chapter Chapter

2 holds, as observed in the terminal Remarks of

2.

Also, the short exact sequence (3) of the proof of

Corollary 2.1 above also holds in these many cases}. Remark 2.

Let

A

be a ring with identity and let

M be a left A-module.

Then for each integer

t

i >0

E;

A.

Let

we have the

short exact sequence: 0-+

{precise ti-torsion in

is the inclusion and

where tion by

tilt.

For each integer

Pj. M T~i

M} -+

i

t M+O,

T T

is induced by "multiplica(1 i + 1) maps

i ;: 0, the sequence

where h + l , k i + l , E: + l ' i i tit, h + is the restriction of i l £i+l is the inclusion. Therefore

into the sequence (li)' by mappings

k + l is "multiplication by i "multiplication by t", and

we have the short exact sequence of inverse systems, indexed by the non-negative integers, of abelian groups (or of left A-modules, if (1)

t

is in the center of

A):

0 + ({precise ti-torsion in M}) i>O + ( . . .

(tiM) i>O

$

M

t

M

t

M)

-+

O.

-+

Throwing through the exact connected sequence of functors (in fact, system of derived fUnctors) lim, 11ml,

i>O

gives a short exact

i>O

sequence of six terms from which we deduce the following Theorem 4. A-module.

Let

A

be a ring, let

Then there is

tEA

and let

(ll

be a left

induced an exact sequence of five terms

of abelian groups (of or left A-modules, if of

M

t

is in the center

A): 0

-+

{infinitely t-divisible elements in

elements in

M} -+ {t-divisible

M}-+[l1ml(precise ti-torsion in

i>O

M)]-+

Chapter 4

552

[lj,m i>O

l

(. ..

(The term Proof:

.t

M

.t

M

"rI'IM"

.t

M) 1

-+

M" 1M -+ O.

is shorthand for

"rI'l (image of

M)".)

Consider the exact sequence of six terms alluded to in

Remark 2 above. lJ,m(tiM) i>O

= n

The third term in this sequence is

tiM

= it-divisible

elements in

M},

that is, the

i>O

the second term in the sequence (1) claimed in the Theorem. second term in the exact sequence of six lim( ... ~ M i>O

.t

M

.t

lim(tiM)

group in

terms is

M), a group that is zero iff

finitely t-divisible elements. is

The

M

has no in-

Clearly, the image of this

{infinitely t-divisible elements}. There-

itO fore,

(if we start with the image of the second term in the

third, of the exact sequence of six terms, then) the exact sequence of six terms determines an exact sequence of five terms, the first four terms of which are the first four terms of the sequence (1).

To co~plete the proof of the Theorem, it is neces-

sary and sufficient to establish a canonical isomorphism:

But in fact, this latter follows from the following elementary general observation, proved in Intro. Chap. 2, sec. 6, Cor. 7.1. Lemma:

If

(Ni,aij)ij~O

is any inverse system of abelian groups

indexed by the non-negative integers such that morphism, all integers

i,j

with

N

~ij

is a mono-

then there is in-

duced an isomorphism ompletion of

C

NO

by the subgroups

N.R<

1.

for the topology given\

N., . 1. (1.mage of

i >0 NO)

)

Cohomology of Cochain Complexes Proof:

553

Consider the short exact sequence of inverse systems in-

dexed by the non-negative integers: 0 ... (N, , a., ,), ~

where

NO

~J

, 0'" NO'" (NO/N, ), 0'" 0,

~,J~

~

~~

denotes the constant inverse system

through the exact connected sequence of functors

NO.

Throwing

' l'tm, 1 l l.m, i>O i>O

and considering the second, third and fourth terms in the resulting exact sequence of six terms, and the fact that the fifth term is Remark 1.

limlN = 0, it"O o

we obtain the Lemma.

By Theorem 4, if

element and if

M

A

is any ring, if

tEA

is any

is any left A-module, then there is induced

a natural mapping of abelian groups (or of left A-modules if is in the center of (2)

t

A): M} ... [lim l (precise ti-torsion in M)] • it"O

{t-divisible elements in

Now suppose that the hypotheses of Theorem 1 hold, and let M=Hn(C*)

for some integer

n.

Then, by conclusion (3) of

Theorem 1, there is induced a natural isomorphism of abelian groups (or of left A-modules if (3)

{t-divisible elements in

t

is in the center of

A):

M}~ [11ml(precise ti-torsion in M)]. i>O

If we study the proofs of Theorems 1 and 4, we see readily that, under the hypotheses of Theorem 1, the homomorphisms (2) and (3) coincide.

In particular, under the hypotheses of Theorem 1, the

homomorphism (2) is an isomorphism.

Substituting into the

exact sequence (1) of Theorem 4, we obtain: Theorem 1'.

Under the hypotheses of Theorem 1, the following

554

Chapter 4

assertion (which is stronger than conclusion (2) of Theorem 1) holds: (2')

Hn(C*)

has no non-zero infinitely t-divisible ele-

ments. A corollary to Theorem l' is Proposition 2'. tion 2.

Suppose that the hypotheses are as in Propos i-

Then the following condition (similar to condition (4)

of Proposition 2) is equivalent to each of the six equivalent conditions (1), ... , (6) (4')

Every

in the statement of Proposition 2.

t-divisible element in

Hn(C*)

is infinitely

t-divisible. Proof:

By Theorem 1',

divisible elements. saying that

"Hn(C*)

Hn(C*)

has no non-zero infinitely t-

Therefore condition (4') is equivalent to has no non-zero t-divisible elements", and

this latter is condition (2) of Proposition 2. Remark 2.

It is easy to verify Theorem l' of Remark 1 indepen-

dently of Theorem 4 by a more direct computational method. By Theorem 1, under the hypotheses of Theorem 1 if any integer and if in

M)

M = Hn (C*) ,

then

M)"

is

liml (precise ti-torsion i>O

is the set of t-divisible elements in

"liml (precise ti-torsion in i>O

n

M.

The group

also appears in Theorem 1 of

of Chapter II -and, by conclusion (3) of Theorem 1 of this chapter, and conclusion (1) of Theorem 1 of Chapter group "liml (precise ti-torsion in i>O (More about it in a later paper).

M)"

2, this

becomes more interesting

Therefore, the following

Proposition is of interest, since in most practical examples it

Cohomology of Cochain Complexes

555

gives a useful formula for this group. Proposi tion 5. Let

M

ment in

Let

A

be a ring with identity and let

be any left A-module such that every M

M

has no non-zero t-divisible

plication by

t): M ... M

t-divisible ele-

(E.g., this is the case

is infinitely t-divisible.

if, either

tEA.

is injective).

element~

2£ if (multi-

Then there is induced a

canonical isomorphism of abelian groups (or of A-modules if is in the center of (1)

[lim

1

t

A)

(precise ti-torsion in

M) ]

i>O

[lim (precise t

i

.

-tors~on

in

""...

M"t1M) ] .

i>O

(Here of course

,,"tiM"

means

"M" t I (image of

(or

M)",

equivalently, the cokernel of the natural homomorphism of abelian groups: for

(In the next Corollary, we write

"/\ n

"At".)

Corollary 5.1.

Let

A

be a ring with identity and let

an element of the center of such that every

A.

Let

M

t

be

be a left A-module

t-divisible element is infinitely t-divisible.

Then the following conditions are equivalent: (1)

liml (precise ti-torsion in

M)

= O.

i>O (2)

M"/M

(3)

u

u

~

has no non-zero t-torsion. A

M ,

tu = 0,

implies there exists

is the image of

of abelian groups: Corollary 5.2.

Let

A

element of the center of

v

v EM

(under the natural homomorphism

M ... M") .

be a ring with identity, let A,

such that

and let

M

t

be an

be a left A-module

Chapter 4

556

such that every t-divisible element in sible.

M

is infinitely t-divi-

Then the group ltml(precise ti-torsion in i>O

M)

is t-adically complete, and has no non-zero t-torsion. (Note:

In the case that

t

is not in the center of

this group need not be an A-module. group, since the group is a sub

~-algebra

of

A

~[tl-module,

generated by

Proof of Proposition 5:

But

t

A,

still acts on the

where

~[tl

is the

t).

We have the short exact sequence of

abelian groups: 0 .... (infinitely t-divisible part of

(1)

M)

->-

M ....

M/ (inf ini tely t-di visible elements)

->-

O.

For each integer

i~O,

sequence of functors

throwing through the exact connected

"~ f\.l~ [Tl /Ti • ~

if. [Tl

[Tl,

Tor 1

i

(,71 [T] /T ''l/ [Tl )

(where each of the three abelian aroucs in the sequence (1)

is regarded as a

plication by

t") ,

is "modding out by

if. [Tl-module by letting

T

and using the facts that

"

till

and that

"To~[Tl( 1

act as "multi-

0 /Ti • if. [Tl " [Tl ,if. [Tl/ Ti • if. [Tl) " ~

is "taking the precise ti-torsion," yields the exact sequence (the first four terms of the indicated exact sequence of six terms) : (2)

0 ->- (precise ti-torsion in (infinitely t-divisible part of

M) 1 .... (precise ti-torsion in M')

->-

->-

(precise ti-torsion in

(infinitely t-divisible part of

t-divisible part of where

M)

M)/t

i

- (infinitely

M),

M' =M/(infinitely t-divisible part).

The fourth group is

Cohomology of Cochain Complexes

557

clearly zero; therefore (2) defines a short exact sequence. i 2.. 0,

short exact sequences (2), for

The

def ine a short exact se-

quence of inverse systems of abelian groups indexed by the nonnegative integers.

Throwing that short exact sequence through

the right exact functor

liml, and using the fact that i>O precise ti-torsion in (infinitely

[ t-divisible part of

= 0,

] M)

since the maps in the indicated inverse systems are epimorphisms, it follows that we have a canonical isomorphism of abelian groups (or of A-modules if

t

is in the center of

liml (precise ti-torsion in

M) ':;.

'+0 1>

where

A)

liml (precise ti-torsion in M'),

itO

M' = M/ (infinitely t-divisible elements).

But since

clearly also

to prove the Proposition for

M it suffices to prove it for

By the hypotheses of the Proposition,

(t-divisible part of M)

(infinitely t-divisible part of

Therefore,

non-zero t-divisible elements.

M).

M'

Mr.

has no

Therefore, to prove the Propos i-

tion, it suffices to prove the Proposition in the case that M has no non-zero t-divisible elements; we assume this for the rest of the proof.

Then

M

is a submodule of

Also, we can regard Z[T]

M as being a

~. Z[T]-module, where

is the polynomial ring in one variable over

quiring that

T'

x=

Z[T]

by

T

and

t

t • x,

all

x EM.

Z,

Then replacing

if necessary, we can assume that

by reA

by

t

is in

Chapter 4

558

the center of Case 1.

M

.i'/M = 0,

A,

= i',

and that Le.,

t

is a non-zero divisor in

is t-adically complete.

M

A.

Then clearly

so the conclusion of the Proposition is that ' 1 ( prec~se , l ~m ti i~O

(1' )

'

-tors~on

,

~n

M) = 0,

which is what we must prove. Choose

P,

a free left A-module, and

phism of left A-modules. under the functor (2)

1/\ :

F"

"/\"

= M,

the image of

1

is an epimorphism

/\

Then

K

is the kernel of an A-homomor-

of t-adically complete left A-modules, and is therefore

phism

pA

t-adically complete. and

rI-

an epimor-

+ M.

K = Ker (1 ).

Let

Then since

1:P +M,

t

is the completion of a free A-module,

is a non-zero divisor in

FA

has no non-zero t-torsion.

of

F/\, K

A; therefore by Lemma 1.1.1 Since

K

is a left sub A-module

also has no non-zero t-torsion.

Therefore, if we

define C l = p/\, cO = K,

i C =0, and define

it-O,l,

dO :C O + C

l

to be the inclusion:

K ~F",

then

C*

is a cochain complex of left A-modules, and the hypotheses of Theorem 1 are satisfied.

Therefore the conclusions of Theorem

1 hold for the cochain complex of left A-modules

particular conclusion (3) for the integer that

C*, and in

n = 1, which implies

Cohomology of Cochain Complexes , ' 1 ( prec~se ti-torsion in l ~m

(3' )

i>l fE-divisible elements in But clearly

Hl(C*)~M.

Hl(C*»

559

~

Hl(C*)}.

And by the assumptions of Case 1,

is t-adically complete.

Hence

{t-divisible elements in

Hl(C*)~M

This observation, the fact that

M M} =0.

as left A-module

and equation (3') imply equation (1'), which proves Case 1. Case 2. tEA

General case.

(I.e.,

A

is a ring with identity,

is a non-zero divisor in the center of

A,

and

M

is a

left A-module having no non-zero t-divisible elements (so that M

is a submodule of

M' = M't»

.

Then we have the short exact sequence of left A-modules

Let

i

be any non-negative integer.

Then if we throw the short

exact sequence (3) through the homological exact connected sequence of functors (in fact, sequence of left

derived functors)

"(A/tiA) €I", "Tor1 (A/tiA, )", noting that A

"(A/tiA) €I N"

~ "N/tiN"

and

i "Tor1(A/t A,N) "",,"precise ti-torsion

A

in

N"

canonically as functors in

N

(i.

e., these two isomor-

phisms are specific natural equivalences of functors), we obtain the exact sequence of six terms (4 ) i

0 .... (precise ti-torsion in

M)""

(precise ti-torsion in M') . . , i , , M') .... M/tiM"" M' /tiM' (prec~se t -tors~on ~n M

(A/tiA) €I (M' /M) .... O. A

Since the natural mapping:

M/tiM .... MA /tiMA

is an isomorphism,

Chapter 4

560

it follows from the exact sequence (4 ) that i (5)

(M" !M)

is a t-divisible module,

and that for each integer

i >0

we have the short exact se-

quence (4

l

~)

0 .... (precise ti-torsion in

M) .... (precise ti-torsion

in

MA ) ....

" ti-torslon " "In (preclse

For each integer

i.::: 0,

"multiplication by

mapping from the short exact sequence

t"

(4 + ) i l

induces a into the short

exact sequence (4 ), and there is therefore induced a short i exact sequence of inverse systems of left A-modules (indexed by the positive integers), (6)

0'" (precise ti-torsion in M) i>O'" (precise ti-torsion in

ri') i>O'"

(precise

M"

The left A-module

ti-tor~ion

in

(M" !M)) i>O'" O.

is t-adically complete, and in par-

ticular has no non-zero t-divisible elements, and therefore also no non-zero infinitely t-divisible t-torsion elements. Therefore (7)

lJ,m(precise ti-torsion in

hO Also, the left A-module

M"

M")

= O.

obeys the hypotheses of Case 1.

Therefore, by Case 1 (and by equation (I') of Case 1), (8)

" " . " 1 ( preclse t i -torslon In I ~m

M1\) = 0 •

i>O Throwing the short exact sequence (6) through the exact connected sequence of functors

"lim" ,

ito

yields an exact sequence

Cohomology of Cochain Complexes

561

of six terms, a portion of which is the exact sequence (9)

-+

l,im(precise ti-'torsion in r.f) -+ lim (precise ti_ i~OI\ dO 1 i!O.1 torsion in (M /M)~lim (precise t -torsion in 1 i i'::'°A M) -+ lim (precise t -torsion in M·) -+ ••• i>O

substituting equations (7) and (8) into equation (9), we see that the map

dO

(from the zero'th coboundary in the connected

sequence of functors) of equation (9) is an isomorphism of left A-modules: (10)

lim (precise ti-torsion in (M' /M))~ i'::'°liml(preCiSe ti-torsion in i>O

M),

which completes the proof of Case 2, and therefore also the proof of the Proposition. Proof of Corollary 5.1.

By equation (5) in the proof of Case

2 of Proposition 5, we have that i.e., every element of fore

M'/M

M'/M

M'/M

is a t-divisible module,

is infinitely t-divisible.

has no non-zero t-torsion iff

infinitely t-divisible, t-torsion.

M'/M

There-

has no non-zero,

This latter occurs (as for

any left A-module) iff

lim (precise ti-torsion in M' /M) = 0. i>O By the conclusion (1) of Proposition 5, this latter occurs iff

condition (1) holds.

Thus, condition (1) of Corollary 5.1 is

equivalent to condition (2) of Corollary 5.1. (2)~(3),

Since obviously

this proves Corollary 5.1.

Froof of Corollary 5.2.

If

N

is any left A-module, then it is

easy to see that the left A-module R = lim (precise ti-torsion in N) i>O

Chapter 4

562 has no non-zero t-torsion.

Also,

R,

being the inverse limit

of t-adically complete A-modules, is itself t-adically complete. Applying this observation to the A-module

N=

M'IM,

and using

the conclusion of Proposition 5, we obtain the conclusion of Corollary 5.2. Remark:

Do Theorem 4, the Lemma following Theorem 4, Propos i-

tion 5, Corollary 5.1 and Corollary 5.2 generalize to abelian categories?

The answer is "yes".

A be

More precisely, let

any abelian category such that denumerable direct products exist and such that the denumerable direct product of epimorphisms is an epimorphism (i.

e.,

and such that the functor "denumerable

direct product" preserves epimorphisms). Theorem 4 (generalized).

Let

t:M'" M be any endomorphism.

Then

M be an object in

A

and let

Then we have an exact sequence of

five terms (1) as in the conclusion (1) of Theorem 4.

(Where,

as usual, "(infinitely t-divisible part of M)", "(t-divisible part of M)", etc., are interpreted in the obvious way, as in Chapters 1 and 2). Lemma (generalized).

In the statement of the Lemma following

Theorem 4, replace the word "abelian groups" by "objects in and "subgroups" by subobjects". (Where of course "completion of the subgroups Theorem l' .

A",

Then the Lemma remains valid. NO

for the topology given by

N;, i> 0" is replaced by "lim (N IN.) .") ... itO 0 ~ (generalized}. Let C* be a (z-Tndexed) cochain

complex of objects and maps in the category be an endomorphism of the cochain complex

A C*.

the hypotheses of Theorem 1 (generalized) hold.

and let

t = t*

Suppose that Then conclusion

(2') of Theorem l' holds (where "infinitely t-divisible part" is defined as in Chapter 1).

Cohomology of Cochain Complexes Proposition 2'

(generalized).

563

The hypotheses being as in Propo-

sition 2 (generalized), then the six equivalent conditions of Proposition 2 (generalized) are all equivalent to condition (4') of Proposition 2'. proposition 5 (generalized). A

gory

and let

t:M .... M

divisible part of

M)

Let

M

be an object in the cate-

be an endomorphism such that the (t-

coincides with the (infinitely t-divisible

n 1m ti=Im (lim ( ...~ M~M~M) .... M». i>O i>O Then conclusion (1) of Proposition 5 holds-(where of course

part).

(I.e., such that

e.g. "(precise ti-torsion in

M)"=Ker (t i )", etc.).

Corollary 5.2 (generalized).

Under the hypotheses of Proposi-

tion 5 (generalized), we have that the object (1)

liml (precise ti-torsion in

M)

PO in the category

A is complete for the endomorphism induced by

t, and also that the endomorphism induced by

t

of the object

(1) is a monomorphism. However, the proof of Corollary 5.1 appears to require a stronger axiom on the abelian category (Intro., Chap. 1, sec. 7). Corollary 5.1 (generalized).

A

Suppose that the abelian category

has, in addition to the previously stated property, the

property that, whenever objects and maps in

(Ai,aij)i,j~O

is an inverse system of

A indexed by the non-negative integers

then a + ,i is an epimorphism, all integers i ~ 0, i l the natural mapping: (lim Ai) .... A is an epimorphism (it is eO i~O quivalent to say that, for every such (A. , a· .). . 0' we have 1 1J 1,J;: such that

that

liml Ai = 0) • i>O

(Otherwise stated:

In stating the general i-

zations of Theorem 4, ... , Corollary 5.2, our assumption on the

Chapter 4

564

A was "(P.l)" of [E.MJ; but for Corollary 5.1

abelian category

we also need the stronger axiom" (P.2)" of [E.M.l.

(To the best

of my knowledge, no one knows whether or not there exists an

A that obeys (P.l) but not (P.2».

abelian category

Then the

three conditions stated in Corollary 5.1 are equivalent; where "precise ti-torsion" means "Kernel of

till, "has no non-zero t-

torsion" means "t induces a monomorphism", and condition (3) becomes: (3)

Let

¢:M

-..r1'

be the natural map.

Then the (precise t-torsion in in the image of

r1')

is contained

¢.

(The reason for the stronger hypotheses on the abelian

A in the generalization of Corollary 5.1 is that in

category

proving that (1)

~(2),

we need to know that the mapping:

(l~m (precise ti-torsion in

r1' 1M»

.... (precise t-torsion in

r1' 1M)

i~O

is an epimorphism, and here the maps in the inverse system in question are epimorphisms; so that we require exactly the stronger hypotheses (P.2) on the abelian category that (2)

~(l)

remains valid if only (P.l) holds, and

requires no hypothesis (not even (P.O» A.

(Where

M,

A.

r1'

abel ian ca tegory

are any objects and A) - use

The proof (2)~(3)

on the abelian category M -..

r1'

is any map in any

the exact imbedding theorem [I. A. C . 1) •

(However, if we modify condition (2) of Corollary 5.1 to read.

"(2')

r/'/M

has no non-zero t--torsion that is infinitely

t-divisible", then the resulting generalization to abelian categories remains valid for abelian categories obeying the milder

Cohomology of Cochain Complexes assumption (P.l).)

A.

categories

tha t

1T i

Also, Lemma 1.1.1 generalizes to abelian

(One must replace

the form

565

"?

[tl (I)/\t"

by objects of

with endomorphism the "shift map"

+l o t =

1T i

projection, all

i.::.O,

'

ITO,ot=O,

i.:::. 0, and where

where

AE A

1T. 1.

t

such

is the i'th

is any object).

Does Theorem 1 remain valid if one deletes the hypothesis til :C n ...,. C

(2), that "multiplication by gers

n?

The answer is "yes".

n

is injective, all inte-

At least, if this hypothesis is

deleted, then most of the conclusions of Theorem 1 continue to hold.

The same is true for Theorem l' of Remark 1 following the

proof of Theorem 4. Theorem 6. C*

be a

Let

That is,

A

be a ring with identity, let

(~-indexed)

tEA

and let

cochain complex of left A-modules such

that (1)

n C

is t-adically complete, all integers

n.

Then

most of the conclusions (1)-(4) of Theorem 1, and also conclusion (2') of Theorem l'

(of Remark 1 following the proof of Theorem

4) continue to hold; that is, we have that: (2)

for every integer

n,

Hn(C*)/(t-divisible elements)

is t-adically complete. (2') For every integer

n,

Hn(C*)

has no non-zero infi-

nitely t-divisible elements. (3)

For every integer

n,

there are induced natural iso-

morphisms of abelian groups (or of left A-modules if t

is in the center of

A)

n

n

H (C*) / (t-divisible elements", H (C*) (3') For every integer group of equation

n,

/\t

.

we have an epimorphism from the

(3) onto

Chapter 4

566

lim [Hn(C*/tic*)].

i>O (4)

Also, for every integer

n,

there are induced natural

isomorphisms of abelian groups (or of left A-modules if

t

is in the center of

(t-divisible part of

A):

Hn(C*))

~ liml[precise ti-torsion in

Hn(C*)].

i>O (5)

Also, for every integer

n,

the subgroup of

Hn(C*)

described in conclusion (4) is contained in the subof

group

Notes 1. that

n H (C*).

Notice that, by Corollary 1.2 of Chapter

(lim l Hn-l(C/tic*)]

3, we have

is naturally identified with a sub-

ito

group of- Hn(C*), as alluded to in equation (5) above. by Corollary 1.2 of Chapter

all integers

n > O.

Also

3, we have the short exact sequence

The epimorphism of equation (3') is induced

by the epimorphism of this short exact sequence. 2.

Let

n

be an arbitrary fixed integer.

(t-adically complete, left A-submodule)

Then if the

Im(dn:C n ... c n + l )

has no

non-zero t-torsion elements, then the proof of Theorem 6 below shows that, for that integer is an isomorphism.

n,

the epimorphism of equation (3')

I.e., that for that integer

subgroups (the one of conclusion (4) and

n,

the two

liml Hn-l(C*/tic*))

i>O described in conclusion (5),coincide. Remark.

However, in the context of Theorem 6, we state no ana-

logue of Corollary 1.1.

(Since Bockstein spectral sequences

567

Cohomology of Cochain Complexes

don't make sense unless hypothesis (2) of Theorem 1 holds - although, by using "percohomology" as we shall define it in Chapter 5, a Bockstein spectral sequence can be defined, and an analogue of Corollary 1.1 can be stated under the (weaker) hypotheses of Theorem 6.

This analogue of Corollary 1.1 is actually

the corollary to another (different - and more shallow) generalization of Theorem 1 that holds under the same hypotheses as Theorem 6,

(if also the element

tEA

is in addition assumed to be

a noa-zero divisor), namely, to Theorem 6' of Remark 4 below. Note 3.

Also, as in the Note to Theorem 1,

(and also Theorem

1', see the preceding Remark), Theorem 6 above generalizes to abelian categories s.t. den. dir. products exist and are exact. That is, let

A be any abelian category such that denumer-

able direct products exist and such that the direct product of denumerably many epimorphisms is an epimorphism (i.e., and such that the functor "denumerable direct product" preserves epimorphisms).

Let

C*

be an arbitrary cochain complex, indexed by

all the integers, in th,:! abelian category

A,

be an endomorphism of the cochain complex

C*.

n C

is complete for the endomorphism induced by

and let

t = t*

Suppose that (1) t,

all integers

n. Then the six conclusions,

(1), (2'), (3), (3'), (4) and (5) of

Theorem 6 all hold. (Where, terms like "complete with respect to an endomorphism", fIt-divisible part", "completion with respect to an endomorphism", etc., are defined as in Chapter 1.)

(The proof of

this Note is exactly the same as that of the Theorem.) Note 4.

Theorem 6 is an improvement of both Theorem 1 and

568

Chapter 4

Theorem 1'.

These latter were used to prove Proposition 2 and

Proposition 2'.

Theorem 6 implies the following partial improve-

ment of Proposition 2 and Proposition 2'. Proposition 2".

Under the hypotheses of Theorem 6, if

any fixed integer, then the conditions (1),

(2),

n

is

(4), and (5) of

Proposition 2, and also the condition (4') of Proposition 2', are all equivalent to each other.

Also, in this case, conditions

(3) and (6) of Proposition 2 are equivalent to each other, and imply the other conditions of Proposition 2. Proof of Theorem 6: of

A

Replacing the ring

generated by the element

t

A

by the subring

if necessary, one immediately

reduces the proof to the case in which the element center of the ring since hypothesis

A.

~[t]

t

is in the

Then by Corollary 1.2 of Chapter

(1) of Theorem 6 holds, for each integer

3, n

we

have the short exact sequence: (5)

o+liml [Hn-l(C*/tiC*)]+Hn(C*)+ i>O lim [Hn(C*/ti(C*)] +0. i>O

Clearly, from equation (5), it follows that conclusion (3') of Theorem 6 follows from conclusion (5) of Theorem 6. Fix an integer plex of

C*

n E;r,

and let

D*

be the subcochain com-

such that

I

ci , n n :er (d : C

i~n

+

C

n+l

),

- 1,

i = n,

i>n+l.

Then the natural map induces an isomorphism: Therefore to establish conclusions (1),

Hn(D*) ~Hn(C*).

(2') and (3) for the co-

Cohomology of Cochain Complexes chain complex for

D*.

C*,

it suffices to establish these conclusions

Notice that

D*

obeys all the hypotheses of the Theon l D + = O. By Note 2 to the Theorem,

rem, and in addition that for such a Hn(D*)

D*,

569

it is asserted that the two subgroups of

described in conclusion (5) coincide.

But, in the short

exact sequence (5' )

established in Corollary 1.2 of Chapter

3, the third group is

a t-adically complete left A-module (being an inverse limit of t-adically complete left A-modules - Hn(D*/tiD*) complete since it is annihilated by

ti,

is t-adically

all integers

i~O).

Therefore, both conclusions (1) and (3) of Theorem 6 for (and therefore also for

C*)

D*

would follow if we knew that, in

the exact sequence (5'), the first group is the t-divisible part of the second group, all

integer~

n.

But this would follow

from conclusion (5) of Theorem 6 as modified by Note 2.

There-

fore, to prove Theorem 6, it suffices to prove both conclusion (2') and conclusion (4) and (5) of Theorem 6 and Note 2 to Theorem 6. Equations (2'), (4) and (5), and Note 2, are together equivalent to: (6)

both For each integer

n,

Hn(C*)

has no non-zero infinitely

t-divisible elements, and the t-divisible part of Hn(C*) ~ liml(precise ti-torsion in i>O and

(7)

For each integer ' 1 ( prec~se , l ~m ti i~O

Hn(C*»,

we have a canonical monomorphism:

n, '

-tors~on

,

~n

570

Chapter 4

Ij,mlHn-l(C*/tiC*). i>O

If

r=rm(dnc*:Cn+cn+l)

is

without non-zero t-torsion, then this monomorphism is an isomorphism. We prove equations (6) and (7) separately. First, every A-module where

T· x

= t • x,

all

N

can be regarded as a

x E'N.

Therefore replacing

if necessary, we can assume that

t E center of

~[T)-module,

A by

A,

~

[T)

and that

t

is a non-zero divisor. Part 1.

Proof of equation (6):

Fix an integer n, and let M = Hn (C*). Let Dl = Ker (dn:C n + Cn + l ) . Then, since by hypothesis (1) of the Theorem n n C and c + l are t-adically complete, we have that Dl is t-adically complete.

Define

the A-homomorphism induced by

DO = Cn - l ,

and let d: DO + Dl be d~:l:cn-l + Cn Then M = Coker (d),

so that we have an exact sequence of left A-modules: d 1 Do +D +M+O,

(8)

where

DO Let

and

Dl

are t-adically complete.

F be a free left A-module and let

morphism of left A-modules. plete, functor,

Dl = (Dl),,'t,

Then since

Dl

TT:F + Dl

be an epi-

is t-adically com-

we have that the image of

TT

under the

"t-adic completion", is an epimorphism of (complete)

left A-modules

.

TT" . F"

+

and modules:

Dl Tf

. 1

1\

=Tf

•

Then we have a diagram of left A-

Cohomology of Cochain Complexes

with

TIl

an epimorphism.

And

0°,0

1

and

Fl

571

are all t-adi-

cally complete. I

Let

F O _ Fl D0 xl'

the fiber product of this diagram.

o

Then we have a commutative diagram:

(9)

Since

TI

1

is an epimorphism, so is

°

TI .

is the kernel of

a homomorphism of t-adically complete left A-modules - namely, FO = Ker y,

where

y :0° x Fl .... 0 1

is the homomorphism of left

A-modules y = TIl

0

TI 1 - do F

where

TI 1:0

0

011 XF .... F

and

° TI 0:0° x F

l

.... D°

are the projections.

o

F

Therefore

7f

'FO,

being the kernel of an A-homorphism of t-adi-

cally complete left A-modules, is likewise t-adically complete. Consider the commutative diagram of left A-modules, with an exact right column and with exact rows:

Chapter 4

572

°t

0

t

O-M~===M-O

t Fl

(10)

1 Tf)

t D~O

1 °i a

'F O

Tf

d

)DO~O

(where the rightmost column is the sequence (8), and where the 1

map

Fl .... M

is the composite:

F l .2!..,.D l .... M).

Since the commuta-

tive square (9) is a fiber product square, it follows by diagram chasing that the leftmost column in the commutative diagram (10) (Proof:

is exact.

F

1

being a composite of two epimor-

-+M,

phisms, is an epimorphism. the composite:

The composite:

°

'FO~DO~Dl""M,

of the right column of (10). zero in

M,

'FO .... Fl .... M equals

which is zero by exactness

Finally,

if

fl E Fl

maps into

then by exactness of the rightmost column (and

commutativity of the top square) in the diagram (10) we have that there exists

dOEDO

such that

Tfl(fl) =d(dO)

in

Dl.

But then, since the commutative square (9) is a fiber product square, there exists a unique element a('fO) =f l

and

TfO('fO) =dO.

'fOE 'FO

In particular,

such that a('fO) =f l .)

Therefore, we have constructed an exact sequence of left Amodules,

such that Fl

and

Fl

are t-adically complete, and such that

is the t-adic completion of a free left A-module.

element of

'FO

A,

tEA

is a non-zero divisor in

A

and is in the center

it follows by Lemma 1.1.1 that the A-module

non-zero t-torsion.

Since the

Fl

has no

Cohomology of Cochain Complexes Let

pO

be a free left A-module and let

epimorphism of left A-modules. complete, if we let

FO

=

FO

A

Then since

573

ljJ:p o . . 'FO

'FO

then the image of

be an

is t-adically ljJ

under the

functor

"t-adic completion" is an epimorphism of left A-modules:

Letting

d = 8

0

ljJ

A

and considering exactness of the sequence (II),

we obtain an exact sequence

°

F _dF 1 .... M .... O,

(13) where

FO

and

and therefore

Fl

are both completions of free left A-modules,

FO

and

Fl

are both t-adically complete, and Define

with no non-zero t-torsion by Lemma 1.1.1. i d F * = 0,

it-O,l,

it-O.

Then

F*

is a

F

i

=

0,

(non-nega ti ve)

cochain complex of left A-modules that obeys the hypotheses (1) and (2) of Theorem 1, and

Hl(F*) ~M.

conclusion (4), for the case (t-divisible part of

n

= I,

But then, by Theorem 1, if follows that

M) ~ liml (precise ti-torsion in itO

And by Theorem 1', conclusion (2'), applied to the case M

has no non-zero infinitely t-divisible elements.

M=Hn(C*), Part 2.

n = 1, Since

this completes the proof of equation (6).

Proof of equation (7).

Fix an integer Case 1.

M).

C

i

= 0,

n,

and let

all integers

dn-l:cn-l .... cn

n M = H (C*).

it-n-l,n,

and

is a monomorphism.

Then we have the short exact sequence of left A-modules (8)

o . . Cn-l

.... C

n .... M .... O.

Chapter 4

574 Let

i

be a non-negative integer.

Then throwing the short

exact sequence (8) through the homological, exact connected sequence of functors (in fact, system of left " (A/tiA)

~ :', "Tor1 (A/tiA, )"

derived functors)

yields an exact sequence of

A/tiA-modules of six terms, a portion of which is the exact sequence

(9)

••• -+-

(precise ti-torsion in

Cn)

(precise ti-torsion in

M)

+

~

(Cn-l/tiCn-l) + Cn /tiC n + ••• Since the kernel of the mapping Hn - l (c*/tic*)

(since

(Cn-l/tiCn-l)

Cn - 2 = 0),

+

(Cn/tiC n )

is

the exact sequence (9) implies

that we have an exact sequence (lOi)

••• -+

ti-torsion in

(precise

(precise ti-torsion in Hn - l (c*/tic*) where

a

cation by

is induced by t"

6.

+

Cn)

+

M) ~

0,

For each integer

induces a map from the sequence

i

~

0,

"multipli-

(lOi+l)

into

(lOi); we therefore obtain a similar exact sequence of inverse systems of left A-modules indexed by the non-negative integers. Throwing that sequence through the right exact functor

liml i>O

yields the exact sequence

(11)

[liml(precise ti-torsion in i!.°l [lim (precise ti-torsion in i>Ol . [IIm Hn - l (e*/t~e*)] + 0.

en)] + M)] +

i>O But since

en

is a t-adically complete left A-module,

(by the

Cohomology of Cochain Complexes hypotheses of this Theorem),

en

575

obeys the hypotheses of Propo-

sition 5, and therefore by Proposition 5 [lim

(12)

l

en)] ~

(precise ti-torsion in

ito

.

[lim (precise t~-torsion in

(eN'/e n »].

i>O n C

But since

eM /C

n

=

is t-adically complete,

M

C

=0

en,

therefore

and therefore the group on the right side of equa-

0,

tion (12) is zero. of equation (12).

Therefore so is the group on the left side Substituting into the exact sequence (11), we

see that the second A-homomorphism in the sequence (11) is an isomorphism (precise ti-torsion in

(13)

M)] ....

Hn-l(C*/tic*)], which proves equation (7). Case 2.

i C = 0, all integers

i

~ n - 1, n.

Then we have an exact

sequence of left A-modules (8)

cn- l

n-l

..2.-C n .... M .... O.

and

Then

Dn-l ,

being the image

of an A-homomorphism of t-adically complete left A-modules, is t-adically complete, and since complete.

Define

the inclusion.

D

Then

i

n n D = C ,

n D

is also t-adically

= 0,

D*

is a cochain complex of left A-modules

obeying the hypotheses of Case 1, and

M~H

n

(D*).

Therefore, by

Case 1, we have a specific isomorphism of left A-modules (9)

(precise ti-torsion in Hn-l(D*/tiD*)].

M)] ""

Chapter 4

576

Let

K

= (kernel of the natural epimorphism:

Ker(d n - l ) =Hn-l(C*),

and for each integer

(Kernel of the epimorphism:

C

i >0

n l n l - ->- D - )

let

K. = 1

(Cn-l/tiCn-l) ->- (Dn-l/tiDn-l)).

Then

we have a short exact sequence: O+K->-C n-l +D n-l +0, and therefore an exact sequence (since

II

(A/tiA) 0

is a right

n

A

exact functor)

Therefore also of i,

let

such that

K.

1

K,

all integers

.K*

1

is a quotient A-module of i > O.

K/tiK,

and therefore

For each non-negative integer

be the unique cochain complex of left A-modules

j 0 iK-- I

all integers

j~n-l,

.K

1

n-l

= K .• 1

Then we

have a short exact sequence of cochain complexes of left A-modules: (10) j~n-l

j = n - 1 ,

a portion of the long exact sequence of cohomology of the short exact sequence of cochain complexes (10) is the exact sequence of left A-modules:

For each non-negative integer the sequence (lli+l)

i,

we have a natural map from

into the sequence (lli)' and we therefore

obtain an exact sequence of inverse systems of left A-modules

Cohomology of Cochain Complexes indexed by the non-negative integers. . h t exact f i l l ~im lll r~g unctor i>O (12)

K.]

~

. Id s y~e

577

Throwing through the

t h e exact sequence

->- [li ml Hn-l(C*/ti(c*)]->i>O

Hn - l (D*/tiD*)] ->-

o.

But we have observed that

Ki

is a quotient A-module of

all non-negative integers

i,

whence

K,

liml K. = O. i>O ~

(13)

substituting equation (13) into the exact sequence (12) we see that the second map in the sequence (12) is an isomorphism of left A-modules (14) Equations (9) and (14) imply equation (7). i C = 0, all integers

Case 3.

cochain complex of that

Di=O,

C*

i > n + 1.

such that

all integers

D

n-l

i;;o!n-l, n.

Then let =C

n-l

Let

0*

be the sub-

,

and such E*=C*/D*.

Then

i < n - 2

and we have a short exact sequence of cochain complexes of left A-modules: (8)

O->-D*->-C*->-E*->-O.

The cochain complex Hn(D*)

(=Hn(C*»

=M.

D*

obeys the hypotheses of Case 2, and

Therefore, by Case 2, we have a canonical

isomorphism of left A-modules

578

(9)

Chapter 4 M») ~ [liml Hn-l{D*/tiD*»).

[liml (precise ti-torsion in

i>O

i>O For each integer

the short exact sequence:

j,

splits (since in fact i f j

tity of c j j map C ->- E

j

and

E = 0;

the map

D

and if,

j~n-2

then

is the identity of

through the additive functor every non-negative integer

is exact.

j

j2:. n - l

II

i

j

C ). (A/tiA)

->-

C

j

is the idenj

D = 0

and the

Therefore, throwing

~ ", we have that for

that the sequence:

Therefore, for every integer

i 2:. 0,

we have the

short exact sequence of cochain complexes of left

(A/tiA)-

modules:

A portion of the long exact sequence of cohomology of this short exact sequence of cochain complexes is the exact sequence of left

(A/tiA)-modules: (10)

• • • -+

2 . an - 2 1 . Hn - (E* /tlE* ) _ H n - (D* /tlD*)

->-Hn-l(C*/tiC*) ->-Hn-l(E*/tiE*) ->- '" all integers

i > O.

· Bu t , Slnce

all integers

i 2:. 0,

and that

Hn - 2 (E*/t i E*) (A/tiA)-module

E n - l = 0,

,

we have that

is a quotient (A/tiA)-module of the

En-2/tiEn-2 (=Cn-2/tiCn-2),

and therefore that

Cohomology of Co chain Complexes (12)

Equations

579

i n 2 H - (E*/t E*)

is a quotient left A-module of the

left A-module

En-2 ,

(11) and (12)

all integers

i.

substituted into equation (10) yields

the exact sequence of left A-modules

For each integer

i > 0,

we have the natural mapping from the

sequence (13 + ) i l

into the sequence (13 ). i

We therefore obtain

an exact sequence of inverse systems of left A-modules indexed by the non-negative integers. "liml" ~ i>O

f unc t or (14)

Throwing through the right exact

. Id s th e exac t Yle

[lj,m l E n i>O

2

sequence

] .... [lj,m l Hn - l (D*/tiD*)] .... i>O

[lj,m l Hn - l (C*/tic*)] .... 0 i>O Since the inverse system Ij,m i>O

l

n 2 E - = O.

n 2 "E - "

is constant, we have that

Substituting into equation (14), we see that the

second map in the sequence (14)

is a specific isomorphism of

left A-modules (15) Equations (9) and (15) Case 4. of

C*

General case.

imply equation (7). Let

D*

be the quotient cochain complex

such that

j.::,n

Let

E*

D*-+C*.

be the kernel of the epimorphism of cochain complexes: Then

580

Chapter 4 j~n

j>n+l.

The cochain complex

D*

obeys the hypotheses of Case 3.

fore, by Case 3 we have a canonical isomorphism of left

ThereA-module~

We have the short exact sequence of cochain complexes of left A-modules (9)

O+E*+C*+D*+O. i E z, the sequence

For each integer

splits (since

Ei

=0

for

i < nand

Di

=0

for

Therefore, throwing through the additive functor

i

~n + 1).

"(A/tjA)

@

II

A

we obtain a short exact sequence of cochain complexes of left (A/tjA)-modules:

all integers

j

~

O.

A portion of the long exact sequence of

cohomology of this short exact sequence of cochain complexes is the exact sequence

But ,

; nc e S ...

Ej

=0

f or

J.

~

n,

...;t follows that

Hn-l(E*/tjE*)

Cohomology of Cochain Complexes n

.

H (E*/tJE*) =

o.

581

Substituting into the last long exact sequence,

it follows that, for every integer

j

~

0, that the natural map

induces an isomorphism:

Throwing through the functor

liml, i>O ral mapping induces an isomorphism: (10)

it follows that the natu-

[liml Hn-l(C*/tjc*) J ~ [liml Hn-l(O*/tjo*) J. i>O i>O

A portion of the long exact sequence of cohomology of the short exact sequence of cochain complexes (9) is the exact sequence n-1

(11)

•••~Hn

n

(E*) -+ Hn (C*) -+ Hn (0*) ...£-H n + 1 (E*) -+

Hn+l(C*) -+ ..• We have that (12)

En = 0,

and therefore

Hn(E*) = 0,

and

Hn + 1 (E*) =Ker (d~~1:En+1-+En+2). E

But

En+l=C n + 1 ,

n+2 _ Cn+2 dn+1 _ dn+l , E* - C* •

Therefore

Ker (d~~l: En + l -+E n + 2 ) =Ker (dg~l: c n + l -+C n + 2 ),

therefore the kernel of the natural mapping: is canonically isomorphic to we let

n+l 1 = 1m (d *: C -+ C ), n

c

n

and

n n H + l (E*) -+ H + l (C*)

n n l 1m (d~*: C -+ c + ).

Therefore, if

then substituting this last obser-

vation and equation (12) into equation (11) yields the short exact sequence (13) where

0-+H n (C*)-+H n (D*).§.1-+0,

n n l 1 = 1m (d~*: C -+ c + )

and

<5

is induced by a coboundary

58;>

Chapter 4

(in fact, into

1m

n l is the natural map from Coker (d~:l: C -

0

(d~*: Cn

-+

n l C + ».

-+

Cn)

The monomorphism in equation (13)

implies that the natural map induces a monomorphism, (14)

Since

C*

(t-divisible part of

n H (C*) )c.->

(t-divisible part of

Hn(D*».

and

D*

both obey the hypotheses of this Theorem, by

equation (6) applied to (15)

C*

and to

(t-divisible part of [llm i>O

l

D*

we have that

n H (C*) ) ""

(precise ti-torsion in

Hn(C*»],

and (16)

(t-divisible part of

n H (D*»

""

[liml (precise ti-torsion in i>O Equations (8),

(10),

(14),

(15)

Hn(D*»].

and (16) imply that we have a

natural monomorphism: (17)

Ilml (precise ti-torsion in i>O

Hn(C*»

~

liml Hn-l(C*/tic*). i>O This completes the proof of equation (7). On the other hand,

is such that

1

by equation (13), if

1 = Im(d~*: Cn

-+

Cn + l

has no non-zero t-torsion, from equation (13),

it follows that the natural map is an isomorphism (precise ti-torsion in

Hn(C*»~

(precise ti-torsion in

Hn(D*»,

all integers

i > O.

Throwing through the functor

liml, i>O

we see

Cohomology of Cochain Complexes

583

that in this case the natural map: (18)

[lim i>O [lim itO

1

(precise ti-torsion in

Hn(C*» 1 ->-

1

(precise ti-torsion in

Hn (D*) ) )

is an isomorphism.

Equations (8),

(10) and (18) therefore imply

Note 2.

Q.E.D.

Corollary 6.1. Let

C*

Let

A

be a ring with identity and let

tEA.

be an (arbitrary, Z-indexed) cochain complex of left

A-modules, and let

c*" = c*"t.

Then for every integer

n,

there is induced a natural mono-

morphism of abelian groups (or of left A-modules if center of

is in the

A), (t-divisible part of

(1)

t

If for some fixed integer

n,

Hn((c*)A)}c.t[lj,m l Hn-l(C*/tiC*)l. i>O Cn + l has no non-

we have that

zero t-torsion, then for that fixed integer

n

the monomorphism

(1) is an isomorphism. If, in addition, the endomorphism "multiplication by til

:C

n

ger

->-

n C

n

is injective, all integers

n,

then for every inte-

there is induced a canonical isomorphism of abelian

groups (or of left A-modules if

t

is in the center of

A)

of

each of the groups in equation (1) to the group (2)

Note:

" 1 (preclse " t i -torslon " " ln I !m i>O

Corollary 6.1 generalizes abelian categories:

Let

A be

an abelian category such that denumerable direct products exist and such that the functor "denumerable direct product" preserves

584

Chapter 4

epimorphisms.

Then

Corollary 6.1 (generalized).

Let

C*

be an (arbitrary, z-

A,

indexed) cochain complex of objects and maps of t'" t*

be any endomorphism of

C*.

Then the conclusions of

(With phrases like nt-divisible" and "in-

Corollary 6.1 hold.

jective" interpreted in the obvious way, as in Proof.

and let

Chapte~

1.)

We have that the natural map is an isomorphism:

(C*/tic*)

~ (C*A/tic*A),

all integers

i.

Therefore, conclusion

(1) of Corollary 6.1 follows from conclusion (3) of Theorem 6, applied to the cochain complex n l c +

c./I.

If for some integer

n,

is with no non-zero t-torsion, then it follows from Lemma n C +l

1.1.1 that

A

also has no non-zero t-torsion.

by Note 2 to Theorem 6, for that integer

n

Therefore

the monomorphism

(1) is an isomorphism. If the endomorphism "multiplication by jective for all integers

n,

t" :C

n

->-

n C

is in-

then the hypotheses for the exact

sequence (I. 8) of [P.P.WC.] hold - i. e., the hypotheses of Theorem 1 of Chapter 2

hold.

rem 1 of Chapter (2)

Therefore we have conclusion (2) of Theo-

2, i.e., we have an isomorphism:

Itml [H n - l (C*)] "';. [liml (precise ti-torsion in

i>O all integers

Hn (C*))]

i>O n.

This completes the proof of the Corollary.

Q.E.D. Remarks 1.

Let

C*

be any

plete left A-modules, where let

tEA.

z-indexed cochain complex of comA

is a ring with identity, and

Then, by Theorem 6, applied to

sion (4), the

C*A = C*A t,

conclu-

leftmost group in conclusion (1) of Corollary 6.1

is canonically isomorphic to

Cohomology of Cochain Complexes (2')

liml (precise ti-torsion in

585

Hn(C*A».

i>O Therefore, an equivalent way to state the second conclusion of Corollary 6.1 is to say that, if is injective, all integers

n,

n t": C

"multiplication by

-+

n C

then the natural map from the

group (2) of Corollary 6.1 into the above group (2')

is an iso-

morphism. 2. be a

Let

A

be a ring with identity, let

(~-indexed)

tEA

and let

cochain complex of left A-modules.

Suppose

that "multiplication by integers valid?

n;

tll:C

n

-+

cn

C*

is not one-to-one for all

then is conclusion (2) of Corollary 6.1, still

The answer is "no", as is seen by the following counter-

example. Example 1.

Let

M

n M"" H (0*)

and is such that (~-indexed)

part of

for some integer

n

and some

cochain complex of complete left A-modules

(Such modules mark 7.)

be any left A-module that is not complete

0*.

exist, see Examples 1 and 2 below, after Re-

M

Then by conclusion (1) of Theorem 6, the (t-divisible M)"I O.

Also, by conclusion (4) of Theorem 6,

(t-divisible part of

M) "" liml (precise ti-torsion in

M) •

So

Therefore (1)

[lim l i>O

(precise ti-torsion in

On the other hand,

MA

M)

1 "I

O.

is t-adically complete, and therefore

has no non-zero t-divisible elements, so by Proposition 5 (2)

Chapter 4

586

[lim (precise ti-torsionin

i>O Let

C*

(~)i\) 1 = o. M

be the cochain complex such that i 'I O.

integers

i C = 0,

cO = M,

all

Then

Therefore, by equations (1) and (2) of this Example, and by Remark 1, the conclusion (2) of Corollary 6.1 does not hold for C* . Also, it is not difficult to construct examples in which all the hypotheses of Theorem 6 hold, yet the monomorphisms described in conclusion (5) of Theorem 6 are not all isomorphisms. Example 2.

Let

field and let

be a complete discrete valuation ring not a

0 t

be a generator for the maximal ideal of

be the completion of a free

Let

rank having for

for

O-module of denumerable

O-basis the elements

the completion of a free O-basis the elements

xi'

i > O.

r i'

Let

O-modules

is isomorphic as

O/tiO,

as

R.

~

O-module to

d

1

:c l

....

Define

Let

CO

be

C

2

R , i

be the comple-

i > O.

where

i;:O,

R.

~

be a generaDefine

c2 all integers all integers

ci=O,

O.

and

by requiring that that

>

and let

O-module, all integers

dO:C°-+C l

i

O-module of denumerable rank having

tion of the direct sum of the

tor of

O.

all integers

Then we have a cochain complex hypotheses of Theorem 6.

i;:O,

and

i > 0.

i'lO,1,2. C*

of

O-modules obeying the

However, if we define

D*

by requiring

Cohomology of Cochain Complexes that

0*

be a quotient cochain complex of

C*

587

and that

i = 0,1, i = 2,

then the element Hl(O*)

X

t

oE C

l

=0

1

defines a divisible element in

unequal to zero.

However, an explicit computation shows that

G = Hl (C*)

is

not finitely generated, has no non-zero t-torsion and is tadically complete.

In particular,

t-divisible elements.

Hl(C*)

Considering the proof of Case 4 of

Theorem 6, we see that (t-divisible part of

~

with and that

has no non-zero

Hl(C*))

(the two groups of conclusion (4) of Theorem 6 n=l),

(t-divisible part of

Hl(O*)

Therefore, the two subgroups of

Hl(C*)

described in conclu-

sion (5) of Theorem 6 are distinct in this case. This example is also interesting for another reason.

It is

easy to see that

.

Hl (C*) ~

!G, 0,

all integers

iEZ.

i = 1, i

~

1,

However, if

0*

is as in the proof of

Case 4 of Theorem 6, then, by the proof of Case 4 of Theorem 6,

588

Chapter 4 [limj HO(C*/tic*)] ~ [limjHO(o*/tio*)], i>O itO

j=O,l. Also by conclusions (4),

(5) and Note 2 of Theorem 6 (or,

in fact, by conclusion (4) of Theorem 1, since

0*

also obeys

the hypotheses of Theorem 1), we have that [lim itO

l

O H (o*/tio*)]~ (t-divisible part of

But, an explicit computation shows that O-submodule, the

O-module

relations: X

o

N

Hl(O*)

with generators In the

i > 0.

Hl (0*))

O-module

contains, as i ~ 0,

xi' N,

part of

N) c: (t-divisible part of

(where the leftmost isomorphism is the one that sends into

X

o E N).

and

the element

is t-divisible, and is not a t-torsion element.

o ~(t-divisible

.

In fact, Hl (0*)) , 1 E0

In fact, in this case, an explicit computation showl

that Hl(C*/tiC*) =0, [lim itO

all integers

i~O;

and therefore

Hl(C*/tic*)] = 0,

and therefore by the short exact sequence of Corollary 1.2 of Chapter 3,

Therefore, in this case, the epimorphism of conclusion (3') of Theorem 6, for (Note:

is the zero map, and is not an isomorphism.

It is easy to observe directly in this case that for all

i>O, HI (c*/tic*)

....

i = 1,

= 0,

so that

0*

is not necessary in the com-

Cohomology of Cochain Complexes

589

putation).

Notice also that the subgroup

of

in this case is strictly larger than the subgroup,

Hl(C*)

(t-divisible part of

[liml HO(C*/tic*») i>O

Hl(C*».

However, if one uses percohomology,

(see near the end of

Chapter 5 for the definitions), then from the exact sequences of percohomology and the fact that

. I

Hl. (C*) =

i = 1,

G,

0,

all integers

i E;Z,

ill,

it follows that i

= 1,

ill,

all integers

i, j 1

with

j

~

0.

.

Therefore, in this case, 1

[lim HO (C*,O/t1))) "" G"" H (C*), i>O and

[ lim ito

1

HOO (C * , 0/ t i 0 ») = 0.

Therefore, in this case, the natural mappings: i [lim H6(c*,O/t O») ... [lim Hl(C*/tic*)], i>O ito [liml Hg(C*,O/tiO)] .... i>O are not isomorphisms (in fact, both are the zero maps from to zero, and from zero to

G,

respectively).

G

Thus, even in

cases when the epimorphism (3') of Theorem 6 is not an isomorphism, it is still suggested, if one replaces

"H*(c*/tic*)"

by the corresponding percohomology groups (see the end of Chapter 5 for the basic definitions),

"HA(A/tiA,C*)",

then

590

Chapter 4

that the corresponding mapping should be an isomorphism. fact, this is true (if the element

tEA

In

is a non-zero divisor),

see Theorem 6' of Remark 4 below. The remainder of this chapter, Remarks 3 through 8, make use of percohomology, a concept that is defined and studied in Chapter 5 below, following the proof of Corollary 1.2 of Chapter 5.

These Remarks, 3 through 8, involve generalizations and

improvements of Theorem 6, Corollary 1.1, Proposition 2", Corollary 2.1, etc., of this Chapter, and therefore these Remarks belong at this point.

The reader is advised to skip ahead to

Chapter 5, following Corollary 1.2, for the definition of the percohomology groups

H~ (M, C*) ,

complex of left A-modules module

M,

C*

n E;Z,

of ~ (;Z-indexed) cochain

with coefficients in

~

right A-

before reading the rest of this chapter, Remarks 3

through 8 below. Remark 3.

Under the hypotheses of Corollary 6.1, let

any cochain complex of left A-modules. tEA

C*

be

Suppose that the element

is a non-zero divisor, and is in the center of the ring

Then, if the reader uses percohomology as defined in the latter part of Chapter

5 (vide infra), then it is easy to see that for

every integer

n,

that we have a canonical isomorphism:

[lj,m l H~-l(A/tiA,C*))""

(2)

i>O [lim l i~O

(Proof:

Choose

modules and

(precise ti-torsion in 'C*

¢*:' C*

+

a C*

;Z-indexed cochain complex of left Aa map of (.,z-indexed) cochain complexes

of left A-modules such that tegers

n,

Hn(C*))).

n H (¢*)

is an isomorphism, all in-

and such that "multiplication by

tit:

'C n

+

'C n

is

A.

Cohomology of Cochain Complexes injective, all integers

n.

Then the second conclusion of Corol-

lary 6.1 applied to the cochain complex sion (2) of this Remark.)

591

'C*

implies the conclu-

Also, the notations being as in this

Remark, we also have the short exact sequence: O-+Hn(C*)At+ [lim H~(A/tiA,C*)J-;. i>O

(1)

[lim (precise ti-torsion in i>O all integers

n.

Hn + l (C*)) J + 0,

Conclusions (1) and (2) can be interpreted as

a generalization of Theorem 1 of Chapter 2 , in the case that n n "multiplication by t,,:c +C is not necessarily injective for all integers Remark 4.

n - however, this generalization is shallow.

Suppose that the hypotheses of Theorem 6 hold (and t":Cn+C n

that e.g., "multiplication by jective for all integers

n)

is not necessarily in-

and that the element

tEA

non-zero divisor and is in the center of the ring

H~(A/tiA,C*) cients in integers

be the percohomology groups of

n

Then let

with coeffi-

as defined at the end of Chapter ~ below, all

A/tiA i,

C*

A.

is a

with

i > O.

Then one can retrieve all of the

results of Theorems 1 and I', if one uses the percohomology

"H~(A/tiA,C*)",

groups

"Hn(C*/tic*)", Theorem 6'.

"H~-l(A/tiA,C*)"

"Hn-l(C*/tic*)".

Let A

More precisely,

be a ring with identity and let

element of the center of the ring zero divisor.

A,

such that

t

t

be an

is not a

be a (z-indexed) cochain complex of n left A-modules such that C is t-adically complete, all integers

n. (1)

Let

in lieu of

C*

Then For every integer

n,

t-adically complete.

Hn(C*)/(t-divisible elements is Also,

Chapter 4

592

(2')

For every integer

n,

Hn(C*)

has no non-zero in-

finitely t-divisible elements. Also, for every integer

n,

there are induced canonical

isomorphisms of left A-modules: (3)

Hn(C*)/(t-divisible elements) "" Hn(C*)"t""

~!m [H~(A/tiA'C*)). ~>O

Also, for every integer

n,

there are induced canonical

isomorphisms of left A-modules (4)

(t-divisible part of

Hn(C*)) ""

liml [precise ti-torsion in i~O

Hn(C*)) ""

I' 1 HnA-l(A/tiA,C*).

it~

Proof:

By Corollary 3.2 of Chapter 5, there exists a

cochain complex

C*

of the left A-modules, such that

is t-adically complete, all integers

n,

has no non-zero t-torsion, all integers

(~-indexed)

(1)

'C

and such that (2) n,

n

'C

and such that we

have a mapping <j>*:'C*+C* of

(~-indexed)

cochain complexes of left A-modules, such that

is an isomorphism, all integers

n.

Conditions (2) and (3) im-

ply that

all integers

n,

i

with

i > O.

Equations (1) and (2) imply

n

Cohomology of Cochain Complexes that the cochain complex

'C*

of left A-modules obeys the hypo-

theses of Theorem 1 and of Theorem 1'. sions (1),

Therefore we have conclu-

(3) and (4) of Theorem 1 and conclusion (2') of

Theorem I' for the cochain complex us to substitute "Hn-l(,C*)" 'C*,

593

"Hn(C*)"

for

in conclusions (1),

'C*.

Equations (3) allow

"Hn(,C*)"

and

equation (4) allows us to replace and

"H~-l(A/tiA,C*)",

for

(3) and (4) of Theorem 1 for

and in conclusion (2') of Theorem I' for

"H~(A/tiA'C*)",

"Hn-l(C*)"

'C*.

"Hn(('C*)/ti(,c*))"

"Hn-l(('C*)/ti(,c*))"

Finally, with

with

respectively, all integers

i.::O,

sions (3) and (4) of Theorem 1 for the cochain complex

in conclu'C*.

Q.E.D.

Remark 5.

If one has the hypotheses of Theorem 6' of Remark 4,

then one obtains generalizations of Corollary 1.1, Proposition 2, Proposition 2' and Corollary 2.1 when one replaces the "cohomology groups mod till described in these corollaries and propositions with the corresponding "percohomology groups".

The

proofs are entirely analogous to that of Theorem 6' of Remark 4 (and, as in the proof of Theorem 6', immediately reduce to the earlier corollaries and propositions, by constructing ¢*: 'C* + C*

'C*

and

obeying the conclusions of Corollary 3.2 of Chapter

5) •

Corollary 1.1'.

Under the hypotheses of Theorem 6' of Remark 4,

we have that both conclusions of Corollary 1.1 hold (where the generalized Bockstein spectral sequence of the endomorphism "multiplication by

till

C*

with respect to

is as defined in

Remark 4 following Theorem 2 of Chapter 1 in the case that "multiplication by

t":C

n

+c

n

is not one-to-one.)

594

Chapter 4

E~

Thus, for example, Proposition 2'".

=

H~ (A/tiA,C*),

all integers

n.

Under the hypotheses of Theorem 6', let

any fixed integer.

n

be

Then the following seven conditions are

equivalent. (1)

Hn(C*)

is t-adically complete.

(2)

Hn(C*)

has no non-zero t-divisible elements.

(3)

The natural mapping: n H (C*)

H~ (A/tiA, C*)

lim i>O A-modules:(4)

(a)

->-

uEHn(C*),

If

then and

(b)

u

then

u

t-u=O,

and

u

is t-divisible,

is infinitely t-divisible

n u E H (C*)

If

is an isomorphism of left

and

t

i

- u"l 0,

all integers

i,

is not t-divisible.

(4') Every t-divisible element in

Hn(C*)

is infinitely

t-divisible.

(5) (6)

Proof:

liml (precise ti-torsion in i~O l~ml Hn - l (A/tiA,C*) = O. A i>O Take

n H (C*) ) = O.

as in the proof of Theorem 6'.

'C*

tion 2 and Proposition 2' for

'C*

Then Proposi-

imply Proposition 2 "'

C*.

for Q.E.D.

And similarly Corollary 2.1 immediately generalizes: Corollary 2.1'. let

n

Under the hypotheses of Theorem 6' of Remark 4,

be an integer such that the seven equivalent conditions

of Proposition 2'"

all hold.

Corollary 1.1', we have that (1)

liml i>O

E~(ti)

=0.

Then, the notations being as in

Cohomology of Cochain Complexes

595

And Remark 1 following Corollary 2.1 similarly generalizes. Remark 6.

An amusing consequence of Theorem 6, Note 2, is that,

under the hypctheses of Theorem 6, if n 1m (d : C n

that

-+-

n l C + )

n

is any integer such

has no non-zero t-torsion, and if

n u E H (C*) ,

then

u

is t-divisible in

Hn(C*)

iff, for every non-negative

n there exists v. E H (C* / (precise ti-torsion)), integer i, l. such that the image of V. under the mapping induced by multil. plication by

is

ti:

u.

Proof:

From the long exact sequence of cohomology of the short

exact sequence of cochain complexes: .

uti"

.

0-+ [C*/ (precise tl.-torsion)] ~C* -+ C*/tl.C* we see that there exists Hn(C*/tic*) gers

i> 0

is zero.

vi

0,

as above iff the image of

Thus, there exist such

iff the image of

-+

u

in

vi

u

in

for all inte-

lim [Hn(C*/tic*)]

is zero.

itO But, by conclusion (3') of Theorem 6 and Note 2, this latter occurs iff Remark 7.

u

Q.E.D.

is t-divisible.

In Remark 3, Theorem 6' of Remark 4, Corollary 1.1'

of Remark 5, Corollary 2.1' of Remark 5, Proposition 2"1 of Remark 5, but not in Remark 6 - in fact, throughout most of the last Remarks - we had to assume that the element zero divisor.

tEA

is a non-

This assumption can be removed, and the entirety

of these Remarks then remain valid, if throuahout, we replace

596

Chapter 4

the percohomology groups

"H~ (A/tiA, C*) ", Here

C*

(where

etc., by

is regarded as being a cochain complex of Z[Tj-modules is the polynomial ring in one variable over

Z [T]

by requiring that center of

"T"

acts as

C*

the action of

"T"

then since "T"

(Then, if

t

is in the

is a cochain complex of left A-modules, and (=

the action of

action of every element of

since

"t".

Z)

H~[T] (C*,Z [T]/TiZ[T]), etc., is also a left A-

A,

module since

A,

"H; (T] (C* ,Z (T] /TiZ [T] ) ", etc.

and

A.

"t")

Also, if

commutes with the t

is in the center of

i "T "

acts as zero in

Z[T]/(T

"t"

act the same on

C*, it follows that the

left A-modules, e.g.

H~(A/tiA,C*)

and

i

'Z[T]),

and

H;[T] (C*,;Z[TJ/TiZ[T]),

i

are actually left (A/t A)-modules.) Remark 8.

One might wonder, under the hypotheses of e.g., Theo-

rem 1, if

n

Hn(C*)

is a fixed integer, whether it is possible for

to possess t-divisible elements; and, if so, whether or

not the t-divisible part of

Hn(C*)

(which by conclusion (2')

of Theorem 6 has no infinitely t-divisible elements) must necessarily be, either a t-torsion module, or else be free of nonzero t-torsion?

The following two examples (of a rather practical

nature from our p-adic cohomology in algebraic geometry) show that all such questions are answered in the negative; otherwise stated, in "practical" examples, pathologies "just as bad as looks conceivably possible" can occur under the hypotheses of, e.g. Theorem 1. Example 1.

Let

A= 0

be a complete discrete val ua tion ring of

mixed characteristic, let

r

be a positive integer and let

Cohomology of Cochain Complexes

597

H=O[T , ... , T }, the polynomial ring in r variables over O. l r n Let B = the n' th exterior power of the module of differentials of the

O-algebra

O-modules. O.

Then

Let

t

Then

B*

is a cochain complex of

be any generator for the maximal ideal of

H"t = ring of t-adically convergent power series over

in r-variables. n> r resp.

H.

Let or

C* = (B*)At.

(Then is is easy to see that

i < 0,

n

and

B , resp.

n .~s a f ree C,

H";-module of finite rank, all integers

n.

However, of

are not finitely generated as

course

O-modules,

and of course the coboundaries are not H-linear in either or

H,

B*

C*). It is easy to see (see Examples in Remark 1 of Chapter II

of [p.p.we.}) that Hn (B*) "" ED (O/tiO) i>l

(1)

all integers n H (C*),

n

(w) ,

1 < n < r.

such that

The cohomology

l
is much more pathological, even though plete and "multiplication by tegers

n

t":

cn

C*

+C n

is t-adically comis injective, all in-

(i.e., the hypotheses of Theorem 1 hold).

In fact,

each of the six equivalent conditions of Proposition 2 all fail n H (C*), all integers

for

n

such that

1 < n < r.

dition of Corollary 2.1 holds for all integers

E~(ti) =0,

fact

all integers

i EO (t ) = O/tiO , all integers 00 using C*

B*

instead of

C*

i >

i2:,0,

n,

(But the consince in

all integers

° - this

nlO,

and

computation follows by

(the two cochain complexes

B*

and

have the same generalized Bockstein spectral sequences for

0

598

Chapter 4 all integers

the endomorphism same

E~(ti)),

i ~ 0,

since they have the

using Theorem 2 of Chapter 1, and equation (1)

of this Example).

In an Example in a Remark in Chapter II of

[P.P.WCJ, I noted that

Hn(C*)/(t-divisible elements) has non-

trivial topological t-torsion, so that n [H (C*) / (divisible elements)] ® K

o

is an uncountable dimensional K-vector space, all integers such that

1 < n
(when

K

is the quotient field of

n

0).

As an application of Proposition 5, we show that the tdivisible part of tion of a free

Hn(C*)

for

l::'n::'r,

is the t-adic comple-

O-module of uncountably infinite rank.

(Which

of course implies that condition (1), and therefore all the equivalent conditions (1), (2), ... , (6) of Proposition 2 fail for C*, as we asserted above). In fact by the second conclusion of Corollary 6.1 applied to the cochain complex (2)

B*,

we have that for every integer

(t-divisible part of [lj,m

l

n

Hn(C*)) ""

(precise ti-torsion O'f

Hn(B*))]

i>O

(Since by definition But, if

C*

=

(B*)").

n M = H (B*), then by equation (1) of this Example,

M

has no non-zero t-divisible elements, and therefore obeys the hypotheses of Proposition 5. (3 )

' 1 [1 ~m i>O

(

Therefore, by Proposition 5,

, , , preClse t i -torslon In

[lim (precise ti-torsion in i>O The

M)] ""

M"/M)].

O-module on the right side of equation (3) is clearly

Cohomology of Cochain Complexes

599

the t-adic completion of a free

a-module of rank equal to the

dimension of the k-vector space:

(precise t-torsion part of rf /M)

where

k=a/tO.

But, if

easy to see that k-vector space.

rf/M

l:::n.:.r,

then from equation

(1)

it is

is of uncountably infinite dimension as

(Proof:

If

u.

is one of the basis elements

1

in the i'th direct summand of the right side of equation (1), all integers

i .:::. 1,

w (a i) i> 1 E k ,

then for every sequence

we

have the distinct t-torsion elements: \,'

La. (t.

i>l

in

1\

The images of these elements in

M.

of cardinality:

(cardinal (k»

cise t-torsion part of space,

i-I

1

~

It follows that the k-vector

(precise t-torsion part of

finite dimension}.

are a subset

of the k-vector space (pre-

0

rf/M).

rf /M

MA/M), is of uncountably in-

Combining this observation with equations

(2) and (3), we see that (t-divisible part of t-adic completion of a free

Hn(C*»

~ (the

a-module of uncountably infinite

rank) . Remark.

Notice that, under the hypotheses of Theorem 1, the

conclusions of Corollary 2.1 may hold, sometimes even when the conclusions of Proposition 2 fail.

E.g., see the preceding

Example. Example 2.

Let

A

be a ring with identity and let

Assume for simplicity that M be any left A-module.

t

is a non-zero divisor in

all integers

A.

Then choose an A-homomorphism

of free left A-modules such that Bn=O,

tEA.

nt-O,l.

M~

Then

a

Coker (d ) I B*

and define

is a cochain complex

Let

I

Chapter 4

600

of left A-modules and

Hl(B*) '" M.

Define

C* = (B*)J\t.

Then the

hypotheses of both parts of Corollary 6.1 are satisfied, so that by conclusions (1) and (2) of Corollary 6.1 we have that (1)

(t-divisible part of

Hl (C*))'"

liml (precise ti-torsion in

M).

i>O

But, by Theorem 4, we have a monomorphism:

(2)

~

(t-divisible part of M) (infinitely t-divisible part of liml (precise ti-torsion in

M)

M).

i>O

Combining equations (1) and (2), we see that if A-module, e.g. such that

M is any left

M has no non-zero infinitely t-divi-

sible elements, then the (t-divisible part of isomorphic to an abelian subgroup (or if of A,

to a left A-submodule) of (3)

Here

(t-divisible part of

t

M)

is canonically

is in the center

Hl(C*), M)

C

Hl (C*).

C* is a (non-negatively indexed)cochain complex of left

A-modules that obeys all the hypotheses of Theorem 1.

Equation

(3) above shows, under the hypotheses of Theorem 1, just how "bad" the divisible part of left A-module

M,

we can build a

of Theorem 1, such that t

Hn(C*)

is in the center of

Hl(C*) A,

can be - namely, for every C*

obeying all the hypotheses

contains, as a subgroup (of if

as a left A-submodule) the group

(or left A-module) (t-divisible part of M) (infinitely t-divisible part of

M)

.

Cohomology of Cochain Complexes (Of course, by conclusion (2') of Theorem 6,

601

Hn(C*)

has no

infinitely t-divisible elements; so that in essence this last counterexample

cannot be improved upon).

Thus, in particular, we have examples of hypotheses of Theorem 1, such that Hn(C*)

C*

obeying the

has e.g. t-divisible,

t-torsion elements that are not infinitely t-divisible. Let

A

be a ring with identity and let

divisor in the center of the ring

A.

t

Then if

be a non-zero C*

is any (z-

indexed) cochain complex of left A-modules, such that (multiplication by

t): c

n

->-c

n

is injective, all integers

n,

then the

natural mapping induced on percohomology:

is an isomorphism, all integers Since

t

c

with

i >

is a non-zero divisor in the ring

(mutliplication by 1.1.1,

n, i

nAt

->-

c

nAt

,

t): c

n

n ->- C ,

o.

A,

(Proof: and since

and therefore also by Lemma

is injective, all integers

n,

we have

that

all integers

n, i

from the fact that e*/tie*.)

with

i > O.

e*At/tic*At

is canonically isomorphic to

Does this result remain true if one deletes the hypo-

thesis that "multiplication by integers

So the indicated result follows

n?

The answer is

discrete valuation ring and t A, and even if

C*

t":

en ->- en

is injective, all

"no", not even if

A

is a complete

generates the maximal ideal of

is non-negative and

en

=

0

for

n;: 3,

and

602

Chapter 4 is finitely generated as A-module, all integers

n,

as

is seen by Example 3, below. Also, if

A

is a ring with identity and if

ment in the center of the ring

A

t

is an ele-

that is a non-zero divisor,

then let (1)


'c* ->c*

be a map of (z-indexed) cochain complexes of left A-modules such that the endomorphism;;;, "multiplication by t", of the left An n modules 'C and c , are injective, all integers n. Then if ¢*

induces an isomorphism on cohomology in all dimensions, then

so does

(Proof:

lS

By Theorem 6' of Remark 4, it suffices to prove that

an isomorphism, all integers

n,

i

with

i > 1.

But by the

observation of the preceding paragraph, it is equivalent to show that

is an isomorphism, all integers

i, n

with

i >

o.

But this

latter observation follows from the universal coefficients spectral sequence of percohomology.) One might ask that, if one deletes the hypotheses on and

I

C*,

that "multiplication by

are injective, all integers

n,

til,

en

-+

en,

and

I

en

C* -+

then does it remain true if

I

cn ¢*

Cohomology of Cochain Complexes

603

is a mapping of cochain complexes as in equation (1) that induces an isomorphism on cohomology in all dimensions, then is it necessarily true that the mapping

¢*At

in equation (2)

necessarily induces an isomorphism on cohomology? "no", not even if

A

is a complete discrete valuation ring,

is a generator for the maximal ideal of n C

both non-negative, tion n,

Hn(C*)

The answer is

=

'C

n

=

°

for

A,

n> 3

C*

and

'C*

t

are

and even if in addi-

is finitely generated as an A-module, all integers

as seen in Example 3 below.

Example 3. a field

Let

A=

(e.g., 0

be a complete discrete valuation ring not

=£, p

p

O.

Let

quotient field of ideal of

a

any rational prime). t

Let

K

be the

be a generator for the maximal

O.

Then let

I

n = 1,2,

Kia,

Cn =

0,

nr!1,2,

and let

d

Then

n

multiplication by

=

( 0, is a

We have that

Let

,cn

=

= 0,1

0,

n

0,

nr!O,l

j

n = 1 n r! 1.

n n C* = (C ,d )nEZ

A-modules.

t,

(non-negative) cochain complex of left

Chapter 4

604

and

'd

n

n=O

t,

multiPlication by =

\ 0,

n

10.

Then

is a

(non-negative) cochain complex of left A-modules, and

'c*"t

= 'C*.

Let

be the unique homomorphism of A-modules that maps the class of

CP*:

1

t'

Let

lEO

into

Then

'C*+C*

is a map of cochain complexes of left A-modules that induces an isomorphism on cohomology in all dimensions.

Notice that

n=l

nIl, all integers

n.

Notice also that

all integers

n.

Therefore

percohomology of

C*

'C*

'Cn

.

H~ (C* ,A/t1A)

0,

with coefficients in any A-module.

=

{A/tA, 0,

for

can be used to compute the

this way, we see that, e.g.,

(1)

is flat over

n = 0,1 n 10,1,

In

Cohomology of Cochain Complexes all integers

n, i

with

i

~

1,

605

and that the natural mappings:

an isomorphism if

n = 1,

the zero map

n=O,

are

for

n = 0, 1.

Therefore

(2)

H~(C*,A/tiA)

lim i>O

all integers (3)

n,

if

--1 A/tA,

n

= 1,

0,

n

t

and

liml HAn (C*,A/tiA) = 0, i>O

all integers

n.

On the other hand, since replace

1,

C*

with

c*/\

C*/\=O,

it follows that, if we

on the left sides of equations (1), (2),

and (3), then the right sides become zero. Therefore, for this cochain complex

C*,

the natural

map:

H~ (C* ,A/tA) .... H! (c*/\, A/tA) is not an isomorphism.

Also, although the map of (z-indexed)

cochain complexes of left A-modules ¢*:

'C* .... C*

induces an isomorphism on cohomology, nevertheless the induced map of

(Z-indexed) cochain complexes on the completions

606

Chapter 4

does not induce an isomorphism on cohomology (since

HI ('C*At) =HI ('C*) "" A/tA,

but

HI (C*At) = HI = 0).

(the zero cochain complex)

CHAPTER 5 FINITE GENERATION OF THE COHOMOLOGY OF COCHAIN COMPLEXES OF t-ADICALLY COMPLETE LEFT A-MODULES

Proposition O.

Let

A

be a ring with identity and let

an element of the center of the ring

A.

complex of the left A-modules such that plete, all integers (0)

i,

(multiplication by

nand

r

ci

C*

be

be a cochain

is t-adically com-

and such that

all integers Let

Let

t

ci

t):

-+

ci

is injective,

i.

be fixed integers with

r > O.

Then the

following conditions are equivalent: is the image of some t-torsion ele-

(1)

ment in

Hn(C*),

then

u

tr-torsion element in (2)

n u E H (C*),

t i . u = 0

H

n

is the image of some precise (C*).

for some

i

~

0,

implies that

tr·u=O. (3)

If

then (4)

u E Hn - l (C* /tC*) dn-l(u)

is such that

is t-divisible in

d n - l (u) E t r • Hn (C*) , Hn(C*).

n If u E H (C*) is a precise t-torsion element such r that u=t ·v, for some vEHn(C*), then u is t-divisible.

(5)

In the (singly graded, cohomological) generalized Bockstein spectral sequence, as defined before Lemma 1 607

608

Chapter 5 of Chapter I, we have that d

n-l r

=d

(6)

n-l = r+l

Let

=d

n-l =0, s

n u E H (C*)

all integers

s>r.

be any t-torsion element.

Then the

following four conditions are equivalent: exists

n v ~ H (C*)

finitely t-divisible; proof:

u =t

such that u

r

. v;

u u

there

0;

=

is in-

is t-divisible.

By conclusion (2) of Theorem 1 of Chapter 4, we have

that there are no non-zero infinitely t-divisible, t-torsion elements in

o

Hn(C*).

Therefore, condition (2) of Proposition

is equivalent to, e.g., condition (5') of Corollary 3.1 of

Chapter 1.

And also, for the same reason, condition (6) of the

Proposition is equivalent to condition (5) of Proposition 3 of Chapter 1.

In addition, condi tions (1), respectively:

(3), (4)

I

(5) of Proposition 0 are identical to conditions (2), respectively: (3), (4),

(1) of Proposition 3 of Chapter 1.

equivalence of conditions (1), (3), (4), (5) and

o

Therefore the (6) of Proposition

follows from Proposition 3 of Chapter 1; and the equivalence

of condition (2) of Proposition 0 with the other five conditions follows from Corollary 3.1 of Chapter 1. Lemma 1.1.1.

Let

A

Q.E.D.

be a ring with identity and let

element in the center of the ring is t-adically complete.

A.

t

be an

Suppose that the ring

Then the following two conditions are

equivalent. (1)

A

(2)

The ring

Proof:

is left Noetherian

(1)~(2)

Suppose that

AltA

is left Noetherian.

is obvious. AltA

To prove that

is left Noetherian.

(2)~(1).

Then let

I

be

A

Finite Generation any left ideal in the ring

A.

609

For each integer

n,::, 0,

define

of left ideals by inHaving defined

duction as follows. n.::. 0,

let

In+l = {x E A: tx E In}'

I=IOc1lc12c ... c1nc", in the ring

AltA

r >0

ideal

I

in the ring

Therefore there exists an in-

left ideal

Then

I

ul, ... ,u

A

is finitely generated by induction on

1=1 =1 =

o

generate I

m let

x EI

elements

a , ... , am E A l

But then

xl E II'

,aIm E A

But then

x

a.l, ... ,a. J

But then

Jm

2

Ul

Let

n

' ... ,

urn E I

Since

be any element. and an element

II = I,

AltA.

Then I claim A.

Then there exist xl E A

x2 E A

j

x. E I. ]

the

such that

there exist elements

By induction on

and elements

generate

as left ideal in the ring

and an element

E1 , 2

EA

=1

1

in the left Noetherian ring

In fact,

all""

We prove that the left

Ir = Ir+l = ...

be elements such that the images

that

I ,I l ,I 2 , ... ,I n , ... O

r.

r = O.

Case I.

Then the sequence

of images of

AltA.

such that

the integer

for some integer

n

is an ascending sequence of left ideals in

the left Noetherian ring teger

I

J

such that

~

1,

we construct elements

such that

610

Chapter 5 x = (a l + (,Lltjajl) )u + (a + ( L tja '2) )u l 2 2 h j~l ] + ... +(a

m

+

Itja,)u Jm m

,L

J~ l

in the t-adically complete ring generated the left ideal Case II. II c A

r> 1.

is

II

I

"r"

for the left ideal

so that by the inductive assumption, there

exists a finite sequence

the ideal

as asserted.

Then the integer

r-l,

that generate

I,

Therefore

A.

vI""

,v

k

E II

of elements of

as left ideal in the ring

in the left Noetherian ring

generated, there exists an integer

AltA

m> 0

tvl, ... ,tv fact,

let

k

x E I.

I

Then since

ideal in the ring

EI

generate

generate

xl E A

I

generate

as left

a , ... ,am E I l

A.

generate b , ... , b

l

k

EA

in Equations (1) and (2)

I

In

such that

there exist elements

as left ideal in

such that A.

imply that

completing the proof that as left ideal.

as left ideal.

AltA, there exist elements

Since

I

I

u ' ... , urn l

in

A,

is finitely

We claim that the elements

of the ideal

and an element

Since also

and elements

such that the images as Ie ft idea 1.

A.

II

u ' ... , urn' l

tv l' ••. , tv k E I

generate

Q.E.D.

Finite Generation

611

The main theorem is Theorem 1. center of

Let A.

A

be a ring and let

Suppose that

A

t

be an element in the

is t-adically complete.

be a cochain complex of left A-modules,

Let

C*

indexed by all the in-

tegers, such that

gers

(1)

m C

(2)

(multiplication by

m.

is t-adically complete, all integers

Let

n

t) :C

m

m ..,. C

be any fixed integer.

m,

and

is injective, all inteThen if

is finitely generated as left A/tA-module, and if the ring

A

is left Noetherian, then is finitely generated as left A-module.

Also, when this is the case, there exists an integer that the six equivalent conditions of Proposition fixed integer (2)

n,

such that

Remark.

a

such

hold for the

and in particular

There exists a fixed integer

that

a

r>

uEHn(C*),

tiu=o

r >0

(depending on

for some

i~O,

n)

implies

tru = O.

Clearly, if the ring

A

is left Noetherian, then con-

clusion (1) of Theorem 1 implies conclusion (2) of Theorem 1. Corollary 1.1.

Under the hypotheses of Proposition 0, let

an integer such that there exists an integer

r >0

six equivalent conditions of Proposition 0 hold.

n

such that the Then also

is t-adically complete, and the natural homomorphism is an isomorphism (1)

Hn(C*) ':';.lim Hn(C*/tic*).

i>O

be

Chapter 5

612 In addition

liml Hn - l (C*/tiC*) = O. i> 0

(2)

Corollary 1.2.

m i Em (t ),

mE:z,

The hypotheses being as in Corollary 1.1, let be the

E",,-term of the generalized Bockstein

spectral sequence, as defined in Theorem 2 of Chapter 1 for the cochain complex

C*

and the endomorphism "multiplication by

Then for the fixed integer (1)

tin.

n

Hn(C*)/(t-torsion)~ lim E~(ti), i>O

and (2)

liml En - l (t i ) = i> 0

o.

Proof of Theorem 1 and of Corollary 1.1:

By Proposition 3 of

Chapter 1 and Corollary 3.1 or Corollary 3.2 of Chapter 1, the condition

(*n)'

and the fact that the ring

A

is left Noether-

ian, imply that the six equivalent conditions of Proposition 0 above hold, for some integer r > 0,

r >

o.

(The boundaries,

being an ascending sequence of

n E~ = H (C* /tC*) , of Theorem 1.

must stabilize).

(A/tA) -submodules of

This proves the last sentence

Assume now that we have the hypotheses of Corollarl

1.1.

Consider condition (5) of Proposition 2 of Chapter 4 (IV. 2. 5) By condition (2) of Proposition 0 above, if we let be the inverse system in (IV.2.5), then in

Hn(C*)),

i.::.r,

and the map

,a .. ). '>0 1J 1,J_ A. = (precise tr-torsion (A.

1

1

ai+j,i=(mulitiplicationb Y

Finite Generation t

j

)

=0

whenever

j

~ r,

i > O.

613

Therefore

" A, = I'*m 1 A, = 0 , I ~m

i>O

i>O

l

l

so that condition (IV.2.5) above holds. ditions Hn(C*)

of Proposition 2 of Chapter

Therefore all the con-

4 hold.

is t-adically complete (condition

of Chapter

In particular,

(1) of Proposition 2

4), and conclusions (1) and (2) of Corollary 1.1

above follow from cor.ditions

(3) and (6), respectively, of

proposition 2 of Chapter 4

This proves Corollary 1.1.

It re-

mains to verify conclusion (1) of Theorem 1, under the strong assumption (*n)' (4)

In fact,

Hn(C*)

is a t-adically complete left A-module

(by Corollary 1.1), and the long exact sequence of cohomology: n-l d_ _"H n (C*) ..!.>H n (C*) implies that

Hn(C*)/tHn(C*)

-+

n Hn (C*/tC*) ~ ...

is isomorphic as

(A/tA)-module

to a submodule of the finitely generated (A/tA)-module Since by hypothesis the ring

A/tA

Hn(C*/tC*).

is left Noetherian, it

follows that (5)

n (A/tA) ® H (C*)

is finitely generated as (A/tA) -module.

A

Conditions (4) and (5) the left A-module In fact, let and

M/tM

is finitely generated.

n M = H (C*) .

Then

is finitely generated as

e , ... ,e E M l h M/tM

Hn(C*)

(on any A-module) easily imply that

be such that the images

as (A/tA)-module.

Then

M

is a complete A-module

(A/tA)-module.

Let

el , ... ,eh E M/tM

el, ... ,e h

generate

generate

M as

614

Chapter 5 (Proof:

A-module.

If

x E M,

x =: aIel + •.• + ahe h (mod tM) , By

inductio~

Xi E M,

i

~

integers

there exist

0,

such that

i > 1.

so that

x

a

(i)

l

then 3 a , ... , a E A such that h l say X = a e + ••. + ahe + tx . h l l l (i)

, ... , ~ (i)

xi = a l

E A,

l

~

(i)

e l + .•• + a h

0,

and all

e h + tx i + l ,

Then

is in the A-submodule of

M generated by Q.E.D.

e , .•. , e ) . l

h

Remarks.

1.

Let

A

be a ring with identity and let

an element in the center of the ring

A.

tEA

be

Suppose also that the

several equivalent conditions of Proposition 0, or of Corollary 3.1 of Chapter 1, hold (equivalently, that conclusion (2) of Theorem 1 holds).

(It suffices, e.g., that

A

ian and that the image of the natural mapping:

be left NoetherHn(C*) -T Hn(C*/tC*

be finitely generated as (A/tA)-module; or weaker that

A

be not

necessarily left Noetherian, and that the image of the t-torsion under the natural mapping: genera ted as

Hn(C*) -THn(C*/tC*)

be finitely

(A/tA)-module (since then

ously stabilizes)).

obvi-

r ~ 0,

Then the proof of Theorem 1 above shows

that the following two conditions are equivalent: (1)

Hn(C*)

(2)

the image under the natural homomorphism

is finitely generated as an A-module, and

Hn(C*) -THn(C*/tc*)

is finitely generated as

(A/tA) -module. 2. (*), n

Suppose that the hypotheses of Theorem 1, except hold, and that

A

is left Noetherian.

Let

n

possibl~

be a fixed

Finite Generation integer.

615

Then the following two conditions are equivalent:

(1)

Hn(C*)

is finitely generated as left A-module.

(2)

The image under the natural mapping: n

n

p :H (C*) .... H (C*/tC*)

is finitely generated as A-module. Proof:

(1)

since

(A/tA)

image of

p

~

(2)

is obvious.

Conversely, assume (2).

is Noetherian, every

Then

(A/tA)-submodule of the

is finitely generated as

particular the image of the restriction

(A/tA)-module, and in T

of the natural map

p,

T:{t-torsion in

Hn(C*)} .... Hn(C*/tC*).

But (Chapter 1, Corollary 1.1) the image of

T

is

Boo (E~)

in

the generalized Bockstein spectral sequence (for the cochain complex

C*

and the endomorphism "multiplication by is finitely generated as

fore

fore there exists an integer

r

t").

There-

(A/tAl-module, and there-

such that

Therefore the conditions of Proposition 3 of Chapter 1 hold, so that by Remark 1 above,

Hn(C*)

is finitely generated as

A-module.

Q.E.D.

Proof of Corollary 1.2: Chapter 4 ter

In fact, the hypotheses of Theorem 1 of

hold, so by conclusion (1) of Corollary 2.1 of Chap-

4, we have conclusion (2) of the Corollary.

lary 1.1 of Chapter

Also, by Corol-

4, we have the canonical isomorphism of

A-modules (3)

[Hn(C*)/(topological t-torsion) 1'" lim

i>O

E~(ti).

Chapter 5

616

To complete the proof of conclusion (1) of this Corollary, it therefore suffices to show that (4)

{topological t-torsion in {t-torsion in

Hn(C*)}

=

Hn(C*)}.

But in fact, by conclusion (2) of proposition 0, (5)

n H (C*) } =

{t-torsion in

{precise tr-torsion in for some positive integer Corollary 1.1,

Hn(C*)

t-adically Hausdorff,

r

Hn(C*)}

(depeding on

n).

But since by

is t-adically complete, and therefore {precise tr-torsion in

Hn(C*)},

being

the kernel of the (continuous) homomorphism of Hausdorff topological groups,

(multiplication by

r t ): Hn(C*) +Hn(C*},

it

follows that (6)

{precise tr-torsion in in

Hn(C*).

Equations (5) and (6) imply that t-adically closed in t-torsion")

Hn(C*)} is t-adically closed

Hn(C*),

{t-torsion in

Hn(C*)}

is

and (by definition of "topological

this implies equation (4), and therefore the Corol-

lary. Can Theorem 1 be generalized to cochain complexes of A-modules that are t-adically complete, but such that the endomorphism, "multiplication by

t"

is not injective?

TheDefini-

tion and Proposition that follows, about percohomologx grouvs, answer

th~s

question in the affirmative.

First, let us introduce some terminology.

A

(~-indexed)

Finite Generation cochain complex spectively: i that e = 0

C*

617

of right A-modules is bounded below, re-

bounded above, iff there exists an integer whenever

cochain complex

C*

i

< N,

respectively:

i > N.

n.

The cochain complex

extremely right flat iff there exists a directed set (ei,aij)i,jEI

such

A (.?"-indexed)

will be called right flat iff every

right flat, for all integers

direct system

N

en

C*

is and a

I

of right flat cochain complexes

of right A-modules, such that each of the cochain complexes is bounded above, for all e* "" lim

HI

is

i E I,

C~

].

and such that

C~

].

as cochain complexes of right A-modules.

Thus, every extremely

right flat cochain complex is right flat: but as we shall see (Example 2 below), the converse is in general false. Definition.

Let

A

be a ring with identity, let

e*

be an

arbitrary (Z-indexed) cochain complex of right A-modules and let

M be any left A-module.

groups of e* n H (e* ,M) ,

with coefficients in

n E Z,

as follows.

Then we define the percohomology M,

which we denote as

or sometimes more precisely

H~ (e* ,M) ,

n E Z,

We give three definitions (Definition 1, Definition

2, and Definition 3) which we prove below yield canonically isomorphic groups (and independent, up to canonical isomorphisms, of any choices made in these definitions). Definition 1.

Let

'e*

be another (Z-indexed) cochain complex

of right A-modules such that the cochain complex

IC*

i5 cx-

tremely right flat, and such that we have a map of cochain complexes

618

Chapter 5 CP*:'C*->C*

such that 'C*

Hn(CP*) is an isomorphism, all integers

(Such a

always exists - e.g., use a double complex that is a projec·

tive resolution of let

n.

'C*

C*, in the sense of [C.E.H.A.], pCJ.363, then

be the associated singly graded cochain complex). (*)

(It is easy to see that below, even if

C*

'C*

may of necessity not be bounded

is non-negative.) n

H~(C*,M) =H ('C*3M),

(1)

Then define

all integers

n.

A

Definition 2. M.

Define

Let

Dn=P

-n

P* ,

be a flat resolution of the left A-module all integers

n

(thus, the cochain comple)

D* is non-positive), and define (2)

n

H~(C*,M) =H (C*I3D*),

all integers

n.

(**)

A

Definition 3.

Choose

is an isomorphism, nit ion 2.

'C*

n c"if,

and

cjJ*: 'C* ->C*

and choose

P*

such that and

D*

Hn(CP*)

as in Defi-

Then define

H~(C*,M) =H n ('C*3D*), A

all integers

n. (*** )

(*) This construction appears in the (lengthy) footnote to Remark 4 following Theorem 2 of Chapter 1, construction number (1) •

(**) Notice that Definition 2 generalizes the one presented,construction (2),in the footnote to Remark 4 following Theorem 2 of Chapter 1. (***) Notice that Definition 3 generalizes the one presented, construction (3), in the footnote to Remark 4 following Theorem 2 of Chapter 1.

Finite Generation

619

Then, by methods similar to those used in the footnote to Remark 4 following Theorem 2 of Chapter 1, it is easy to show

H~(C*,M)

that the above three definitions of

give canonically

isomorphic definitions, and are independent of all choices. Proof:

We first prove a lemma.

Lemma A. Let

A

be a ring with identity, let

R*

be a

dexed) cochain complex of right A-modules and let

(Z-in-

p*: S* .. T*

be

a mapping of (Z-indexed) cochain complexes of left A-modules. Suppose that (1)

Rn

is flat as right A-module, all integers

n,

and

that (2)

Hn(P*): Hn(S*) -+Hn(T*) gers

also

(3)

(4)

n.

Suppose

that either and

is an isomorphism, all inte-

T*

R*

is bounded above; or that both

are bounded below.

S*

Then

Hn(R* ~ p*): Hn(R* ~S*) ->Hn(R* ~T*) A A A is an isomorphism of abelian groups, all integers

~.

If we delete hypothesis (3), but instead strengthen hypo-

thesis (1) to read: (lxtr)

R*

is extremely flat,

then again conclusion (4) of the Lemma holds. Proof. Case 1.

In addition,

S*

and

T*

Then there exists a positive integer for Sn

n.

n > N + 1;

= Tn = 0

for

and such that either n < -N - 1.'

are both bounded above. N n R

such that

=0

for

Sn = Tn = 0

n ~N + 1;

or

620

Chapter 5 Then

R* 13 S*

and

A

groups. R*

(9

R* 13 T*

are double complexes of abelian

A

The first spectral sequence of the double complex:

S*

is such that

A

EP,q = R P 13 sq o A'

dP,q=RP@d q o A S*'

all integers

p,qE?'.

(E.g., see Introduction, Chapter 2, section 10.

Or [P.P.W.C.],

Chapter I). RP

Since

(5)

is flat as right A-module, it follows that q Ei' q "" RP 13 H (S *), A

all

p, q E l' •

Similarly, the first spectral sequence of the double complex of abelian groups

R*

(9

T*

is such that

A (6 )

EP,q ~ RP 13 Hq (T*) , 1 ~

all

p,qEl'.

A

The cohomological, doubly graded spectral sequences (5) are confined to the region: -N.::.q'::'+N,

p,q '::'N,

and have for abutments:

and (6)

or to the region Hn(R*(>:lS*),

nE?',

and

A

Hn(R* @T*),

n EU',

respectively.

Since the indicated regions

A

are such that, for every (p,q)

on the line

n,

p + q =n

there are only finitely many such that

it follows

(see Introduction, Chapter 2, section 10) that the filtrations on these abutments are finite.

p~

induces a mapping from the

spectral sequence (5) into the spectral sequence (6).

By hypo-

thesis (2) of the Lemma, and by equations (5) and (6), the map induced by

p*

betweeen these two spectral sequences induces

an isomorphism along

and therefore along the abutments,

completing the proof of Case 1.

Finite Generation

Case 2. and

General Case.

T*(N)

621

For each integer

N

let

be the sub-cochain complexes of S*

S*(N), and

T*,

re-

spectively, such that

n'::'N - I, n = N, n~N+l,

n'::'N - I, n = N,

n>N+l. Then the restriction

p*(N)

of

p*

to

S*(N)

is a mapping

of Z-indexed cochain complexes of left A-modules from into

T*(N),

and the cochain complexes:

and the map of cochain complexes:

R*, S* (N),

p*(N)

S*(N) T* {N)

obey the hypotheses

(I), (2) and (3) of the Lemma, and also obey the hypotheses of Case I, for all integers

N.

Therefore, by Case I, we have that the

homomorphism of abelian groups n n H (R* 0 p* (N) ): H (R* ~ S* (N» A A is an isomorphism, all integers limit in equations (4 ) N

as

N -+ +

n. 00

-+

n H (R* 0 T* (N) ) A

Passing to the direct we obtain that the homomor-

phism (4) is an isomorphism, all integers

n,

(since both coho-

mology of cochain complexes of abelian groups, and tensor products, commute with direct limits over directed sets), which proves Case 2. Proof of Note.

Let

I

be a directed set and

(R~,ao 0) 0 ~

~J

0EI

~,J

a direct system of right flat cochain complexes, each bounded above, such that

R* '" lim Rr •

iEr

~

622

Chapter 5

Then for every

Ri'

i t= I,

hypotheses of the Lemma.

8*,

T*

and

iEI

T*

-+

obey the

Therefore

is an isomorphism, for all integers limit for

p* :8*

n.

Passing to the direct

in equations (4 ), we obtain equation (4). i Q.E.D.

Remark:

If we simply delete hypothesis (3) of Lemma A, then

the Proof of Case 1 of Lemma A, together with Prop. 4 of Introduction, Chapter 2, section 8, shows that the induced mapping of cochain complexes of abelian groups: q ([

(RPOS )]

•

p+q=n

A

•

P2.0

[

IT

(R

p+q=n p>O

P

0 sQ) ] )

---i>

A

nE~

induces an isomorphism on cohomology in dimension integers

n.

n,

for all

(See also Introduction, Chapter 2, section 10.)

We now resume the proof that Definitions 1,2, and 3 of Hn(C*,M) E.g., let

yield canonically isomorphic groups, all integers 'C*

and

as in Definition 2.

~*

be as in Definition 1 and let

Then (taking

cochain complex such that

o T = M,

D*

n. be

R*= 'C*, S*=D*, T* = the i

T = 0,

all

i

~ 0)

by the

Note, Lemma A, we have that the natural mapping is an isomorphism: n n H (' C* 0 D*) ~ H ( 'C* ~ T*) , A A

623

Finite Generation all integers

n.

Otherwise stated, Definitions 1 and 3 give

canonically isomorphic groups. the ring equals

A

On the other hand (replacing

by the opposite ring

y.x

in

A, all

such that

x,y€A),

as in Definition 3 then letting

if

'e*, ¢*

R* = D*,

and

p* = ¢*

D*

are

in the Ler:una,

we have that the natural mapping is an isomorphism: Hn ( I e *

@

D*) ':;. Hn (e *

A

all integers

@

D* ) ,

A

Otherwise stated, Definitions 2 and 3 give

n.

canonically isomorphic groups. Finally, Definition I is independent of the choice of 'e*, ¢*. any

D*

(Proof:

Given another such, say

as in Definition 2.

1, using either

'e*, ¢*

or

'e*, ¢*

then choose

Then we have shown that Definition "e*, T*,

to Definition 2 using the fixed independent of

"e*, T*,

D*.

is canonically isomorphic Therefore Definition I is

up to canonical isomorphism).

Similarly,

Definition 2 (respectively: 3) is likewise independent of the choice of

0*

(respectively:

If we fix

e*

and let

of

'e*,¢*,D*).

Q.E.D.

M vary, then the assignment:

M~H~(e*,M)

is an exact connected sequence of functors: and the

assignment:

e*-tHn(e*,M)

is also such a sequence.

every (z-indexed) cochain complex of right A-modules for every left A-module

Also, for e*

and

M we have the universal coefficients

spectral sequence, a left half plane spectral sequence with

E~,q = Tor~p (H q (e*) ,M)

and abutment

H~ (e* ,M) .

(The abutment

always has a discrete filtration such that the union of all the filtered pieces is all of

H~(e*,M)}.

(We sometimes call this

spectral sequence the first spectral seguence of percohomology).

Chapter 5

624

Proof: If

C*

A-modules, then regard

M,

-n

complex of right

as a cochain complex in the usual

,

d

n

=d

Then for any left A-module

-n

-n

A

define

C*

n C =C

way, by defining

(~-indexed)chain

is a

Hn (C*,M) = HA (C* ,M),

all integers

n.

A

Then if

denotes the abelian category consisting of all non-negative chain complexes of right A-modules, then by [RP.WeJ,chapter I, section 1, Theorem 2, pg. 115, we have that

A has enough projectives.

Also, if we fix a left A-module

then the assignment:

n is a system of left

~

M,

0,

A.

derived functors on

(Proof:

clearly an exact connected sequence of functors.

If

It is C* E A

is

projective, then by [P.P.W.C.], Chapter I, section 1, Theorem 2, we have that

Ci "" Di Ell Di - l ,

i

~

0,

where

Di

is projective,

i

~l,

in such a way that the boundary map:

d :C -+ C _ corresponds i i i l D Ell D projection~D inclusion~D E& D i-I i-l i-I i-2' i

to the composite: all integers i

~

1.

It follows readily that

Hn(C*@M) = 0,

n~l.

A

But since

C.

is projective,

1

i

~

0,

C.

is flat, all integers

1

so that by Definition 1 ( (c- n ) nE.?' is non-pos., so bdd. abOVE

i ~ 0, A

H (C*,M)=H (C*@M), n

n

all integers

all integers

A

n

~

the axioms (Hl),

1,

all

(H2),

C* E A

n.

Therefore

that are projective.).

From

(H3), Chapter I, section 2, of [RP.we.],

pg. 117, it follows that, in the terminology of Chapter I, n ~ 0,

section 2, of [PE.WCJ that the assignment:

are the right hyperderived functors of the right exact functor ~

N®M

from the category of right A-modules into the category

A

of abelian groups.

Therefore,

[P~WCJ,Chapter

I, section 2,

Theorem 1, pg. 118, we have two spectral sequences.

The second

625

Finite Generation of these spectral sequences is a first quadrant homological spectral sequence:

If now regard

C*

is any non-positive cochain complex, then we

C*

as a non-negative chain complex in the usual fashion, -n (by requiring that C = C ,etc. ) . Then the indicated homon

logical spectral sequence becomes a third quadrant cohomological spectral sequence, with A (Hq(C*) M)= Hn(C* M) EP,q=Tor 2 -p , A" This proves the assertion in the case that Suppose now that

C*

is a

~-indexed)

C*

is non-positive.

cochain complex of right

A-modules that is bounded above (i.e., such that there exists an integer

N

such that

i C = 0,

i

all

~ N + 1) •

Then making appro-

priate dimension shifts, we obtain the indicated spectral sequence from the case that C*

be an arbitrary

modules.

~-indexed)

For each integer

complex of

C*

C*

N,

is non-positive.

Finally, let

cochain complex of right Alet

C*(N)

be the subcochain

such that

ci ,

I

=

1m (d

i < N,

N- l CN-l ;

~

CN) ,

i

= N,

i > N.

0,

Then we have the indicated spectral sequence for the cochain complex limi t at

C'(N)' N ++00

for each integer

N>

o.

Passing to the direct

gives the desired univeral coefficients spectral

Chapter 5

626

sequence for

C*,

a left half plane cohomological spectral

sequence, such that the filtration on the abutment is discrete (i.e., such that zero is a filtered piece of the n'th group H~(C*,M)

of the abutment, all integers

n),

and such that the

union of the filtered pieces of the n'th group of the abutment, H~(C*,M),

Example 1. flat over

H~(C*,M),

is all of If

C*

A,

all integers

is extremely flat over

Q.E.D.

n.

A,

or if

M

is

then

H~ (C* ,M) '" H~ (C* 13 M),

all integers

n.

A

(Whether or not the hypotheses of Example 1 hold, the groups on the right side of this equation were called traditionally the cohomology of

C*

call the ones

H~(C*,M)

efficients in

M,

Example 2. of

Let

with coefficients in

n.

percohomology groups of

C*

with co-

to avoid any confusion.) A = '1' / 4;;",

('1' / 4'1') -modules wi th

all integers

M, which is why we

Then

and let n

C = '1' / 44' , C*

C* d

n

be the cochain complex = (mul tiplica tion by 2),

is flat over

A.

Let

M=O'/2O'.

Then it is easy to see that n H (C*) = 0,

all integers

Therefore (e.g., taking

n.

'C* = (the zero cochain complex) in

Definition I, or if one prefers using the universal coefficients spectral sequence) we have

H~ (C* ,M) = 0,

Hn(C* 60 0'/40'

for all integers

0'/20') "'7/20',

n.

all integers

However,

n.

Finite Generation Therefore, the condition tion 1 (and likewise on weakened to "flat".

627

"extremely flat" on C*

'C*

in Defini-

in Example 1) cannot in general be

Also, by Example 1, it follows that

an example of a flat cochain complex of

C*

is

(7/47)-modules that is

not extremely flat. Example 3.

Suppose that

exists an integer

N>0

A

Tor + ( ,M) N l

M

is a left A-module such that there

such that

,,0.

Then in Definition I, the condition on extremely flat", Proof:

can be weakened to,

'C*, "'C*

that "'C*

is

is flat".

As in the proof of Proposition 2.1, P9. 110 of [C.E.H.A.]

(with "flat" replacing "projective" and "Tor replacing "Ext"), one sees

. 1 y th a t

eas~

" Tor A+ ( , M) = - 0" N l

is equivalent to:

"There exists an acyclic, flat homological resolution M

such that

0*

for this

l

P*

Now let

Then if

A-modules and

'C*

CP*:' C* .... C*

right A-modules such that integers

n,

i>N+I".

C*

of

Therefore, constructing

as in Definition 2 above, we have that

is bounded below. A-modules.

for

P. = 0

P*

0*

be any cochain complex of right

is any flat cochain complex of right any map of cochain complexes of Hn(CP*)

is an isomorphism, for all

then by Definition 3 above, we have that

H~(C*,M) =Hn('C*~D*),

(1)

A

nEz. (where

But, by Lemma A, with M*

i"l 0), since

R*='C*,

5*=0*

is the cochain complex such that M*

and

0*

and

T*=M*

o

M =M,

are both bounded below, we have that

the natural map is an isomorphism:

628

Chapter 5 (2)

H

n

n ( I C* 0 D*) "';. H ( I C* 0l M*) , A A

all integers

n.

Equations (1) and (2) prove the Example. Q.E.D.

Example 4. let

Let

C*

be a cochain complex of right A-modules and

M be a left A-module. p Tor (C ,M) = 0,

(1)

wi th

q

~

q

1,

Suppose that for all integers

p,q

and in addition that

either

(2a) C*

is bounded above

or

(2b) There exists a positive integer

N

such that

Tor~+l ( ,M) ::: O. Then the natural mapping is an isomorphism, H~ (C* ,M)

"';.

Hn (C* 13 M) , A

for all integers Note.

n.

If we delete the hypotheses (2a),

(2b), but instead

strengthen (1) to read (lxtr)

C*

is isomorphic to the direct limit, over a

directed set, of cochain complexes each of which is bounded above, and all obeying hypothesis (1), then the conclusion of this Example continues to hold. proof:

Let

P*

be a flat resolution of

(2b) holds, then we can choose i > N + 1. 2, and let

Let M*

D*

P*

M.

If condition

such that

be constructed from

P*

P. = 0 l

for

as in Definition

be the cochain complex such that

M

o =M,

i M = 0,

Finite Generation

629

Then exactly the same argument as the Proof of Lemma A,

itO.

Case 1, shows that the mapping of cochain complexes: C* 13 0* A

-+

C* 13 M* A

induces an isomorphism on cohomology.

(And if we have the

hypotheses of the Note to the Example, then the Proof of the Note to Lemma A proves this assertion.)

But by Definition 2,

n

H~(C*,M) =H (C*I9D*), A

so this latter observation proves the Example. Remarks 1.

Notice, by Definition 1, respectively:

2, that for every integer

n

M~>H~(C*,M), respectively:

C*, respectively:

M,

Definition

we have that the assignment:

C*~>H~(C*,M), preserves direct

limits over directed sets, and is also a half where

Q.E.D.

exact functor;

runs through the abelian category

of cochain complexes of right A-modules, respectively: the abelian category of all left A-modules, and C*,

cjJ*: C*

0* -+

Also, it is clear from Definition 1, that if

C*

are cochain complexes of right A-modules, and if 0*

is a map of cochain complexes such that

an isomorphism, for all integers

H~ (cjJ* ,M): H~ (C* ,M)

-+

n H (cjJ*)

is

n, then

H~ (0* ,M)

is an isomorphism, for all integers

M.

respectively:

is fixed. 2.

and

M,

n,

and all left A-modules

630

Chapter 5

Lemma 2.1.1. ~-indexed

Let

A

be a ring with identity, let

C*

cochain complex of right A-modules and let

left A-module.

Let

'C*

be a

right A-modules, and let

~-indexed

¢*:' C* -+ C*

A

Tor + ( ,M) ::: 0 N l A n Tor ('C ,M) = 0, p

cochain complex of

Hn(¢*):Hn(,C*) -+Hn(C*)

n,

such that

for some integer

all integers

M be a

be a map of cochain com-

plexes of right A-modules, such that is an isomorphism, all integers

be a

and such that

N':' 0,

n,

all integers

p':' 1.

Then there are induced canonical isomorphisms,

H~ (C* ,M)

':-

n H (' C* t9 M) ,

all integers

n.

A

Proof:

Follows immediately from Remark 2 above and Example 4.

Corollary 2.1.1.1.

Let

A be a ring with identity and let

be a non-zero divisor in the center of the ring a

A.

(z-indexed) cochain complex of right A-modules, and

Let

t C*

be

'C*

be a (z-indexed) cochain complex of right A-modules and let ¢*: 'C* .... C*

be a map of (z-indexed) complexes of right A-modules is an isomorphism, all inte-

gers

n,

and such that the right A-module

t-torsion, all integers

n.

'c n

has no non-zero

Then there are induced canonical

isomorphisms, n

H~ (C* ,A/tA) Q:' H (' C* @(A/tA», A

Finite Generation all integers Proof:

631

n.

Since the element

with identity

A

t

is in the center of the ring

and is a non-zero divisor, it is easy to see

that, for every right A-module

C,

we have that

C/tC,

i = 0

I

A

Tor. (C,A/tA) '= (precise t-torsion part of ~

C) ,

0,

(E.g., see Chapter proof).

i :> 2.

8,

Lemma 2, in the case

r =1

for a

Therefore the Corollary follows form Lemma 2.1.1.

Example 5.

Let

C

a cochain complex

be any right A-module. C*

Regard

C

as being

of right A-modules by defining

n'=O

Cn = {C, 0,

Let

i '= 1

n

Ji O.

M be any right A-module.

The~e.g.,

by Definition 2,

A

n

HA (C*,M) '" Tor __ (C,M), n all integers ring

A,

integers

n.

Notice that these depend very strongly on the

and that the groups may be non-zero for some negative n

(but in this case vanish for all positive integers

n) •

Remark 3. If M

is a cochain complex of left A-modules and if

is a right A-module then we have similarly the percohomology

groups where AO

C*

H~ (M, C*). Explicitly, these are defined to be H~O (C* ,M) , AO is the opposite ring of A, such that X ' Y in

equals

y' x

Proposition 2.

in Let

A. A

be a ring with identity and let

be a non-zero divisor in the center of

A

such that

A

tEA is

632

Chapter 5

t-adically complete.

Let

C*

be a cochain complex of left

A-modules, indexed by all the integers, such that m C

(1) Let

n

is t-adically complete, all integers

be a fixed integer.

n (* ) H (A/tA C*) n

A

and if the ring

Then if

is finitely generated as

'

m.

(A/tAl-module,

A is left Noetherian, then is finitely generated as A-module;

and there exists a fixed integer tha t conditions (2), Proof.

r > 0

(depending on

n)

such

and (6) of Proposition 0 hold.

(4)

The proof of the Proposition makes use of Corollary 3.1

below. Let

'C*

be a

(z-indexed) cochain complex of left A-modules

such that (multiplication by t): integers

n,

'C

n

-+ 'C

n

is injective, all

and such that we have a mapping

q,*: 'C* -+C* of cochain complexes of left A-modules such that isomorphism of left A-modules, all integers n C

q,*

(C*ll\t=c*.

Therefore, if

under the functor

Corollary 3.1 below,

"At"

(q,*ll\t

('1>*)l\t

n.

n, we

denotes the image

(=t-adic completion), then by

induces an isomorphism on coho-

mology in all dimensions,

all integers

is an

Then, since

is a t-adically complete left A-module, all integers

have that of

n.

Hn(q,*)

By Corollary 2.1.1.1 above,

Finite Generation

H~(A/tA,C*) '" Hn('C*/t.'C*), But

'C*/t.'C*'" ('C*)At/ t . (,c*/,t.

But, by Corollary 3.1, integers

n.

~(A/tA,C*)

('Cn)At

633

all integers

n.

Therefore

has non non-zero t-torsion, all

But by hypothesis the left (A/tA)-module is finitely generated, for a fixed integer

n.

Therefore by equation (4) the left (A/tA)-module Hn(A/tA) 0 ('C*)At) A

is finitely generated for the fixed integer (Z-indexed) cochain complex

('C*)At

n.

But then the

of left A-modules obeys

all of the hypotheses of Theorem 1, including

(*n)'

Therefore,

by Theorem 1, is finitely generated as A-module, and we have that (9)

There exists an integer

r'::' 0,

depending on

n,

such

that all the conditions of Proposition 0 hold for the cochain complex

('C*)At,

and in particular conditions (2), (4) and (6)

of Proposition 0 hold for

('C*)At.

Considering the equation

(3), we see that equations (8) and (9) imply, respectively, the conclusion (1) and the latter conclusions of this Proposition. Q.E.D. Remark 1.

Slightly more can be said under the hypotheses of

Proposition 2 above.

(See Remark 2 following Proposition 0'

at the end of this chapter).

Chapter 5

634 Remark 2.

Let

C*

A-modules, where left A-module.

be a A

(Z-indexed) cochain complex of right

is any ring with identity.

Let

M

be any

Then in addition to the "universal coefficients"

spectral sequence above, there is also induced a spectral sequence, confined to the lower half plane, such that

We call this spectral sequence the second spectral sequence of percohomology.

Suppose that either (1) the cochain complex

is bounded above (i.e., that there exists an integer

A

q 2:D + 1.

H~ (C* ,M)

of

H~(C*,M)

,

all integers is

either

C*

holds,

H~ (C* ,M) ,

EP,q

(finite, descending) filtration n,

such that the associated graded

(The condition, for an integer

H~(C*,M)

H~(C*,M)

be finite, means, as usual,

is a filtered piece, and also

that the zero group is a filtered piece, of First, consider the case in which

Then making the usual notation shift: C*

A-modules.

(I.e., when

n E: Z, is an abutment for the second spectral

that the whole group:

regard

all integers

is bounded above or condition (2) above on the Tor's

that the filtration on

Proof:

n,

of this spectral sequence.

sequence of percohomology). n,

all integers

q

Then there exists a

on

such

or (2) that there exists an integer

n

Tor (C ,M) =0,

such that

D

i.:: N + 1)

for

that

N

C*

C

H~(C*,M». C*

= C- n

is non-positive. d

= d- n

n ' n as being a non-negative chain complex C*

we can ' of right

But then, we have observed, in the course of estab-

lishing the universal coefficients spectral sequence (see just after Definition 3 of percohomology), that in the notations of the proof of the universal coefficients spectral sequence,

Finite Generation for any fixed left A-module A

C*....N> H (C*,M), n

n.:: 0,

M,

635

the assignment:

from the category of non-negative chain

complexes of right A-modules into the category of abelian groups, are the right hyperderived functors of the right exact functor: N- N 6 M

from the category of right A-modules into the category

A

of abelian groups.

Therefore by

[P.P.W.c.J, Chapter I, section 2,

Theorem 1, pg. 118, we have two spectral sequences.

The first

of these is a first quadrant, homological spectral sequence such that E

1 A =Tor (C ,M), p,q q p

A Hn(C*,M),

and abutment

Therefore

n>O.

Rewriting in cohomological

notation, this gives the desired cohomological spectral sequence (a third quadrant cohomological spectral sequence) in the case that

C*

If now

is a non-positive cochain complex of right A-modules. C*

obeys hypothesis (I), i.e., is bounded above, then

shifting degrees we again obtain the spectral sequence (0), (with abutment n E: Z). C*

Hn(C* M) A

'

,

n EZ,

having finite filtration, each

That proves the assertion under hypothesis (1).

is any (Z-indexed) cochain complex of right A-modules, then

define

C(N)

to be the sub-cochain complex such that

ci

,

C~N) = 11m (dN- l :

N l

c -

+

cN~

0,

Then

C'(N)

obeys condition (1), all

i
i=N i>N+l. N E: Z ,

so we have the

second spectral sequence of percohomology for gers

If now

N.

Passing to the direct limit as

half plane spectral sequence such that

N+ +

C'(N)' OD

all inte-

gives a lower

Chapter 5

636

EP,q = HP (Tor A (C* ,M», -q

2

all integers

p,q.

If condition (2) should hold, then the

second spectral sequences of percohomology of efficients in

M

C(N)

are all confined to the strip

(See Note 3

the spectral sequence. Notes 1. i.e., if

nEZ,

Notice that, if condition i

c

=

°

for

i.:::.N + 1,

M

each

below~

Q.E.D.

(1) should hold for

for some integer

is confined to the region

It

serve as an abutment for

second spectral sequence of percohomology of cients in

-D.2. q.2. 0.

H~(C*,M),

follows readily that in this case the groups with a finite filtration, each

with co-

then the

N

C*

C*--

with coeffi-

p'::'N, q.::.O.

(This can

be turned into a first quadrant homological spectral sequence after re-indexing). 2.

A

If

n

C*

Tor (C ,M) = 0, q

some integer

is such that condition (2) holds (i.e., all integers

O},

n,

q.:::. D + 1,

all integers

for

then the second spectral sequence of perco-

homology is confined to the region

-0,::, q.::.

°

(an "infinite

horizontal strip"). 3.

Notice that, in discussing the universal coefficients

spectral sequence, and the second spectral sequence of percohomology, that we took direct limits of spectral sequences of abelian groups over the directed set of positive integers.

Per-

haps some discussion is in order. Let (IN)

D

be a directed set, and for every

(EP,q (N), dP,q (N), ,p,q (N» r r r p,q,rEz, r>2

NE 0 be a

let (whole plane),

doubly graded cohomological spectral sequence of abelian groups defined for

r 22,

with

E -term 00

EP,q(N). 00

(,** (N) r

denotes

Finite Generation the isomorphism from the cohomology of r ~2.)

Suppose also that, for every

637

E;*(N) nEz,

E;~l(N),

onto

Hn(N)

is a fil-

tered object with decreasing filtration together with isomorphisms:

all integers

p,q(oZ, n=p+q,

Suppose that, whenever the spectral sequence and a map

SN,M

(IN)

in

NED.

0,

we have a map

into (lM),of bidegree

aN,M

(2 ) into N Suppose also that the maps aN,M'

) M

all

such that

N < M < R.

TP,q=limTP,q(N), r NED r then

(2

aN,N = identity of (IN)'

and such that M,N,R ED

from

(0,0) ,

of filtered objects from

that is compatible. are such that

M> N

all

Then if we define

all

p,q,rEz

(EP,q dP,q, p,q) r ' r 'Tr p,q,rE;z, r>2

with

r>2.

is a (whole plane), doubly

graded, cohomological spectral sequence, and of course is the direct limit of the spectral sequences

E~,q

denotes the

Eoo-term of this spectral sequence, then it is

not in general true that

It is true that the permanent boundaries in limit of the permanent boundaries in wi th

r':' 2,

EP,q r

E~,q(N),

all

but the permanent cycles in

the direct limit of the permanent cycles in

is the direct p,q,r EZ

is not in general EP,q(N) r

for

NED.

Chapter 5

638

However, if (4)

For every pair of integers integer

r=r(p,q)

such that

all

there exists an

depending on the pair

r'.:.r(p,q)

d~:q(N) = 0,

p,q,

p,q

implies that

NED,

then the permanent cycles in

EP,q(N)

e.g.

2

the same as the (r-2)-fold cycles,

where

for all

NED

r = r(p,q).

are

Therefore,

if condition (4) should hold, then we get equation (3). Always, if we define (5) Then

~

n n H = lim H (N), NED

all integers

n,

is a filtered abelian group, and

(6)

all integers

p,n.

Thus, if equation (3) should hold (which is

always the case if condition (4) should hold), then we have isomorphisms: (7)

all integers

p,q,n

filtered objects

Hn,

with

n = p + q.

n E;Z,

Thus, in this case, the

serve as an abutment in the usual

sense of the direct limit spectral sequence (EP,q dP,q p,q) r ' r ,T r p,q,rE;z, r>2. Example 1.

If for every integer

n,

there exists an integer

n n p = p(n) such that F H (N) = 0 for all NED, then F H = O. P P n Therefore in this case the filtration on H is discrete, all

Finite Generation integers

639

n.

Example 2.

If for every

NED,

we have that

n U F H ", Hn. pEZ p

then

Example 3. (8)

Suppose that For every integer E"'E(n) that

E

n,

such that p' g' 2

'

there exists an integer

p'

~p(n),

all

(N) '" 0,

p' +g' "'n,

implies

NED.

Then clearly condition (4) above holds (namely, take r(p,g) =E(p+g) -p),

and therefore we have eguation (3), and

therefore also eguation (7), above. The filtration on

(9)

n,

all

Hn(N)

N),

for some integer

n H

is discrete, all integers E P ' ,g' (N) = 0,

In fact, by condition (8),

such that

p' + g' = n,

follows that

all

p'

~p(n).

p' ,g'

Fp,Hn(N) =Fp'+lHn(N),

By eguation (2 ) , it N p' ~l?(n).

all integers n

FE(n)H (N) = 0,

Passing to direct limits, it follows tha t

n

FE (n) H = 0,

all

NED.

all in-

n.

Example 4.

Thus, we see that, if conditions (8) and (9) hold,

and if also the union of the filtered pieces of of

all

n.

00

Therefore, eguation (9) implies that

tegers

depending on

p

then also

(10) The filtration on Proof:

is discrete, all integers

NED,

(This means that nand

If also

n H (N)

(each integer

n,

each

holds (i. e., the f il tered groups

NED), Hn,

n E ;Z

Hn(N)

is all

then equation (7) ,

serve as an abut-

Chapter 5

640

(EP,q dP,q"P,q) r ' r r p,q,rE;;Z,

ment for the spectral sequence

and also conditions (8) and (9) are inherited by the direct limit spectral sequence,

(as well as the condition that the n H

union of the filtered pieces of ger

n E;Z,

n H (N)

if

is all of

Hn,

has this property for all

each inte-

N EO,

see

Example 2 above). (Graphically, condition (8) says that, the spectral se(EP,q(N), dP,q(N},'tP,q(N}) r r r p,q,rE;z, r>2

quences

are zero out-

side of a region:

~c:.;Z

the line

has the property that, "if one proceeds in a

p +q = n

such that, for each integer

x;z,

south-Easterly direction" along the line a

£ = £(n)

such that all spots

outside the region

R

p +q = n,

"South-East" of

(i.e., are such that

n,

one reaches (.!2.,n -.!2.)

EP,q (N) = 0, 2

are

all

N EO).

(1)

Example 5.

In establishing the universal coefficients

spectral sequence for sequences of half plane

p

E (n)

(when

C* (N),

=

~

o.

I, (2)

C*, for

the universal coefficients spectral N ~ 0,

Therefore the hypotheses of Example 4 hold all integers

(;Z-indexed) cochain complex

A-modules, and a left A-module 0

(2)

n).

In establishing the second spectral sequence of

percohomology, for a

integer

are all confined to the left

M,

C*

of right

such that there exists an

such that A n Tor (C ,M) = 0, q

all integers

q

~O

+ I,

all integers n,

the second spectral sequences of percohomology of the cochain complexes -0

~

q

~

o.

C*(N)

are confined to the horizontal strip

Therefore the hypotheses of Example 4 hold once again.

Finite Generation Ern) = n + D + I,

(Take

all integers

641

n).

Therefore, in both cases (1) and (2) above, the respective direct limit spectral sequences are such that the direct limit abutment is compatible,

(i.e., obeys condition (7», and the

direct limit abutment has a filtration that is both discrete and such that the union of the filtered pieces is all of for each integer (4)

n.

In establishing the universal coefficients spectral

sequence for a C*,

Hn,

(~-indexed)

cochain complex of right A-modules

and in establishing the second spectral sequence of per co-

homology for a

(~-indexed)

cochain complex

C*

of right A-

modules obeying the appropriate "Tor" condition, we used the fact that the functor: C*- H~ (C* ,M),

n E ~,

commutes with direct limits over directed sets, for each left A-module

M.

This follows most easily from Definition 2 of

percohomology, since tensor product commutes with direct limits ~emark:

The two spectral sequences of percohomology can be con-

structed alternatively-without passing to direct limits of spectral sequences-by using Intro., Chapter 2, section 10, Ex. 1). (5) on

Of course, the reader should note that the filtrations

H~(C*,M)

induced by the universal coefficients spectral se-

quence, respectively by the second spectral sequence of percohomology, are of course in general different. Example.

Let

t

ring with identity

be a non-zero divisor in the center of the A

and let

M =A/tA.

Then

TO~ (C,A/tA) = 0, A

642

Chapter 5

all integers

n

~

1

TorA(C,A/tA)

and

2,

all right A-modules

C.

~

(precise t-torsion in

C),

Therefore, in this case, the second

spectral sequence of percohomology just described in the case M = A/tA

becomes the long exact sequence: n-l n n !L.-H +l(TOr1(c*,A/tA)) ->-H~(C*,A/tA) ->-Hn(C*/tC*)~ ... , and the other maps are edge homo-

(in which the map morphisms, and where whose n).

"Tor!(c*,A/tA)"

is the cochain complex

n'th term is "(precise t-torsion in

Cn)",

all integers

This long exact sequence can be of use, e.g., in estab-

lishing condition Corollary 2.1. have that

A

(*n)

of Proposition 2.

For example,

If under the hypotheses of Proposition 2, we is left Noetherian, and that

Hn+l(precise t-torsion in

C*)

(A/tA)-modules, then conditon

Hn(C*/tC*)

and

are finitely generated as (*

n

of Proposition 2, and

)

therefore the conclusions of Proposition 2, hold. (The proof of Corollary 2.1 follows from the long exact sequence of the last Example, and from Proposition 2.) Proposition 3.

Let

A

be a ring with identity and let

be an element of the center of the ring is a non-zero divisor.

Let

C*

and

A.

D*

n.

of cochain complexes of left A-modules.

Suppose that

tEA

be (Z-indexed) cochain n C

complexes of the left A-modules such that t-adically complete, all int..egers

tEA

Let

and

Dn

¢*:C* ->- D*

are be a map

Then the following two

conditions are equivalent: (1)

Hn(¢*):Hn(C*) ->-Hn(D*)

is an isomorphism, all integers

n. (2)

The induced map on percohomology modulo

t:

Fini te Generation

H~ (A/tA, CP*): H~ (A/tA, C*) -+ H~ (A/tA, 0*) integers Note:

643 is an isomorphism, all

n.

If we drop the hypothesis that the element

tEA

is a

non-zero divisor, then the Proposition remains true, if in condition (2) we replace

throughout with

IIF; IT] (

,71. IT]/T • 7l. IT]) ".

Proof.

Since

tEA

exact sequence:

is a non-zero divisor, from the short

0 -+ A ~A -+ (A/tA) -+ 0

we deduce the long exact

sequence of percohomology: n-l n d_ _~Hn (C*) .....!..>H n (C*) -+ H~ (A/tA, C*)...£.....> ... (note that

since

A-module), and similarly for (1)~(2).

0*.

It remains to show that

A

is flat as right

Therefore by the Five Lemma (2)~

(1).

Assume (2);

to prove (1). Choose 7l.-indexed cochain complexes modules such that we can insist that

'Cn, 'Cn,

'On ,on

'C*,

'0*

of left A-

are flat left A-modules (in fact, be free left A-modules), and such

that we have maps of (71.-indexed) cochain complexes of left Amodules:

'C* -+C*,

'0* -+0*

that induce isomorphisms on coho-

mology: Hn ('C*) ~ Hn (C*), Hn ('0*) ~ Hn (0*), and such that we have a map

all integers

'cp* : 'C*-+'O*

of

n,

7l.-indexed co-

chain complexes of left A-modules such that the diagram:

'cp* >'0*

'C*

j C*

CP*

1

> 0*

Chapter 5

644 is commutative. integers

n,

n C

Then since

is t-adically complete, all

the image of the map

'C* -+ C*

under the functor,

"t-adic completion", is a mapping of (Z-indexed) cochain complexes of left A-modules: ('C*)/\ -+C*. ('D*)/\

Similarly for

and

D*.

We therefore deduce a commuta-

tive diagram: '¢*

'C*

.J.

( 'C*) /\

(1)

- - - > , D*

J. /\

(' *)/\

!C*

dJ

> (' D*)

.L.

¢* >

D*

The image of this diagram under the additive functor

n H

is a

commutative diagram: n H (' ¢*)

n H ('C*) (2)

""

>

1

Hn ( ( , C* ) /\ )

I

all integers

1

Hn «,¢*)/\)

> Hn ( ( , D* ) /\ )

J

n H (¢*)

V

n H (C*)

>

n > O.

Hn ('D*)

""

Hn(D*)

The composite of the left (respectively:

right) column in the commutative diagram (2) is an isomorphism since the mapping of cochain complexes: '0* -+ 0*)

'C* ..,. C*

(respectively:

was chosen so as to induce an isomorphism on cohomology

in dimension

n,

all integers

upper (respectively:

n.

From commutativity of the

lower) square in diagram (2), and the fact

that the composite of each column in the diagram (2) is an isomorphism, we see that to prove that the mapping

n H (' ¢*)

(re-

Finite Generation n H (¢*»

spectively:

645

is a monomorphism (respectively: epimor-

phism) it suffices to prove that the mapping monomorphism (respectively: epimorphism).

Hn«,¢*)A)

is a

Therefore, to com-

plete the proof of the Proposition, it suffices to prove that Hn«,¢*)A)

is an isomorphism, all integers

Since

'Cn

is flat as left A-module,

zero t-torsion, t":

'cn

->-

'Cn

"

'cn

has no non-

(since the map "left multiplication by

can be obtained by throwing the monomorphism of

right A-modules "left multiplication by functor

n.

t": A ->- A,

through the

n @IC ,,). A

('C*)A/ti«'C*)A)"" 'c*/ti(,c*),

Also so that (3)

Hn ( ( , C* ) A/ t i ( ( , C* ) A ) ) "" Hn ( ( , c * ) / t i ( , C*) ) Hn«A/tiA)@ ('C*», A

all integers the map

'C*->-C*

and such that n.

n, i

with

i > O.

The cochain complex

are such that

Hn (C*) ->- Hn (C*)

'C

n

'C*

and

is flat as left A-module,

is an isomorphism, all integers

Therefore by Definition I of percohomology, (4)

n H «A/tA) 0 ('C*»

"" H~ «A/tA) ,C*),

A

all integers

n.

Similarly for

'0*.

Therefore, by equation

(2) in the statement of the Theorem, we have that the mapping: (5)

(A/tA)

~

(' ¢*):

(A/tA)

~

A

A

('C*) ->- (A/tA)

@

('0*)

A

induces an isomorphism on cohomology in all dimensions. (multiplication by

t):

'C*->- 'C*

Since

is a monomorphism, we have

Chapter 5

646

the short exact sequence of cochain complexes of left A-modules: 0+ [('c*/t('c*)]"ti-l">[('c*)/ti(,c*)] + [('c*)/ti-l(,c*)] +0 all integers

i ':'1,

a portion of the long exact sequence of

cohomology of which is the exact sequence: n-l _d_>Hn ((A/tA) 0 ('C*»"ti-l">Hn«A/tiA) 0 ('C*»

(6)

A

Hn ( (A/ti-lA) ®

+

A

n

('C*»~ ... ,

A

all integers "D*"

i > 1.

replacing

We have a similar long exact sequence with

"C*" ,

for each integer

i.:: 1,

and

' ¢*

in-

duces a map from the long exact sequence (6) into the corresponding long exact sequence for

'D*.

Since the mapping (5)

induces an isomorphism on cohomology, from the Five Lemma and the map of long exact sequences induced by

'¢*

from the se-

quence (6) into the corresponding long exact sequence with "'0*"

replacing

"'C*",

we have, by induction on

i,

that the

mapping Hnl(A/tiA) ~ ('¢*»: Hn((A/tiA) ® ('C*» A A is an isomorphism of left with

i> O.

(A/tiA)-modules, all integers

This last equation and equation (3) for

the analogue of equation (3) for (7)

+Hn((A/tiA) ® ('D*» A

D*,

n, i

C*,

and

imply that the mapping

Hn«(A/tiA) 0 ('¢*)/\): Hn«A/tiA) 0 ('C*)/\) + . A A Hn ( (A/tlA) 0 (' D*)/\) A

is an isomorphism, all integers

n, i

with

i > O.

But then,

the short exact sequence of Corollary 1.2 of Chapter 3.applied to the complete cochain complexes of left A-modules

('C*)/\

Finite Generation and

('D*)A

647

implies that the mapping

is an isomorphism, all integers

n.

Q.E.D.

In the proof of the preceding Proposition, the commutative diagram (2) implies that mand of all

In fact, it is true that

Hn«'C*)A).

n E 7l.,

where

is canonically a direct sum-

Hn(C*)

(' C*)A

Hn(C*)ll;jHn«'c*)A),

is constructed as in the proof of the

preceding Proposition.

This follows from

corollary 3.1.

be a ring with identity and let

Let

A

an element in the center of the ring divisor.

Let

C*

be any

~-indexed

A

t

be

that is not a zero

cochain complex of t-adi-

cally complete left A-modules, and let

'C*

and

¢*:'C*+C*

be a (z-indexed) cochain complex of left A-modules and a map of z-indexed cochain complexes of left A-modules, such that (multiplication by t): 'C* -+ 'C* is injective, and such that Hn (¢*): Hn(,C*) .... Hn(C*) is an isomorphism of left A-modules, all integers

n.

Let

be the t-adic completion of

('c*t

'C*.

Then (1)

(multiplication by

t):

('C*)I\-+(,c*)"

is injective, and the mappings:

are isomorphisms of left A-modules, all integers n. n has no non-zero t-torsion, by Lemma 1.1.1 Proof. Since 'C following Theorem 1 of Chapter non-zero t-torsion, all integers of the Corollary.

4, we have that

n.

('Cn)A

has no

This proves conclusion (1)

Therefore, since also

t

is a non-zero

Chapter 5

648

divisor in the ring

A,

by Corollary 2.1.1.1 following Example

1 above, (after Definition 3 of "percohomology"), we have that the natural mapping of left (A/tA)-modules is an isomorphisl (3)

But

H

n (A/tA, (' c*)I\) ':;. Hn (A/tA) ~ (' C*)I\), A A

(A/tA) ~ ('C*)I\ RO(A/tA) ~ ('C*). A A (4)

all integers

Therefore

Hn«(A/tA) ® ('C*)I\)R:Hn(A/tA) ® ('C*», A A

Since by hypothesis the mapping,

¢*:

isomorphism, all integers

'C*+C* i,

all integers

"multiplication by

is injective, and since the element and since the mapping

n.

tEA

t":

'C* + 'C'

is a non-zero divisor,

is such that

n H (¢*)

is an

we have by Corollary 2.1.1.1

that (5)

n H (A/tA, C*) R: Hn ( (A/tA) ® 'C*) , A A

all integers

n.

Equations (3), (4) and (5) imply that the mapping of left (A/tA) -modules, (6)

H~ (A/tA, (' ¢*)I\): H~ (A/tA, (' c*)I\) + H~ (A/tA, C*)

is an isomorphism of left (A/tA)-modules, all integers Since by hypothesis

C*

complete left A-modules,

n.

is a cochain complex of t-adically (and since clearly also

('C*)I\

is a

cochain complex of t-adically complete left A-modules), the fact that the mappings (6) are isomorphisms, all integers

n,

and Proposition 3 applied to the mapping:

completes the proof of the Corollary.

Q.E.D.

Finite Generation Corollary 3.2.

Let

A

be a ring with identity and let

an element in the center of the ring zero divisor.

Let

C*

n.

A

such that

t

t

be

is not a

be a (z-indexed) cochain complex of n C

left A-modules such that gers

649

is t-adically complete, all inte-

Then there exists a (z-indexed) cochain complex

of left A-modules and a map

¢*:

'C* .... C*

'C*

of (Z-indexed) cochain

complexes of left A-modules such that (1)

(multiplication by

t):

'C* .... 'C*

is injective, such

(2)

that n 'C is t-adically complete, all integers

n,

and

such that is an isomorphism of left A-modules, all integers Proof:

Let

'0*

be a

A-modules such that gers

n

n.

(Z-indexed) cochain complex of left

'on

has no non-zero t-torsion, all inte-

(in fact, we can even construct

is a free left A-module, all integers

0*

n),

such that

'On

and such that we

have a mapping of cochain complexes of left A-modules

p*:

'0* .... C*

such that is an isomorphism of left A-modules, all integers

n.

'C* = ('0*)",

and

Then by the Corollary 3.1, if we define ¢* =(p*)

",

then

'C*

and

¢*

obey conclu-

sions (1), (2) and (3) of this Corollary.

The hypothesis in Corollary 2.1 that the element a non-zero divisor in

A

can be eliminated.

Q.E.D.

tEA

be

Chapter 5

650

More precisely, Theorem 4.

Let

A be a ring with identity and let

element in the center of the ring cally complete.

Let

C*

such that

A

tEA A

be an

is t-adi-

be a cochain complex of left A-modules

indexed by all the integers, such that m C

(1) Let

n

is t-adically complete, all integers

be a fixed integer. Hn(C*/tC*)

(2)

m.

Then if

is finitely generated as left (A/tA)-

module, and . ' . Hn+l ( preclse t- t orSlon ln

(3)

C*)

is finitely generated

as left (A/tA)-module, and if the ring

A

is left

Noetherian, then (1)

Hn(C*)

is finitely generated as left A-module,

and there exists a fixed integer

r >0

(depending on

n)

such

that conditions (2), (4) and (6) of Proposition 0 all hold. Proof:

Let

A[T]

over the ring

A,

be the polynomial ring in one variable

T

and let

B=A[T]"T be the completion of

A[T]

for the T-adic topology.

is an element of the center of the ring divisor in

B.

Also

B

is also left Noetherian.

B

B/TB = A

T

and is a non-zero

is T-adically complete. (Since

Then

The ring

B

is left Noetherian

by hypothesis, this latter observation follows from Lemma 1.1.1, with

"B"

and

"T"

replacing

"A"

and

At"

respectively).

Finite Generation

651

Then we have an epimorphism of rings: B+A by requiring that is isomorphic to

"T"

map into

A).

"t".

(In fact,

B/ (T-t) • B

Therefore every left A-module

comes a left B-module.

Clearly a left A-module

M

M beis finitely

generated (respectively: t-adically complete) as left A-module if and only if

M

is finitely generated (respectively: T-adi-

cally complete) as left B-module. Now let C*,

C*

be as in the hypotheses of Theorem 4.

considered as a cochain complex of left B-modules, obeys

all the hypotheses of Corollary 2.1 (with and

Then

"T"

replacing

"t").

"B"

replacing

"A"

Therefore, by Corollary 2.1, the

conclusions of Proposition 2 hold for the left B-module

Hn(C*).

Therefore the conclusions of Theorem 4 hold for

con-

Hn(C*)

sidered as left A-module. Remarks:

1.

Q.E.D.

Proposition 0, at the beginning of this chapter,

of course also admits a generalization, with "percohomology groups mod t"

replacing

"Hn(C*/tC*),

all integers

i".

Then

one does not have to assume hypothesis (0) of Proposition O. More precisely, Proposi tion 0'.

Let

A

be a ring with identity and let

be an element of the center of the ring

A.

chain complex of left A-modules such that complete, all integers (0') Let

nand

The element r

i. tEA

Let C

i

C*

tEA

be a co-

is t-adically

Suppose also that is a non-zero divisor.

be fixed integers with

r > O.

Then the six

652

Chapter 5

conditions (1), (2), •.. , (6) of Proposition 0 continue to be

"If (C*/tC*)"

equivalent, if we replace

"~(A/tA,C*)"

by out.

Also,

"H~-l (A/tA,C*)",

and

n l "H - (C*/tC*)"

and

respectively, through-

in condition (5), we replace "Theorem 2" by "Remark

4 following Theorem 2" (i.e., the generalized Bockstein spectral sequence referred to is that of

'r.*,

complex of left A-modules such that torsion, all integers

i,

where 'C

i

'C*

is a cochain

has no non-zero t-

and where we have a map of cochain

complexes of left A-modules

¢ *:

'C* -+ C*

is an isomorphism, all integers

such that

Hi (¢ *)

i).

And, Corollary 1.1 and Corollary 1.2 similarly generalize. That is, Corollary 1.1'. n

Under the hypotheses of Proposition 0',

be an integer such that there exists an integer

r> 0

let such

that the six equivalent conditions of Proposition 0' hold. Hn(C*}

is t-adically complete.

And equations (1) and (2) of

Corollary 1.1 continue to hold, if we replace respectively

"H

n-l

"H~-l(A/tiA'C*)" Corollary 1.2'.

i

(C*/t C*)"

Then

"Hn(C*/tic*)",

n i "HA(A/t A,C*)",

by

respectively

in equations (1), respectively (2). The hypotheses being as in Corollary 1.1',

equations (1) and (2) of Corollary 1.2 hold. Proof of Proposition 0', Corollary 1.1', and Corollary 1.2': By Corollary 3.2, there exists

'C*

a cochain complex of left

A-modules such that (multiplication by tive and such that

'Cn

t):

'C

n

-+ 'C

n

is injec-

is t-adically complete all integers

and such that there exists

'¢*:

'C* -+C*

n,

a map of cochain com-

plexes of left A-modules that induces an isomorphism on cohomology in all dimensions.

Then, if

nand

r are fixed integers

653

Finite Generation then Proposition 0 applies to the cochain complex implies Proposition 0' for the cochain complex

H~(A/tiA'C*)

e.g.,

=Hn('C*/t

Corollary 2.1.1.1.)

i

• 'C*),

Also, Corollary 1.2 applied to C*

2.

which

(Since

i~O, 'C*

by im-

by definition of the

generalized Bockstein spectral sequence of 'C*

C*.

all integers

mediately implies Corollary 1.2' for

lary 1.1 for

'C*,

C*.

implies Corollary 1.1' for

Finally, CorolC*.

The latter conclusions of Proposition 2 and of Corol-

lary 2.1 can be sharpened to read, "then the six equivalent conditions of Proposition 0' hold". 3.

Suppose that all the hypotheses of Proposition 0', ex-

cept possibly the hypothesis (0'), hold. that conditions (2),

Then it is still true

(4) and (6) of Proposition 0 are equivalent.

(The proof is similar to that of Theorem 4).

Also, if all the

hypotheses of Proposition 0', except possibly hypothesis (0'), hold, then one still obtains all the conclusions of Corollary 1. 2I

•

Tha t i s ,

Corollary 1.2".

Let

A

be a ring with identity and let

an element of the center of the ring complex of left A-modules such that all integers

i.

Let

nand

r

A. C

i

Let

C*

be

be a cochain

is t-adically complete,

be fixed integers with

Then if the equivalent conditions (2),

t

r> O.

(4) and (6) of Proposi-

tion 0 all hold, then equations (1) and (2) of Corollary 1.2 hold. corollarly 1.1". have that

Hn(C*)

The hypotheses being as in Corollary 1.2", we is t-adically complete. "H n ;l

[T)

(C*;l [T) /Ti .;l [T) ) "

,

of Corollary 1.1, and replaces

And, if one replaces in equation (1)

"Hn-l(C*/tiC*)"

by

654

Chapter 5

n l "HZ -[T] (C* , Z [T]/Ti·Z [T])"

in equation (2) of Corollary 1.1,

then the so modified equations in Corollary L 1 continue to hold. The proofs of Corollary 1.1" and Corollary 1.2" are similar to the proof of Theorem 4, and easily reduce, by the construction in Theorem 4, to Corollary 1.1' and Corollary 1.2', respectively.

4.

Under the hypotheses of Corollary 1.1'

(or of Corol-

lary 1.1"), suppose that we do not change the "cohomology groups mod tin

of equations

(1)

and (2) of Corollary 1.1 to the cor-

responding percohomology groups. main valid?

Then do these equations re-

(I.e., under the hypotheses of either Corollary

1.1' or of Corollary 1.1", do equations (1) and (2) of Corollary 1.1 hold as written, without making any substitutions with percohomology?)

The answer is "no", not even if

plete discrete valuation ring not a field, if

0,

of the maximal ideal of C* is such that

i

C = 0

for

n = 1 i

~

O.

C*

is a generator

0,1,2.

4, the second Example in

Remark 2 following the proof of Corollary 6.1.

properties, over

t

is a com-

and if the cochain complex

In fact, consider, in Chapter

we constructed a cochain complex

A= 0

C*

In that Example,

having the above indicated

obeys the hypotheses of Corollary 1.1'

(and therefore also those of Corollary 1.1") for the integer n = 1,

but, as we have seen in that Example of Chapter

4, the

natural mapping: HI (C*)

-+-

[li:m HI (C*/tic*)] i>O

is not an isomorphism, but is the zero map from a non-finitely generated

O-module

G~ 0

onto a zero module.

have seen in that Example of Chapter 4,

Also, as we

Finite Generation

655

O [lj,Inl H (C* /tiC*) ] "" G " O.

PO Therefore, for the cochain complex Chapter 1.1"

C*

of that Example of

4, the hypotheses of Corollary 1.1' and of Corollary

hold, yet equations (1) and (2) of Corollary 1.1 fail as

written.

Therefore one must use the indicated percohomology

groups in Corollary 1.1' and in Corollary 1.1". Example 1.

Let

0

be a complete discrete valuation ring

having a quotient field

o

K

of characteristic zero,

can be of mixed characteristic) and let

class field of bra, let

O.

A= A® k

be a Noetherian X

Suppose that

(e.g., it suffices that have,

A

and let

o

over Spec(k).

Let

X

k

(however,

be the residue

(commutativ~

O-alge-

be a scheme simple and proper

X

is embeddable ([PACJ) over ~.)

be projective over

A

Then we

([p~CJ), the lifted (~t)-adic, and the lifted (~)-adic,

cohomology groups of H H

h

h

all integers

X,

(X,~t),

A

(X,~

),

h > O.

(Here,

AA = AA t,

tor for the maximal ideal of Assume for simplicity that

O.

(At

where

t

is any genera-

is as defined in [P.P.WC.]).

A is flat over

O.

Then Theorem 1

of this chapter immediately implies that

is finitely generated as Proof:

In fact, let

m

of which are affine open.

~A-module, all integers be any covering of Then

X

h> O. the elements

656

h

~

Chapter 5

where

0,

D*

is a certain cochain complex of sheaves of

~-modules,

all of which are t-adically complete.

where

C* (ur, X,D*)

C*

=

(Since

Di

integers

Hh(x,r~),

i

c

t):

Also,

-+

c

i

0,

is flat over i.:. 0).

ci

is a cochain complex such that

t-adically complete, all integers (multiplication by

Therefore

i.

is

From the definitions,

is injective, all integers since

C* /tC*

A

is flat over

is such that

the hypercohomology of

X

0,

Hh (C* /tC*)

all R;

with coefficients in the

cochain complex of sheaves of differential forms over is therefore finitely generated as

i.

A

=

A,

(h/th)-module.

and The

Example therefore follows from Theorem 1 applied to the cochain complex

C* ,

Example 2.

the ring

A

and the element

tEA.

The hypotheses and notations being as in Example 1,

suppose we delete the hypothesis that

"A

is flat over

0".

Then the conclusion of that Example remains valid. Proof:

We use Noetherian induction on the ring

assertion is false for

A

exists an ideal

A,

in

X

over

A=

~/t~,

If the

then there

such that the assertion is false

X x Spec (A/(I +t~)) and such that the asserSpec (A) tion is true for ~/I' and X x Spec (A/ (I' + tA)) , all Spec (A) ideals I' in A such that. 11- I'. Replacing A by ~/I, for

~I

I

and

A.

and

we can assume that the assertion is true for J;;i{O}

If

in ~

~/J,

all ideals

A. has no non-zero t-torsion, then the assertion follows

from Example 1.

So we may suppose that there exists

x E (pre-

Finite Generation A)

cise t-torsion of

such that

x

to.

657

Then, we have the short

exact sequence:

(1) where

Q

A s O->-~ ->A->-- ->-0 Q xA

,

is the image in

A

of the annihilator ideal of

x

in

~

(Q~~

s

is the unique homomorphism of A-modules that sends the coset

since

x

is a precise t-torsion element), and where

of 1 into the element then

C

n

x E A.

is flat over

sequence (1) over

A

A,

If we let all integers

o ->- C* ~ (~) A

be as in Example 1,

n,

so tensoring the

yields the short exact sequence of cochain

complexes of A-modules (in fact, of (2 )

C*

AA-modules):

A

->-

C*

->-

Q

C* 0 (~) A

->-

O.

However, from the explicit construction in [P.A.CJ , A A A stands for the conwhere "D*(--=-)" C* 0 (x~) '" C* (ill , X,D* (~) ) , xA " A II A struction analogous to D* in Example 1 with the ring taking the place of

"~",

A

Xspe'b (A) Spec (~)"

and

-

Therefore, by the inductive assumption on finitely generated as A Q

is not a proper quotient ring of

we have a contradiction (since then A

~

A,

A- ( =

be a counterexample).

Therefore the ring

on~, Hh (C*

~,

0 (A/Q) )

Hh (C* ~ is A h> O. Also, if

~

then

-)module, and

X.

A xA- )

== A =

Hh(X,~) =Hh(X,f

finitely generated as

quotient ring of

A,

~A-module, all integers

replacing

~

A)

and is

is supposed to

A

A/Q

is a proper

and therefore by the inductive assumption is finitely generated as

A

~

-module, all

A

integers

h:;- O.

Therefore, from the long exact sequence of co-

homology associated to the short exact sequence of cochain complexes (2), we deduce that

Hh(C*)

is finitely generated as

658 A

~

-module, all integers

completes the proof.

Chapter 5 h,

which by equation (1) of Example 1

CHAPTER 6 THE HIGHEST NON-VANISHING COHOMOLOGY GROUP

Let

A

be a ring with identity (respectively:

an abelian category).

Let

element in the center of t:M+M

ject and let A-module

M

endomorphism that if in

A

M

M

A.

(Respectively:

be a map in

t:M

Every element of divisible part of t

M)

->

M

M M

be an

Let

is t-divisible iff iff the

Notice, therefore, if

is an endomorphism of (1)

M be an ob-

(respectively:

is an A-module (respectively:

and if

t

Then we say that the

is an epimorphism).

lowing conditions are equivalent:

(3)

A).

is t-divisible.

t:M ... M

A be

M be an A-module and let

(respectively: the object

every element of

Let

M

is an object

M),

then the fol-

M is t-divisible (2)

is t-divisible (respectively:

(2) The t-

(as defined in Chapter 1) is all of

is an epimorphism.

(Notice, also, that if

M

A-module - or, respectively, if the abelian category

M).

is an

A obeys

the axiom (P. 2) of [E. M.l - then it is also equivalent to say that "every element of

M

is infinitely t-divisible" - or, respec-

tively, that "the infinitely t-divisible part of in Chapter 1) is all of Lemma 1.

Let

A

M").

be a ring with identity (respectively:

be an abelian category), let

C*

of left A-modules (respectively: and let

t

M (as defined

be a

~-indexed

659

A

A

cochain complex

of objects and maps in

be an element in the center of

Let

A)

(respectively:

660

Chapter 6

and let C*).

t*: C*

->-

C*

be an endomorphism of the cochain complex

Suppose that (l)

(multiplication by all integers

Suppose also that

n

t): C

i

->-

c

i

is a monomorphism,

i. is an integer such that i > n,

(2)

Hi(C*/tc*)!=O'

"I 0,

i = n,

and that (3)

For each integer object)

Hi(C*)

i

~n

+ 1,

the left A-module (or

has no non-zero submodule (or non-

zero subobject) that is t-divisible. Hi(C*) =0,

(4)

!

all integers

Then

i~n+l,

Hn(C*) "10.

Also,

Hn(C*)

is not a t-divisible module, and the natural

mapping is an isomorphism of (5)

(A/tA)-modules,

n n H (C* /tC*) '" (A/tA) ~ H (C*) , A

(where the right side of equation (5) is interpreted as in Chapter 1, in the abelian-category-theoretic case, as the "Cokernel of the endomorphism

Hn(t*)

of

Hn(C*)").

Note 1.

If we replace the hypothesis (3) by the weaker hypo-

thesis:

(3')

Hn+l(C*)

has no infinitely t-divisible precise

t-torsion elements, then the conclusions of the Lemma remain valid, providing only that one replaces the sentence: Hi (C*) = 0,"

in equation (4), with

"Hi (C*)

is at-divisible

Highest Non-Vanishing Cohomology Group A-module having no non-zero t-torsion". can instead replace

"Hi(C*)"

"Hi(C*)/(t-divisible elements)"

661

(If one wishes, one

in conclusions (4) and (5) by and

"Hn(C*)/(t-divisible ele-

ments) ", respectively.) proof.

From the long exact sequence of cohomology

i-l i d . t d ••• - - » H1 (C*) -+ Hi (C*) -+ Hi (C* /tC*) - » and equation (2), we deduce that for "multiplication by Hi(C*)

t"

of

is t-divisible.

i.:':.n+l,

Hi(C*) =0,

Hi(C*)

i

~

n + 1, the endomorphism

is surjective, so that

But therefore by equation (3),

as required.

Also, a portion of the same

long exact sequence is the sequence d n- l t dn --»Hn(C*) -+Hn(C*) -+Hn(C*/tC*) -»Hn+l(C*)-+

Since

n l H + (C*)

follows that

=

0,

and by equation (2),

Hn(C*) l' 0,

t): Hn(C*)-+Hn(C*)

Hn (C* /tC*) l' 0,

it

and in fact that (multiplication by

is not surjective, i.e., that

not a t-divisible module.

Hn(C*)

is

Also, from this fragment of the coho-

mology sequence, we deduce equation (5).

Q.E.D.

The proof of the Note to Lemma 1 is similar. Remarks 1:

Suppose in Lemma 1 (that we are in the module case

and) that we delete hypothesis (1) of Lemma 1. the element

tEA

is a non-zero divisor.

(2) and (6), we change "Hn(C*/tC*)" C*

to

"Hi (C*/tC*)"

to

Suppose also that

The, if in equations "H!(A/tA,C*)"

and

"H~(A/tA,C*)" - that is, the percohomology of

with coefficients in

A/tA,

as defined in the last defini-

tion of Chapter 5 - then the rest of the Lemma continues to hold.

662

Chapter 6

The proof is the same. 2.

Suppose in Lemma 1 that we are in the abelian category-

theoretic case and that (again) we delete hypothesis (1) of Lemma 1. 0*

Then let

0*

be defined as in Chapters 1 and

2 (i.e.,

on = Cn 6l Cn + l ,

is the z-indexed cochain complex such that

n n all integers n, where d :0 n ->- on+l is such that d (u,v) = n n+l .- Cin ; +l , ) (d (u) + tv,.d (v) ), all u E d\ v and in equation (2) replace

"Hi (C*/tC*)"

by

"Hi (0*)",

then (again) Lemma 1

remains valid (in the abelian category-theoretic case).

The

proof is the same. 3.

Obvious generalizations of Lemma 1 are possible, where

the ideal 4.

tA

in the ring

A

is replaced by an arbitrary ideal.

Notice that if, under the hypotheses of Lemma 1 (or of

the more general Lemma described in Remarks 1 and 2 above), we delete the hypothesis (3), then one could still conclude that: is an

A[t-l]-module (or, in the abelian-

category-theoretic case, that all integers Hn+l(C*)

i

~n

+ 2,

Hi(t*)

is an isomorphism),

and that

is a t-divisible left A-module (or, respecn l "H + (t*) -divisible")

tively, a t-divisible (more presicely, object) . Let

A

be a ring with identity.

Jacobsen radical of

A

Then recall that the

is the intersection of all left maximal

ideals in

A - or, equivalently, of all right maximal ideals in

A.

is an element in the center of

If

t

A,

such that

t-adically complete, then it is easy to see that Jacobsen radical of

A.

t

A

is

is in the

663

Highest Non-Vanishing Cohomology Group Proposition 2.

Let

A

be a ring with identity and let

an element in the center of of left A-modules.

Let

C*

all integers

and that we have a non-negative integer Hn(C*/tC*)

(2)

elements. (3)

n

such that

i':'n+l, s

such that

is generated as left (A/tA)-module by

s

Suppose, in addition, that Either (a)

module and

t

or (b) Hn(C*)

be

be a cochain complex

Suppose that we have an integer

Hi (C*/tC*) =0,

(1)

A.

t

Hn(C*)

is finitely generated as an A-

is in the Jacobsen radical of The ring

A;

A is t-adically complete, and the A-module

has no non-zero t-divisible elements.

And finally, suppose that hypotheses (1) and (3) of Lemma 1 hold (we'll see that it follows that all the hypotheses of Lemma 1 hold).

Then

(1)

Hi (C*) = 0,

(2)

Hn(C*)

all integers

i':'n + I,

and

is generated as left A-module by

s

elements.

Also, the natural homomorphism is an isomorphism (3)

(A/tA)

@

n n H (C*) "" H (C* /tC*) .

A

Also, (4)

then

If

ul, ... ,u

ul, ... ,u h

generate

of as

h

(A/tA)-module.

are any elements of Hn(C*) in

Hn(C*),

h;:O,

as A-module iff the images

n H (C*/tC*)

generate

Hn(C*/tC*)

Chapter 6

664 Notes: tEA

1.

As in Remark 1 following Lemma 1, if the element

is a non-zero divisor, then hypothesis (1) of Lemma 1 need

not be assumed, if, e.g., in hypotheses (1) and (2) of Proposition 2, and in conclusions (3) and (4) of the Proposition, we replace "Hi(C*/tc*)"

by

"H~lA/tA,C*)'" cients in

A/tA,

"H!(A/tA,C*)",

and

"Hn(C*/tC*)"

the percohomology groups of

C*

by

with coeffi-

as defined in latter part of Chapter 5.

(Also, by Remark 2 following Lemma 1, even if zero divisor, then if we define

0*

tEA

is a

as in Remark 2 following

Lemma 1, then again hypothesis (1) of Lemma 1 can be deleted, if in hypotheses (1) and (2) of Proposition 2 and in conclusions (3) and (4) of Proposition 2, we replace "Hi (0*) " , 2.

and

n

"H (C* /tC*) "

"Hi (C*/tC*) "

by

n

by

"H (0*) ") .

And as in the Note to Lemma 1, if we replace hypo-

thesis (3) of Lemma 1 with the weaker hypothesis (3') of the Note to Lemma 1, and if we modify hypothesis (3) of this Proposition by replacing in both (3a) and (3b) that

"A

"Hn(C*) (note

by t~at

"Hn(C*)/(t-divisible elements)" (3b) so modified states simply

is t-adically complete")

then the Proposition conti-

nues to hold, where in conclusions (1), (2) and (4) throughout we replace

"Hi(C*)"

elements)"

and

and

"Hn(C*)"

by

"HiIC*)/(t-divisible

"Hn(C*)/lt-divisible elements)" respectively.

Also, the weakening of hypotheses in Notes 1 and 2 can be made simultaneously, if the modifications in the statement of the Proposition indicated in both Notes are made simultaneously. Proof.

Since the hypotheses of Lemma 1 hold (or, in the more

general case in the Note to Proposition 2, since the hypotheses of Remark 1 or 2 following Lemma 1 hold), we have that

Highest Non-Vanishing Cohomology Group

which is conclusion (1) of Proposition 2.

665

Consider the portion

of the long exact sequence of cohomology n d - l

(4)

••• _ _ >H n

By conclusion

(1)

t (C*) .... Hn (C*)

-+

dn t Hn (C*/tC*) -> Hn + l (C*) .......

n l H + (C*) = O.

of Lemma l,

the long exact sequence (4), we have that

Therefore, from (A/tA)

(;9

n H (C*) ""

A

which proves conclusion (3) of the Proposition. be such that the images ul, ... ,u h

in the group (5)

(""Hn(C*/tC*)

of

by conclusion (3) of

the Proposition) generate the group (5) as (A/tA)-module. let

Then

M = Hn(C*).

Case 1.

Hypothesis (3a) of the Proposition holds.

a finitely generated left A-module, and therefore is also a finitely generated A-module. in

(A/tA) QM

generate

A it follows that

(A/tA) 0 N =

o.

Then

M

N=~AUl+

is ... AUh)

Since the images of

(A/tA) 0M

as

(A/tA)-module,

A Since

N

is a finitely generated

A

A-module, and since

t

is in the Jacobsen radical of

follows by Nakayama's Lemma that i.e.,

ul, ... ,u

h

generate

M

N=O.

But then

as A-module.

A,

it

M=Aul+ ... +AU , h

This proves conclu-

sion (4) of the Proposition, in Case 1. Case 2.

Hypothesis (3b) of the Proposition holds.

Then by (3b), since M

N

is a submodule of

A-module, and in

N=M"'.

has no non-zero t-divisible elements,

N.

Thus

u ' .•. , u EN h l [(A/tA)

Let

(;9

N

is a t-adically complete

are such that the images of

NJ = N/tN

generate

N/tN

as

(A/tA) -

A

module.

But then, since

A

is also t-adically complete (the

very elementary) argument in the last part of Theorem 1 of

666

Chapter 6

Chapter S implies that x EM

Now, let

ul, ... ,u h

be any element.

is generated as A-module by exist

a , ... , a E A l h

since

M

in

M.

such that

x = a 1u 1

N,

N

Then

{ul, ... ,u }, h

is an A-submodule of

x EM

generate

x EN, and since

{ul' ... ,u h }.

N

we have that there

+ •.• + a h u h

But

N.

in

it follows that

being arbitrary, it follows that

as A-module by

as A-module.

M

is generated

So again in Case 2, we have proved

conclusion (4) of the Proposition. Thus, in all cases, conclusion (4) of the Proposition is proved.

It remains to establish conclusion (2).

hypothesis (2) of the Proposition, we have that generated, as be

s

(A/tA)-module, by

such elements.

si tion, there exist

s

elements.

In fact, by Hn(C*/tC*)

is

Let

Then, using conclusion (3) of the Propon u l ' ... , Us E H (C*)

under the natural mapping.

that map into

vI'···' v s

But then, by conclusion (4) of the

Proposition (since the images

VI' ... ,v

as (A/tA)-module), we have that

s

generate

u ' ... ,u l s

generate

Hn(C*/tc*) Hn(C*)

as A-module. The proof of Note 2 to the Proposition is similar. Remarks 1.

Q.E.D.

If the hypothesis (3a) of Proposition 2 holds, and

if the hypotheses of Lemma 1 hold, then there exists an integer s > 0

such that the hypothesis (2) of Proposition 2 holds.

(Namley, (Proof.

s

=

number of generators of

Since

n (A/tA) ~ H (C*).)

Hn+l(C*) = 0,

n H (C*)

we have that

as A-module.) Hn(C*/tc*) ""

(A similar observation holds if we also relax

A

hypothesis (1) of Lemma 1, as in the Note following Proposition 2.)

Highest Non-Vanishing Cohomology Group Remark

2.

667

If the hypothesis (3b) of Proposition 2 holds, and

more strongly if

Hn(C*)

is t-adically complete, but if we

delete the hypothesis (2), then the conclusion (4) continues to hold (as do the conclusions (1) and (3), but of course not conelusion (2», and the proof is similar. Also, a strengthening of conclusion (4) can then be stated: (4')

A subset

S

of

left A-module generated by iff the image of

S

in

Hn(C*) S

has the property that the

is t-adically dense in

Hn(C*/tC*)

generates

Hn(C*)

Hn(C*/tC*)

as left (A/tA)-module. (And also, as usual, one can delete hypothesis (1) of Lemma I, if one replaces

"Hi (C*/tC*)"

"H~(A/tA,C*)'"

and

and

"H n (C*/tC*)"

by

"Hi (A/tA,C*)"

the corresponding percohomology groups, as

defined in the latter part of Chapter 5 , throughout.

And the

analogue of Note 2 of proposition 2 also holds valid.) Remark

3.

Suppose that we have all the hypotheses of Proposi-

tion 2, except for hypothesis (3)

(and that we also do not have

hypothesis (3') of the Remark following the proof of Corollary 2.2).

Then "how false" does conclusion (4) of Proposition 2 (or

as modified in the Remark following the proof of Cor. 2.2 below) become? pose that in

That is, let u l ' ... ,u h

M/tM(~E~)

as A-module?

E;

M= Hn(C*)/(t-divisible elements). M are such that the images of

generate

M/tM.

Then do

ul, ... ,u h

Sup-

u l '.··, u h generate

M

(This would be conclusion (4) as modified by the

indicated Remark following Corollary 2.2 below). The answer is, in general, no (see Example below). if we let

N = the A-submodule of

M generated by

However,

u l '·· .,u h '

668

Chapter 6

then the proof of conclusion (4) of Proposition 2 shows that

where

N

and

NAt,

(as

At A -module) by

Thus,

ul' .•. ,u ' h is an AAt-module also generated

is an A-module generated (as A-module) by the completion of u ' ... u . l h

N,

Also, we have that

is "not very far away" from being generated by

M

Example.

Let

A

be any ring with identity and let

non-zero divisor in the center of

A such that

zero infinitely t-divisible elements. not t-adically complete. even if

A

Z'.

than the zero ideal.

has no nonA

is

(Examples of such rings are legion, E.g.,

A=Z'.(p)'

localized at any prime ideal And

be any

Suppose also that

is a discrete valuation ring.

ring of integers

A

t

A = k [T] (Tk [T] )

where

(p) k

the

other

is any

field, the localization of a polynomial ring in one variable over a field

k

Then let

be any ring such that

A'

at the (prime) ideal generated by the variable).

as subrings and such that Or i f

A

we can take

(A I )A = AA .

A'iA,

such that

AcA'cA"

(E.g., we can take

A' =AA,

is a discrete valuation ring that is not Henselian, A'to be the localization of any finitely generated

etale A-algebra (that is not A-isomorphic to A) at any maximal ideal that contains the maximal ideal of

A.

Or again, if

A

is a discrete valuation ring that is not complete, we can let A'

be any discrete valuation ring that is unramified over

that "I A,

A,

and that has a trivial residue class field extension

Highest Non-Vanishing Cohomology Group over

A).

ci

Then define

. r-/. °,

= 0,

cO = A' •

l

Then

669 C*

is a

(non-negative) cochain complex of A-modules, and obeys all the hypotheses of Proposition 2, with (Also

C*

n = 0,

except hypothesis (3).

does not obey hypothesis (3') of the Remark following

the proof of Corollary 2.2 below).

But

HO(C*) =HO(C*)/(divisible elements) =A', HO (C* /tC*) = (A/tA)

@

HO (C*)

A>

and

A/tA.

A

The element

IE HO (C*)

O (A/tA) @ H (C*)

has the property

that the image in

generates this free (A/tA) -module of rank one,

A

yet I does not generate as A-module. A'tA').

(Since

A'

HO(C*)

(=HO(C*)/(t-divisible elements)

is not A-isomorphic to

A,

since

Thus the, somewhat illuminating, observation in the

above Remark, about "how true" conclusion (4) of Proposition 2 remains if we delete hypothesis (3)

(and also do not assume

hypothesis (3') of the Remark following the proof of Corollary 2.2 below), is in essence "best possible". Corollary 2.1.

Let

A be a ring with identity and let

element of the center of

A.

Let

complex of left A-modules, and let

C* n

be a

(~-indexed)

be an integer.

t

be an

cochain Suppose

that hypotheses (1) and (3) of Lemma 1, and hypothesis (3) of Proposition 2, hold. (1)

Suppose also that

Hi (C*/tC*) = 0,

all integers

i >n +1

and that (2) Then

Hn(C*/tC*)

is simply generated as

(A/tA)-module.

Chapter 6

670

(1)

Hi(C*) =0,

all integers

(2)

Hn(C*) ~A/I

as left A-module, where

determined) left ideal in

I

I

is a

(uniquely

A,

Hn(C*/tC*)~A/(I+tA)

(3)

i~n+l,

as left (A/tAl-module, where

is the ideal in conclusion (2), and An element

(4)

uEHn(C*)

A-module iff the image of Hn(C*/tc*) Note:

u

generates

in

Hn(C*)

Hn(C*/tC*)

as left

is a generator of

as left (A/tA)-module.

In Corollary 2.1 above (and also in Corollary 2.2 below)

they hypothesis (1) of Lemma 1 can be dropped, if zero divisor in "Hn(C*/tC*)"

A,

and if we replace

with

"H!(A/tA,C*)"

and

t

"Hi (C*/tC*)"

"H~(A/tA,C*)"

is a nonand respec-

tively throughout (percohomology groups as defined in the latter part of Chapter 5).

(And, even if the element

divisor, if we define

D*

tEA is a zero

as in Remark 2 following Lemma 1, then

Corollaries 2.1 and 2.2 continue to hold if we delete the hypothesis (1) of Lemma 1, and replace "Hn(C*/tC*)"

with

"Hi(D*)"

and

"Hi (C*/tC*)" "Hn(D*)"

and

respectively).

The

proof is the same. Corollary 2.2.

(E~,d~) iEZ

Under the hypotheses of Corollary 2.1, let i E if., be the (generalized) Bockstein

and

r>O spectral sequence as defined in Chapter 1 of the z-indexed cochain complex tion by Let

C*

with respect to the endomorphism, "multiplica-

t". I

be the ideal described in conclusion (2) of Corol-

lary 2.1, and let

J = {x E A:

There exists an integer

r> 0

such

Highest Non-Vanishing Cohomology Group that

t

r

tx E J,

0

x E I}.

Then the ideal

implies

J

671

has the property that

x (; A,

x E J-equivalently, that the quotient ring

has no non-zero t-torsion.

J

can be characterized as being the

smallest left ideal in the ring

A

(In particular,

such tha t I c J.

elements in the ring

having this property and J

contains all the t-torsion

Then, ~or

A).

A/J

E~

of the generalized

Bockstein spectral sequence, we have that En"" A/ 00

(1)

+

(J

tA)

as left (A/tA)-modules. For each integer

r

~

(Thus,

0, And

J =

U J

of conclusion (1) is

Then

).

r>O r (2)

J

En"" A/ (J r

+ tA)

r

Proof of Corollary 2.1.

The hypotheses of Corollary 2.1 imply

those of Proposition 2, with Proposition.

s =1

in hypothesis (2) of the

Therefore we have conclusions (I), (2), (3) and (4)

of Proposition 2, with

s = 1.

Conclusions (1) and (4) of Propo-

sition 2 immediately imply conclusions (1) and (4), respectively, of Corollary 2.1. Hn{C*) that A.

Conclusion (2) of Proposition 2 implies that

is simply generated as left A-module, or equivalently Hn(C*) ""A/I

Then

as left A-module for some left ideal

I={aEA: aox=O,

uniquely determined. 2.1.

all

xEHn(C*)},

I

in

and is therefore

This proves conclusion (2) of Corollary

Finally, conclusion (3) of Proposition 2 and conclusion

(2) of Corollary 2.1 imply conclusion (3) of Corollary 2.1. Q.E.D.

672

Chapter 6

Proof of Corollary 2.2.

The short exact sequence (*) of the

conclusions of Theorem 2 of Chapter 1, for the integer for the cochain complex of A-modules "multiplication by (*)

0+

to,

C*

nand

and the endomorphism

is the short exact sequence

n (A/tA) 0 [H (C*) / (t-torsion) 1 + En A

+

00

{t-divisible, precise t-torsion elements in n l H + (C*)} +

o.

But by conclusion (1) of Corollary 2.1, we have that

Hn+l(C*) = 0,

and therefore the third group in the above short exact sequence is zero.

Therefore the short exact sequence (*) becomes an iso-

morphism of left A-modules (5)

(A/tA)

@

A

n [H (C*) / (t-torsion) 1 ':! En. 00

By conclusion (2) of Corollary 2.1, we have that

The t-torsion in the left A-module

A/I

is

J/I,

as defined in conclusion (1) of this Corollary.

where

J

is

Substituting

this observation and equation (2) into equation (5) yields that (6)

(A/tA)

@

[(A/I)/(J/I) 1"" En, 00

A

or equivalently

which is conclusion (1) of Corollary 2.2. In one of the Remarks (Remark 3) following Theorem 2 of Chapter 1, we have established analogously to the short exact

Highest Non-Vanishing Cohomology Group sequence (*), for every integer

r

~

0,

673

the short exact sequence

(A/tA) ® [H n (C*) / (precise tr-torsion) 1

0-+

A

r-l

t

-+

H

. r • {preclse t -torsion elements in

n l

+ (C*)}

-+

o.

In the same way that equation (6') is derived from the short exact sequence (*), from equation (*r) i = n + 1,

of Corollary 2.1 with

(and by conclusion (1)

and by conclusion (2) of Corol-

lary 2.1), we deduce that

A/

(J

r

+ tAl "" En r

as left A-modules, all integers

r

~

0,

proving conclusion (2) Q.E.D.

of Corollary 2.2. Remark:

Suppose, in the statement of Proposition 2 (or of Corol-

lary 2.1, or of Corollary 2.2), we replace the hypothesis (3) by the weaker hypothesis (3')

Hn(C*)/(t-divisible elements) obeys the hypothesis (3) of Proposition 2.

(3')

Let

(More precisely:

M=Hn(C*)/(t-divisible elements).

Then Either (3'a) the element or (3'b)

tEA

M

is finitely generated as left A-module and

is in the Jacobsen radical of

The ring

A

A,

is t-adically complete.

(This is

the weakening of hypothesis (3) of Prop. 2 discussed in Note 2 to Proposition 2). Suppose also that we replace hypothesis (3) of Lemma 1 by

674

Chapter 6

the weaker hypothesis (3') of the Note to Lemma 1. Then the conclusions (1) - (4) of Proposition 2 (respectively: (1) - (4) of Corollary 2.1;

(1) and (2) of Corollary 2.2) continue

to hold, where conclusion (1) of Proposition 2 (resp: conclusion (1) of Corollary 2.1) must be changed to: (1')

Hi(C*)

is a t-divisible left A-module and has no non-

zero t-torsion, and in conclusions (2) and (4) of Proposition 2, (respectively: conclusions (2) and (4) of Corollary 2.1), one replaces

"Hn(C*)"

with

"Hn(C*)/(t-divisible elements)".

(No

change required in conclusion (3) of Proposition 2 (respectively: (3) of Corollary 2.1;

(1) and (2) of Corollary 2.2).

(And like-

wise the Notes to Proposition 2, and to Corollary 2.1, remain equally valid when condition (3') above replaces hypothesis (3) of Proposition 2, and condition (3') of the Note to Lemma 1 replaces condition (3) of Lemma 1.

Likewise, Theorem 3 below re-

mains valid when the above condition (3') replaces hypothesis (3) of Proposition 2, and also (3') of the Note to Lemma 1 replaces (3 )

0

f Lemma 1. Likewise for Corollary 3.1 below (where of course one must

replace

"Hn(C*)"

in condition «4) (a)) of Corollary 3.1 by

"Hn(C*)/(t-divisible elements)", when weakening hypothesis (3) of Proposition 2 to hypothesis (3') of this Remark, and replacing (3) of Lemma 1 by (3') of the Note to Lemma 1.) And similarly for Corollary 3.1' in Remark 1 following Corollary 3.1 (where (as usual) lary 3.1' must be changed to

"Hn(C*)"

in condition (2) of Corol-

"Hn(C*)/(t-divisib~ elements)",

when one replaces hypothesis (3) of Proposition 2 by the weaker hypothesis (3') of this Remark, and also replaces (3) of Lemma 1

675

Highest Non-Vanishing Cohomology Group by (3') of the Note to Lemma 1.)

Similarly for Remark 2 following

corollary 3.1. Theorem 3.

Under the hypotheses of Corollary 2.1, suppose that

hypothesis (2) is strengthened to (2.1)

Hn(C*/tC*)

is a free (A/tA)-module of rank one.

Then the left ideal

I

Corollary 2.1 is such that ~:

1.

described in conclusion (2) of I c: t • A.

The Note following Corollary 2.1 is equally appli-

cable to Theorem 3. 2.

Under the hypotheses of Theorem 3, conclusion (3) of

Corollary 2.1 is essentially equivalent to the conclusion of Theorem 3. Proof.

By conclusion (3) of Corollary 2.1, we have that

(3) as

Hn(C*/tC*) "" A/(I + tAl

(A/tA)-modules.

But by hypothesis (2.1) of Theorem 3,

n

(2.1)

H (C*/tC*) "" (A/tA)

as left (A/tA)-module.

Equations (3) and (2.1) imply that (A/tA) "" (A/ (I + tA» as left (A/tA)-modules, or equivalently that ideals in A. Corollary 3.1.

as left Q.E.D.

Under the hypotheses of Theorem 3, the following

five conditions are euqivalent. (1)

IctA,

as left A-module.

Chapter 6

676 (2)

(3)

(4)

d n - l = 0,

all integers

r

(a)

Hn(C*)~A/I

r> O.

as left A-module, where

left ideal contained in of and

(b)

I

is a

{t-divisible elements

A}.

u EA

a t-torsion element implies that

u

is t-

divisible (and therefore also infinitely t-divisible) in (5)

Let

J

A.

be the left ideal described in conclusion (1)

of Corollary 2.2. t-divisible. ments of

A}).

Then the left A-module

(Or, equivalently,

J

is

Jc: {t-divisible ele-

(Another equivalent condition:

J c: tAl .

When the five equivalent conditions of this Corollary hold, then necessarily

{t-torsion elements of

elements of

and therefore

A};

A} c:

{t-di visible

{t-torsion elements of

A}

is

a t-divisible left A-module. Note:

The Note following Corollary 2.1 is equally applicable

to Corollary 3.1. Proof:

By hypothesis (2.1) of Theorem 3,

free (A/tA)-module of rank one.

is a

Therefore conditions (1), (2) and

(3) of Corollary 3.1 are equivalent from the general theory of spectral sequences.

By conclusion (1) of Corollary 2.2, we have

that

En~ A/ (J + tAl 00 as left (A/tA)-module.

Therfore condition (1) of this Corollary

677

Highest Non-Vanishing Cohomology Group holds if and only if (5')

JCtA.

(5")

JC it-divisible elements of

If

then certainly (5') holds.

x=ts.y,

where

(1)

and

s >0

x

is not t-divisible.

~

of Corollary 2.2) .

yEA.

Therefore

tion.

Therefore every element of

holds.

Thus

(1)

y


E;

J.

Y It tAo

yEA,

(t S • y E J, s

the property that conclusion

Conversely suppose that (5') holds.

suppose that

x E J,

Then i f

A},

Since

But the ideal

0, yEA, But

t

J C tA, J

implies S

Then

s

~

has

(see

y E J)

.y=xEJ,

Y E tA,

J

0,

is a contradic-

is t-divisible, i.e.

(5")

(5' ) ~ (5") .

As noted in the footnote to conclusion (1) of Corollary 2.2, J=> it-torsion elements of holds,

A}.

Therefore, if condition (5")

it-torsion elements of

A} C it-divisible elements of

It follows readily that

it-torsion elements of

divisible left A-module.

A}

is a t-

This proves that the equivalent con-

ditions (1), (2), (3), (5') and (5")

all imply the observations

of the last paragraph of Corollary 3.1; and also that (5") condition ((4)(b».

If

(5")

holds, then since

(5"), we have condition ((4) (a».

ICJ,

Therefore (5")=> (4).

versely, assume that condition (4) holds, and let there exists an integer condition ((4)(a» yEA,

x

A}.

i >0

such that

is t-divisible.

x=t Since

and since every t-torsion element in

(by condition ((4) (b»), it follows that

i

y

yE J.

.yEI.

~

by ConThen Then by

x=ti.y, i>O, A

is t-divisible

is t-divisible.

678

Chapter 6

Therefore every element of holds.

(5")

holds.

is t-divisible, i.e. condition

J

Therefore (4)=(5").

Finally, suppose that (5")

Then we know that «4) (b»

holds, i.e. that every

t-torsion element is t-divisible. sible element in

A

is infinitely t-divisible, and therefore

that given any left ideal that

(t

s

It follows that every t-divi-

K

• y E K, s..:: 0, yEA,

in

A

(=

implies

left A-submodule) such

y E K) ,

contained in the t-divisible elements of sible as left A-module.

Therefore (5")

A

the ideal iff

I

is

K

is t-divi-

is equivalent to condi-

Hon (5). The two parenthetical parts of (5) are, repsectively, and (5').

(5")

Therefore the non-parenthetical part, and each of the

two parenthetical parts, of condition (5) are indeed equivalent, completing the proof of Corollary 3.1. Remarks: severe.

1.

Condition «4) (b»

of Corollary 3.1 seems rather

A more general situation is covered in the following

corollary. Corollary 3.1'. A' =

Under the hypotheses of Corollary 2.1, let

AI (t-torsion elements).

(I.e.,

A'

is the largest quotient

ring (or quotient left A-module) such that the image of non-zero divisor (or such that "multiplication by injective).

t

is a

t":A' +A'

is

Then the following several conditions are equivalent.

(1)

En 1'0 A' ItA' co

(2)

Hn(C*) ~A/I

ideal contained in

as left

AltA-modules.

as left A-modules, where

I

is a left

{topological t-torsion elements of I

image of

is contained in the t-divisible elements of

in

A'

A

(Or,

equivalently, where I

is a left ideal in

A}.

such that the

Highest Non-Vanishing Cohomology Group

679

A' .)

(3)

Let

J

be the left ideal in

(1) of Corollary 2.2.

of

A}.

A described in conclusion

Then JC {topological t-torsion elements

(Or, equivalently, the image of

in the t-divisible elements of When this is the case,

J~

J

in

A'

is contained

A'). {t-torsion elements in

A},

and

n H (C*/tC*) '" A/ (K + tA), where

K

ments in Note:

is a left ideal in

A

contained in

it-torsion ele-

A}.

The Note following Corollary 2.1 is equally applicable to

Corollary 3.1'. The proof is similar to that of Corollary 3.1, and is left as an exercise to the reader. 2.

Under the hypotheses of Corollary 2.1, we always have

that En", A/K 00 where tAo

K

is a

as

(A/tA) -module,

(uniquely determined) left ideal in

A

containing

Another condition equivalent to the three conditions of

Corollary 3.1' of Remark 1 is (4) Proof:

K ~ {t-torsion elements of

A} + tAo

From the isomorphism, deduced from the short exact se-

quence (*) of Theorem 2 of Chapter 1, n (A/tA) ~ [H (C*) Itt-torsion) ] '" E~, A

680

Chapter 6

we see that under the hypotheses of Corollary 2.1, we necessarily as (A/tA)-modules, then necessarily

have that, if

K::J {t-torsion elements in

A)} + tA.

Therefore condition (4) is

equivalent to condition (1) of Corollary 3.1' of Remark 1). 3. if

A

To what extent is Corollary 3.1' "best possible"? is a ring with identity,

any left ideal contained in A},

t E center of

A,

and if

Hn(C*) ""A/I, where

D*

n C

flat over

i~n+l,

Hi(C*) =0,

I

is

{topological t-torsion elements in

then does there exist a Z-indexed cochain complex

left A-modules, with each

I.e.,

A,

C*

of

such that

and such that

Hn(D*)""A'/tA',

is as in the parenthetical part of the Note to Corol-

lary 2.1; and where answer is "Yes".

A'

is defined as in Corollary 3.1'?

In fact, define

C+ l = A,

cO = A (10),

The

where

(x,) 'EI

is some set of generators for the ideal I, and del dO:CO .... C to be the map that maps the i'th element of the

110

fine

canonical basis of the left A-module, Define

C

i

= 0,

i 10,1.

Then

A (10)

all

into

n = +1,

and the reader

will verify the indicated assertions. 4. let

A

To what extent is Corollary 2.1 "best possible"? be a ring with identity, let

t E center of

A

I.e.,

be such

that every precise t-torsion element is t-divisible (it follows readily that every t-torsion element is infinitely t-divisible see the arguments near the end of Chapter 1), and let left ideal contained in can we find a

Z-indexed cochain complex

modules and an integer i ~n + 1,

it-divisible elements of

and such that

n

such that Hn(D*) "" A/tA,

C*

be any then

of flat left A-

Hn(C*) ""A/I, where

parenthetical part of the Note to Lemma I?

A},

I

D*

Hi(C*) =0, is as in the

The answer is again

Highest Non-vanishing Cohomology Group

681

"yes" and in fact the construction of Remark 3 above gives such a cochain complex 5.

C*

(such that

C

i

= 0,

i

1- 0,1)

with

= +1.

n

Suppose that the hypotheses of Corollary 3.1 hold, and

that the equivalent conditions hold, but that we are in the situation of the Note to Corollary 3.1. "multiplication by t": and replace "Hn(D*)"

c

i

-+

"Hi(C*/tC*)"

c

(I.e., we do not assume that

is injective, all integers

by

throughout, where

Lemma 1.)

i

"Hi(D*)", D*

and

i,

"Hn(C*/tC*)"

by

is defined as in the Note to

Let us also assume that hypothesis (3) of Proposition

2 is replaced by the weaker assumption hypothesis (3') of the Remark following Corollary 2.2. corollary 3.1 is that elements in A

A}.

Then one of the conclusions of

{t-torsion elements in

A}c{t-divisible

This is a strong condition on

A.

E.g.,

if

is a Noetherian commutative ring, then this implies (consider-

ing condition (3') of the Remark following Corollary 2.2) that t

is a non-zero divisor in

zero t-divisible elements in

A,

for the t-adic topology, since A

(Since there are then no non-

A.

since t

A

is then Hausdorff

is in the Jacobsen radical of

by hypothesis (3') «a) or (b»

of the Remark following Corol-

lary 2.2).

6.

At the other extreme of Theorem 3, Corollary 3.1 and

Corollary 3.2 is the following amusing observation.

Assume that

we have the hypotheses of Corollary 2.1, and let

be the left

ideal in conclusion (2) of Corollary 2.1. conditions are equivalent: integer

r >0

such that

(1)

t

r

En

E I.

00

=

En = 0 00

iff the element

I

Then the following (2)

There exists an

(More precisely, in fact, the

following conditions are equivalent: The proof:

o.

I

(lr)

E~

= 0;

(2r)

t

r

E I) .

n 1 E A/ (I + tAl = H (C*/tC*) = En

o

Chapter 6

682

n u E H (C*)

is a permanent boundary iff there exists an element mapping into a generator of the (A/tA)-module that

u

Hn(C*/tC*)

such

is a t-torsion element (this follows from the short

exact sequence (*) of Theorem 2 of Chapter 1). element

u (: Hn(C*),

by conclusion (4) of Corollary 2.1

a generator of the simply generated A-module Therefore such a

Given such an

n u E H (C*)

a t-torsion element,

3r ~ 0

(The equivalence of (lr)

is

Hn(C*/tC*) ~A/I.

exists iff the element

i. e., iff

u

such that

1 E A/I t

r

is

E I.

and (2r) is similar, with the short

exact sequence (*r) in the Remarks (Remark 3) following Theorem 2 of Chapter 1 replacing the short exact sequence (*) of Theorem 2 of Chapter 1). Theorem 4.

Let

the center of left A-modules.

A A.

be a ring with ide and let t be an elt. of Let

Let

C* n

be a

~

-indexed) cochain complex of

be a fixed integer.

Suppose that the

following technical conditions hold. Condition (1) of Lemma 1, Condition (3) of Lemma 1, and

Condition (3) of Proposition 2. Suppose also that

A

(E.g., this is the case if (since

t

has no infinitely t-divisible elements A

is a Noetherian commutative ring

is in the Jacobsen radical of

condition (3) of Proposition 2 holds,

A,

since by hypothesis

(the weaker condition (3')

of the Remark following Corollary 2.2 suffices for

this))~

Suppose also that (1)

Hi (C*/tC*) =0,

i~n+l,

and that

(2) Hn(C*/tC*)

is a free (A/tA)-module of rank one.

Highest Non-Vanishing Cohomology Group

683

Then

and where

(1)

Hi(C*) =0,

(2)

n H (C*) "" A/I

I

i .::. n + 1,

as left A-module,

is a left ideal in

uniquely determined).

A

contained in

tAo

(I

is

In addition, the following several condi-

tions are equivalent: (1)

and

(a)

t

is a non-zero divisor in

(b)

Hn(C*)

is a free A-module of rank one (An equiva-

lent form of

If

«l)~»:

conclusion (2) above, then (2)

En"" (A/tA)

(3)

E~ = E~

(4)

dn- l r

00

(1') Let

= 0, J

A.

as left

I

is the left ideal in I = {o}) •

(A/tA) -module

all integer

r> O.

be the left ideal in the ring

Corollary 2.2.

Then

J=

A

described in

{O}.

Also, when these equivalent conditions hold, A

has no non-

zero t-divisible elements. Notes: 1.

The technical condition, condition (1) of Lemma 1,

can be eliminated, if we assume that A,

t

is a non-zero divisor in

and if in conditions (1) and (2) of the Theorem, we replace

"Hi(C*/tC*)"

and

"Hn(C*/tC*)"

by the percohomology groups (see

the definition in the latter part of Chapter 5) and

"H~(A/tA,C*)"

respectively.

volving the cochain complex

D*,

"H!(A/tA,C*)"

(Also, an observation inas in the parenthetical obser-

vation in the Note to Lemma 1, can be made). 2.

Also, the technical conditions, condition (3) of Propo-

sition 2, and condition (3) of Lemma 1, can be weakened, respec-

Chapter 6

684

tively, to condition (3') of the Remark following Corollary 2.2, and to condition (3') of the Note to Lemma 1, if we replace conclusion (1) of this Theorem with" (1')

Hi(C*)

is at-divisible

left A-module, and has no non-zero t-torsion, all integers i~n+l",

and if we also weaken conclusion (2) of this Theorem,

by replacing

"Hn(C*)"

by

"Hn(C*)/(t-divisible elements)",

and also modify condition «1) (b» replacing 3.

"Hn(C*)"

by

of this Theorem, again by

"Hn(C*)/(t-divisible elements)".

Both the modifications indicated in Notes 1 and 2

above, in this Theorem, can be made simultaneously. Proof.

The hypotheses of Corollary 2.1 hold, and therefore we

have conclusions (1) and (2) of Corollary 2.1, which proves conclusion (1) and the first part of conclusion (2) of this Theorem. Also the hypotheses of Theorem 3 hold, and therefore the conclusion of Theorem 3, that

letA.

conclusion (2) of this Theorem.

This completes the proof of Since the hypotheses of Theorem

3 hold, the hypotheses of Corollary 3.1 also hold, and therefore we have that the five conditions, (1) - (5), of Corollary 3.1 are equivalent.

But, condition (2) of this Theorem is condition (1)

of Corollary 3.1; condition (3) of this Theorem is condition (2) of Corollary 3.1; condition (4) of this Theorem is condition (3) of Corollary 3.1.

Since by hypothesis the ring A has no infi-

nitely t-divisible elements, condition «4) (b» is equivalent to condition «1) (a» If condition «1) (a» sion elements.

of Corollary 3.1

of this Theorem.

holds, then A has no non-zero t-tor-

Therefore an element of

sible iff it is t-divisible.

A

is infinitely t-divi-

But by hypothesis, the ring

no non-zero infinitely t-divisible elements.

Therefore

A A

has

has

Highest Non-Vanishing Cohomology Group no non-zero t-divisible elements.

Thus, under condition «1) (a»

of this Theorem (equivalently condition «4) (b» 3.1) the ring

A

fore i f «1) (a»

685

of Corollary

has no non-zero t-divisible elements. of this Theorem (equivalently,

Corollary 3.1) holds, then condition «4) (a» holds iff condition «1) (b»

There-

«4) (b»

of

of Corollary 3.1

of this Theorem holds.

Therefore,

condition (1) of this Theorem is equivalent to condition (4) of Corollary 3.1.

Finally, note that condition (5) of Corollary

3.1 is equivalent to condition (I') of this Theorem (Proof: Obviously,

(I') of this Theorem implies (5) of Corollary 3.l.

Conversely, if condition (5) of Corollary 3.1 holds,then t-divisible as A-module. finitely t-divisible.

Therefore every element of

But by hypothesis,

infinitely t-divisible elements.

Therefore

A

J

is

is in-

J

has no non-zero J = 0;

Le., condi-

tion (5) of Corollary 3.1 implies condition (1') of this Theorem). Therefore, conditions (1), (2), (3), (4) and (I') of this Theorem are respectively equivalent to conditions (4), (1), (2), (3) and (5) of Corollary 3.1.

Therefore, by Corollary 3.1, conditions

(1), (2), (3), (4) add (I') of this Theorem are equivalent. Finally, we have observed that condition «1) (a» Theorem implies that the ring elements.

A

of this

has no non-zero t-divisible

Therefore the five equivalent conditions of this

Theorem all imply that the ring

A

has no non-zero t-divisible

elements. The proof of Note 2 is similar.

Q.E.D.

This Page Intentionally Left Blank

CHAPTER 7 POINCARE DUALITY

Recall the following two definitions: Definition 1.

Let

A

be a commutative ring with identity.

graded A-algebra is a sequence indexed by all the integers,

A n C ,

nEz, of A-modules, together with homomorphisms of A-modules: n n m C 0 em .... e + ,

called the multiplication (or sometimes the cup

A

product), all integers

n,m,

additive group of the A-module tion

C 0 C .... C A n,mEZ),

all

mutative).

(-1)

e

en @em .... e n +m, A is a ring with identity (but not necessarily com-

The graded A-algebra

n+mv U u,

is non-negative iff Definition 2.

e =

(deduced from the multiplications

skew commutative, iff u E en, uUv =

e en, then the nEz together with the multiplica-

such that if

Let

v E em

all integers e A

i

= 0,

e*

is anti-commutative, or implies that

n,m.

The graded A-algebra

all integers

i < O.

be a commutative ring with identity.

a differential graded A-algebra (abbreviation, "d.g.a. over or

"d.g.a")

is a graded A-algebra

morphisms of A-modules that v E em

n n d +l • d

= 0,

n d n : C .... e n + l ,

all integers

impl ies tha t

all integers

n,m. 687

n,

e*,

C*

together with homo-

all integers and such that

n,

such

u E en,

Then A"

Chapter 7

688

Of course, if

C*

is a differential graded A-algebra, then

is a graded A-algebra.

(Bh(C*))hEZ

Let us pose another definition.

Definition 3. let

C*

Let

A

be a commutative ring with identity and

be a graded A-algebra.

iff there exists an integer

n

Then

obeys Poincare duality

C*

such that conditions (PDl) and

(PD2) below hold.

(RD~)

For the fixed integer

n,

n C

we have that

is a

free A-module of rank one. For every integer

i,

i

i

¢ : C ... Bom (C

let

A

morphism of A-modules, that to every

n i

v EC -

I

phism that to every right" I

'l

and let

u EC

cp

and

all

A

vEC n - i .

tegers

alized:

= u u v,

be the A-homomoru

on the

Then

C*

over cpi

A

is skew-

= (_l)n

• '
¢i,

all

i,

so that (RU2) can be stated only for

all in-

2.

The definition of "Poincare duality" can be gener-

i.

we can weaken Axiom (RD2), by requiring only that

.

n

~ ~2

'

and that

This weaker axiom (equivalent if

"cpi

is an isomorphism C*

mutative) would suffice for Proposition 1 below. n

(u)] (v)

are isomorphisms of A-modules.

Of course, if the d.g.a.

is an isomorphism if if

i

the homomorphisms of A-modules,

commutative (as is usually the case), then integers

[¢

,

associates "cupping with

For every integer i, ,i i n-i n cp,: C ... Bom (C , c ), 1.

be the homo-

associates the A-

A

i

['cpi(u)](v)=vUu,

Remarks:

u

ll

n i Ci ... Bom (C - I en)

(RD2) i

i

u EC ,

homomorphism "cupping on the left with all

n-i , Cn )

is skew-com(Of course, if

is even, then the "weaker" axiom would agree with (p.n2) for

i =!!) • 2

Poincare Duali ty

689

Recall also Definition 4.

Let

A

commutative) and let module

M

be a ring with identity (not necessarily M

be a left A-module.

Then the left A-

is reflexive iff the natural homomorphism of left A-

modules: M -> HOm (Hom (M,A) ,A) Aright Aleft is an isomorphism. Example 1.

Let

A

be a principle ideal domain and let

finitely generated A-module. reflexive iff Example 2.

m

ring,

Let

C*

M

is

is locally free of finite rank (i.e., flat). A ==

i aim,

A==7L.,

are both reflexive.

Proposition 1. let

be a

a is a discrete valuation a, and i is any positive

where

Then every finitely generated A-module is reflexive.

Example 3. w

Let

Then it is easy to see that

is the maximal ideal of

integer.

7L.

M

M

Let

A

the ring of integers. (And, in fact,

Then

7L.(w)

and

Hom7L. (7L. (w) ,7L.) =.zw).

be a commutative ring with identity and

be a graded A-algebra that obeys Poincare duality.

Then (1)

For every integer

i,

(2)

If the graded algebra

the A-module C*

ci

is reflexive.

is non-negative, and if

is not the zero ring, then the integer

n

A

in the definition of

Poincare duality is uniquely determined, and in fact is the largest integer Proof: by axiom

j

such that

c j t- o.

Since the graded A-algebra

(P.Ol) for every integer

of A-modules:

C* i

obeys Poincare duality, we have the isomorphism

Chapter 7

690 cp

i

i

: C "" Hom (C

n-i

A

and similarly for

'cpi.

n ,C )

But by Axiom (P.D.l),

Cn""A

as A-module.

Therefore we have the isomorphisms cpi and ,cpi : Ci "" Hom (C n-i ,A) , all integers (3) A respectively. deduced from cpi and 'cpi, Cn - i

Applying this equation to

i,

and substituting into equation

(3), we have the composite isomorphism Hom (. cp n-i ,A) A

(4)

0

cp

i : Ci""-+ Hom (Hom (C i ,A) ,A) . A A

Checking the construction of the isomorphisms (3), we see that the isomorphism (4) is the natural mapping (from any A-module into its bidual), described in Definition 4. module

C

i

is reflexive, all integers

i.

Therefore the AThis proves equation

(1) •

Next, suppose that the graded A-algebra and that wi th

A

n - i < 0,

C

=

o.

i

Cn - i

= o.

i

C "" Hom (C A

and since the graded algebra

tive, i t follows that i

Then if

then by equations (3),

i > n,

i > n,

is not the zero ring.

C*

is non-negative is an integer

n-i C*

,A).

But since

is non-nega-

Therefore by equation (3) ,

That is C

Since

i

= 0,

all integers

en", A¥ 0

i > n.

as A-modules, conclusion (2) of the Propo-

sition follows. Remarks 1. we have

In all practical applications that I know of in which

C*

a graded A-algebra that obeys Poincare duality,

a commutative ring with identity), it is always true that (1)

The graded A-algebra

C*

is non-negative,

(A

691

Poincare Duality (2)

C

i

is finitely generated as A-module, all integers

i,

and

(3)

The graded A-algebra

C*

is anti-commutative.

However, of course, it is quite easy to construct an example of a graded A-algebra (e.g. with

A=Z)

that obeys Poincare duality

in the sense of Definition 3 above, yet violates all of the above three conditions.

E.g., one can use Example 3 following Defini-

tion 4 to construct such an example, with

n = 0:

ci=o,

and

if-l,O, or 1,

W

<j>: Z "" HOrrz (Z a:Zw+Z w

(w)

,Z)

CO=Z,

c-l=:?!:(w)

Define c+l=Zw.

Let

be the natural isomorphism, and let

be an automorphism of the abelian group

ZWf~l

(e.g.,

an automorphism arising from a non-trivial permutation of the factors, even one of infinite order.) uU v= (<j>(v»

Then

C*

(u),

and

Then define

v U u = <j> (a (v) ) (u) ,

all

uEC

-1

,

is a graded z-algebra obeying Poincare duality in the

sense of Definition 3, yet

C*

does not possess any of the

properties (1), (2) or (3) above. 2.

A generalization of Definitions 1,2, and 3 is possi-

ble, in which the assumption identity"

is replaced by:

"A "A

is a commutative ring with is a ring with identity".

E.g., in Definition 1, replace the phrase "A-module" by "abelian group" and replace

"®"

with

"®"

throughout.

And

A

add the hypothesis, that we are given a homomorphism of rings with identity:

A + cO.

The definitions of "skew-commutative", "non-negative" and Definition 2 ("differential graded A-algebra") then generalize without change. (Definition 3 can also be generalized to the case in which

692

Chapter 7

the ring with identity

A

is not commutative, and one then ob-

tains the generalization of Proposition 1.

However, such a gen-

eralization of Definition 3 makes it necessary to distinguish carefully between

"right" and "left"

Poincar~

duality.

There

are at the moment no known (or even forseeable) applications of such a generalization.

Therefore we choose not to pose such a

generalization of Definition 3.) Example 1.

Let

X

(respectively: over a field tic

2).

be a topological manifold that is orientable

that is not orientable). k

Let

(respectively: over a field

k

Then the usual singular cohomology

POincare duality as graded A-algebra. then if

A be an algebra of characteris-

H*(X,A)

Also, if

X

obeys

is orientable,

A is any ring without non-zero integer torsion (i.e.,

such that

and i f we define

ACA0~),

Ch=Hh(X,A)/(integer

Z

torsion)

(i.e., the image of

Hh(X,A)

in

Hh (X,A)

®~),

then

Z

the (non-negative, skew-commutative) graded A-algebra Poincare duality.

C*

obeys

(This observation follows readily from the

usual universal coefficients theorem of topology, and from Theorem 3 below.) 2.

Let

over a field

k.

X

be a complete, non-singular algebraic variety

Let

a finite algebraic field

extension of the field (1)

H*(X,r~),

k.

Then

the hypercohomology of

X

with coefficients

in the cochain complex of sheaves of differential forms over obeys Poincare duality as graded (2) nal prime

If

k

mology [<:N'JJ

ko-algebra.

is algebraically closed, and if

t- characteristic h

A

H (X,Zq)'

of h > o.

k,

k,

q

is a ratio-

then consider the q-adic cohoDefine

h

h

A

C = H (X,Zq) / (q-torsion) .

Poincar~

Duality

693

Then by Corollary 3.1 below, the graded

i-algebra

C*

q

obeys

Poincare duality. (3)

If

k

is of characteristic

pI- 0,

then let

0

be a

complete discrete valuation ring of mixed characteristic having k

for residue class field.

Let

Hh(X,O)

be the lifted p-adic

cohomology of X, "using the ~" as defined in [P.ACJ. Let h c = Hh (X, 0) / (p-torsion), all integers h> O. Then the (nonnegative, skew-commutative) graded Poincar~

duality.

O-algebra

C*

obeys

(The proof in [P.AC.] strongly uses Corollary

3 .1 below.) (4)

An observation similar to (3) above applies to our

p-adic cohomology using the bounded Witt vectors [BWY.]; the proof is similar. Let

3.

X

A

be a commutative ring with identity and let

be a scheme simple and proper over

then

B

is an etale covering of

(I)

If

A

A.

A.

0

Let

B=H (X,OX);

And,

is an algebra over a field of characteristic

zero, then The graded B-algebra H*(X,rA)-the hypercohomology of

(T)

X

with coefficients in the cochain complex of sheaves of differ-

ential forms over

A - obeys Poincare duality (over

B).

(A similar observation holds in the analytic case, where X

is simple and proper over a Stein manifold "r~",

replaces

covering of

A,

and where

A = r (Y, OY) }.

Y,

Again,

and observation (T) holds.)

and B

"ry"

is an etale

(In characteristic

zero, and also in the analytic case, this follows from the main theorems in [P.ACJ.) If

A

is an algebra over a field of characteristic

p" 0,

Chapter 7

694

A

and if

is a local ring, such that the p'th power of the maxi-

mal ideal of

is zero, then again by methods of [P..ACJ I can

A

(See [P.AJ:J).

prove conclusion (t)

(2) (i. e .,

If

A

is any commutative ring without integer torsion

such tha t

A

C

A Q q,; ) ,

then let

Z

C~Hh(X,rA}/(integer torsion) (i. e.,

the image of

Hh (X, r

A)

in

Hh (x,r ) ® IJ)} , A

all integers

Z

h.

Then the non-negative, skew-commutative, graded B-algebra

C*

obeys Poincare duality.

(3)

If

A

(~/pZ)-algebra,

is a

i-algebra

suppose that we have a X

p

where

A

cal condition of being embeddable over (e.g., it suffices that [1,

X

integers

~/\P-adic cohomology of h> O.

Then

cO

Poincar~

duality.

that obeys the techni-

~,

Let

in the sense of [P.AJ:J

ch =

Hh (X,~/\p) ,

X as defined in [P.N:J, all

is an etale covering of

(non-negative, skew-commutative) graded obeys

Let

be either projective or liftable over

(or else obeys weaker conditions}).

the lifted

is a prime, then

such that

A

be a scheme simple and proper over

p

~/\p,

cO QIJ)-algebra

and the C* ®IJ)

Z Z (This is proved in [EAJ:J, and makes use

of Theorem 3 below, and also of Theorem 1 of Chapter 5 of this book. )

(In the special case that the scheme

proper lifting over

~,

X admits a simple,

this was first proved in my seminar on

"Zeta Matrices" in 1969-1970, for which lecture notes will eventually be available.

A portion of that material appears in

[P.C.TJ ) . Suppose that we have a differential graded algebra over the commutative ring

A

that obeys

Poincar~

C*·

duality.

Then

Poincare Duality

695

when can we conclude that the cohomology

Hh(C*),

is a graded A-algebra, obeys Poincare duality?

hEZ,

which

A weak result

(although, see Example following Proposition 2, pretty close to best possible), is given in Theorem 2. that

A

Let

A

be a commutative ring with identity such

is injective as an A-module.

strong condition, cf.

Remarks below).

(Note: Let

This is a very C*

be a skew-

commutative differential graded A-algebra that obeys Poincare duali ty, and let

n

Definition 3 hold.

be an integer such that

O=~D.l)

and (P.D.2) of

Then condition (2) below implies condition

(1) :

The graded A-algebra

(1)

duality for the same integer (2)

(a)

then (b)

h Ez ,

obeys Poincare

n.

d n - l : Cn-l-+c n

If

in the d.g.a.

and

Hh (C*),

is the (n-1) 'st coboundary

(in fact, cochain complex)

C*,

dn-l=O.

l d- : c-l-+CO,

the (-1) 'st coboundary, is iden-

tically zero. If the d.g.a.

C*

should be non-negative (or weaker, if

n n n d : C ... C + l is the zero map), then conditions (1) and «2) (a» above are equivalent, and each is also equivalent to the following condition: (2') Proof:

Hh(C*)

is a free A-module of rank one.

First, we show that condition (2) implies condition (1).

For every A-module assignment:

M~>M*

M,

let

M* = Hom (M,A). Then the A is an additive, left-exact, contravariant

functor from the category of A-modules into itself, and since by hypothesis

A

is injective as A-module, we have that the

696

Chapter 7

contravariant functor: Let C*

is also an exact functor.

M~>M*

and

be an integer,

i

is a d.g.a.

vEC

n-i-l

Then, since

.

(Definition 2), we have

d n - l (u U v) = d i (u) U v + (-1) iu U (d n - i - l (v) ) . n d - l = O.

But by hypothesis (2),

Therefore, from the above

equation, we deduce that d i (u) U v = (-1) i+lu U d n - i - l (v) ,

(1)

v ~

cn - i - l

,

all integers

i

all

u EC ,

all

i.

By Axiom (ED.l) , we can fix an isomorphism of A-modules:

c

n

", A.

Identify the A-modules

fixed A-isomorphism.

n C

and

A

by means of this

cpi

Then the isomorphism

of Axiom (EU2)

is an isomorphism of A-modules

cpi: Ci ':;' (C n - i ) *,

(2) (d n - i - l ) *:

(C n - i ) *

-+-

all integers

(C n - i - l ) *

(Le. ,

dn- i - l

all integers

i.

(c] n-i-l) * = the

under the contravariant functor,

M~>M*),

Then I claim that the diagram

(C n - i ) *

(3)

Let

denote the transpose of the

A-homomorphism image of

L

(dn-i-ll*

>(C n - i - l )*

ai

CPil t ci

d

i >

,i+l

ci +l

of A-modules either commutes or anti-commutes, for each integer L

(More precisely, the diagram (3) commutes when i is odd,

and anticommutes when Proof.

Let

u E ci

i

is even.)

and let

l v E (C n - i - ).

Then

Poincare Ouality [(d n - i - l ) * (¢i (u»

697

1 (v)

(h .... [¢i (u) (d n - i - l (h»

1) (v)

n i l (h .... [u U d - - (h) 1) (v)

= u U (d

n-i-l

(v»,

while

d

i

(u) U v.

Therefore, considering equation (1), indeed the diagram (3) commutes (resp.:anticommutes) according as (resp.:

is even

odd). oi = (C n - i ) *

Therefore, if we let

all integers is a

i+l

and let i,

then

i i (0 ,0 liEd:

(d:-indexed) cochain complex of A-modules, and the sequence i (¢ ) iEZ

of A-homomorphisms

is a map - in fact, by Axiom (P.O.2),

an isomorphism - of (Z-indexed)-cochain complexes of A-modules: i (¢ )iE::.l

(C*,d*)"" (0*,0*).

Therefore we have the induced isomorphism on cohomology

all integers But since integers

i,

i. oi

=

(C n - i ) *,

and

the cochain complexes

oi

=

(-1) i+l (d n - i - l ) *,

(0*,0*)

all

and

have equal (not merely isomorphic, but equal) cohomology groups. equation (4) becomes an isomorphism

Therefore

698

Chapter 7

(5)

M~>M*

But since the functor

is exact, we have the canonical

isomorphism (6)

Hi(((C n - i )*,

(dn-i-l)*)iE.z)""Hi((Cn-i,dn-i-l)iE.z)*=

Hn-i(C*,d*)*, all integers

i.

The composite of the isomorphisms (5) and (6)

is an isomorphism of A-modules (7)

Hi(c*)~HOmA(Hn-i(C*)'A)'

NOw, since by hypothesis (d- l ) * = O.

have that .

~

= n,

((2) (b))

d

n-l

=d

n

= 0,

Cn""A

we therefore

n n d: C

-+

n+l C

n n. H (C*) = C. S~nce

whence

poincare duality for the integer we have that

d- l = 0,

From the commutative diagram (3) with

we deduce that the coboundary

But then

i.

all integers

n,

is zero. C*

obeys

by Axiom (P.U2) for

C*

as A-modules; and, in fact, earlier we have

identified the two by means of a fixed such isomorphism,

n C = A.

Therefore, (8)

Hn(C*)""A

as A-module.

This proves that the graded A-algebra Axiom (P.Ul) for the same integer (fixed) identification

n C =A

n.

Hi(C*),

i E.z,

obeys

Finally, substituting the

into equation (7) gives an iso-

morphism of A-modules:

If we trace the construction of the isomorphism (9), we see that it is the function that sends

u E Hi (C*)

into the A-homomorphism:

Poincar~

v+uUv, u").

all

vEHn-i(C*)

Therefore Axiom

Duality

(Le.,

699

"cupping on the left with

is verified, and we have proven

(P.D2)

that condition (2) implies condition (1). We have already observed, that if the d.g.a.

C*

(Proposition 1, conclusion (2)),

is non-negative, then the integer

n

such that (pnJ) and (ED2) hold is uniquely determined, and C i = 0, C*

all integers

i

such that

is non-negative, then

i > n.

n dn:C n +C + l

In particular, if

is the zero map, as

asserted parenthetically in the conclusions of Theorem 2.

To

complete the proof of the Theorem, we must show that if n n l n d : C +c +

is the zero map, then conditions (1), «2) (a)) and

(2') are equivalent. In fact, equations (1)-(7) in the proof of the first paragraph of the Theorem use only hypothesis «2) (a)) of the Theorem. Therefore, if «2) (a)) holds, then the diagram (3) is either commutative or anticommutative, all integers cular, for

i = n.

i,

and, in parti-

Therefore if condition «2) (a)) holds, then

By hypothesis, the d.g.a. over the commutative ring obeys

Poincar~

C

i

and in particular for ponds to

d

O CO"" (C )**.

that

d

-1

=

is a reflexive A-module, all integers i =-1

and

i = 0.

iff

we have seen that

l (d- ) ** =

(d- l ) * = 0.

d-

iff

l

C- "" (C- l )**,

l (d- ) = 0.

It follows

But, if condition «2) (a)) holds,

(d-l)*=O

tion «2)(a)) holds, then

°

i,

cores-

But

under the natural isomorphisms: Therefore

°

C*,

duality, and therefore by Proposition 1, part

(1), we have that

-1

A,

Therefore, if condi-

iff l

=

°

iff

Hence, if

Chapter 7

700

then condition «2) (a)) implies condition «2) (b)).

Otherwise

stated: (10)

If

d n = 0,

then condition (2) is equivalent to

condition «2) (a)) . We have already seen that condition

(2)~condition

Obviously condition (1) implies condition (2').

(1).

Therefore, con-

sidering equation (10), to show that, under the hypothesis, "dn=O",

that conditions (1),

it remains to show that: "dn=O",

«2) (a)) and (2') are equivalent,

condition (2'), and the hypothesis

imply condition (2).

In fact, if

n n-l H (C*) = Coker (d ),

then

n C .

A-module of the free A-module of rank one

a quotient

A quotient module

of a free module of rank one is free of rank one iff the submodule that one divides by is the zero submodule. HnCC*)

is a free A-module of rank one iff

n l d - = O.

n

"d = 0",

That is, under the hypothesis

(2') holds iff condition «2) (a)) holds.

Example 1.

iff

condition

This observation, and

equation (10) above, completes the proof that, if conditions (1),

Therefore

n Im(d - l ) = 0

d

n

= 0,

«2) (a)) and (2') are equivalent.

Let

A

be a field.

Then

A

then Q.E.D.

is injective as

A-module, and therefore the key hypothesis of Theorem 2 holds. Example 2.

0

Let

maximal ideal of

be a discrete valuation ring, let 0,

Then the quotient ring

and let A = O/M

m

m

M be the

be a non-negative integer. is injective as A-module, so

again the key hypothesis of Theorem 2 is obeyed by such a commutative ring

A.

It is also not difficult to describe quotient rings of

Poincare Duality regular local rings of dimension

> 2

701

that are injective as

modules over themselves. However, as we have observed prior to Theorem 2,

Remark 1.

the hypothesis in Theorem 2 that is extremely restrictive. is so).

"A

(Of course, if

A

is a field this

But most commutative rings with identity do not have

this property. Can the hypothesis A-module" be eliminated? A

is injective as A-module"

"A is injective as left

The following counterexample, in which

is a discrete valuation ring (this is about the "simplest"

case not covered by Theorem 2) shows that such is not the case (even if

C*

Example 3. field.

is non-negative). Let

Let

~

A=O,

a discrete valuation ring that is not a

be the residue class field of

a generator of the maximal ideal of

0

and let

s

be

O.

l

Define cO = 0, C = free O-module of rank 1 with basis 2 3 {x}, C = free O-module of rank 2 with basis {h,k}, C = free O-module of rank 1 with basis one with basis

{u}

i C* = (C ) i~

Make

and into a

{z},

ci=o,

4 C = free

all integers

(non-negative) graded

O-module of rank

i~O,1,2,3,4. O-algebra by

requiring that 1 E cO x

2

=h

be a two-sided multiplicative unit,

2

=k

2

= z

2

= xh = hx = xk = kx =

hz = zh = kz = zk = 0, xz = - zx = kh = hk = u. Then

C*

is a skew-commutative non-negative graded

and we see that the graded Define

i d : Ci

-+

i l C +

O-algebra

C*

O-algebra,

obeys Poincare duality.

to be the unique homomorphism of

O-modules,

702

Chapter 7

each integer d

i.:: 0,

such that

o = d 3 = 0,

dl(x) = s • h,

d 2 (h) = 0, d 2 (k) = s • z.

and

(Thus, necessarily or

i.::3.)

a-algebra

d

i

all integers

= 0,

Then the reader will C*,

i+l i dOd

together with the coboundaries

all

i <0

i d ,

iEZ,

thus

a-algebra

(I.e. , (C*,d*). '+ ' 1 i, and that d J (u U v)

, all lntegers

= 0,

di(u)Uv+(-UiuUdj(v), Note that, by

such that

verify easily that the graded

defined, is a differential graded that

i

i,jEz,

all

i

u E C,

all

a-linearity, it suffices to verify these identi-

ties for elements of

{l,x,h,k,z,u}).

Thus, all the hypotheses of Theorem 2 above, except the hypothesis that A

is injective as A-module, hold, for the d.g.a.

C*

A = 0,

(2 ')

over the ring -

with

and also the condition

n = 4. "d

n

Also, conditions (2) and

= O"--hold.

But, if we com-

pute cohomology, we see that O H (C*) "" 0,

Hl(C*) =0, 2 H (C*)""Ik,

H3 (C*) "" k, H4 (C*) ""

and of course 3 H (C*)

a,

Hi(C*) =0,

i<-l

are not reflexive as

or

i>5.

Thus,

H2 (C*)

and

a-modules, and therefore, by Propo-

sition 1, part (1), the (skew-commutative, non-negative) graded

Poincar~

Duality

703

does not obey Poincare duality over

O.

Therefore, although conditions (2) and (2') - and the condition "d

n

=

0",

and the condition

"C*

is non-negative" - all hold,

condition (1) of Theorem 2 fails. "A

is injective as A-module" in Theorem 2 cannot be deleted

(even in the case that Remark 2.

A is a discrete valuation ring).

Suppose that all of the hypotheses of Theorem 2 hold

for a given d.g.a. tity

Therefore, the hypothesis

C*

over a given commutative ring with iden-

(including the condition that

A,

A-module).

A

is injective as

Suppose also that conditions (1), (2), and (2') of

Theorem 2 hold.

Thus,

C*

obeys Poincare duality over

A,

and

so does the d.g.a. i If each c is projective finitely generated (*) as A-module, all integers by the A-modules

i E:Z, Hi (C*) ,

then is the same property inherited i E Z?

(Of course, this is the case if every A-module is projective.)

A

is a field, since then

But, the following example shows m A=O/M ,

that this is not the case, even if

m~2,

(the ring

(*) I f A is a commutative ring with identity, and if M is an A-module, then the following three conditions are equivalent: (1) M is projective and finitely generated as A-module.

(2)

M

is flat and of finite presentation as A-module.

(3)

The quasicoherent sheaf M over the topological space Spec (A) of 0Spec(A) -modules over Spec(A) is locally free of finite (not necessarily constant) rank.

'"

704

Chapter 7

of Example 2), the quotient of a discrete valuation ring not a field by any positive integral power greater than or equal to two of the maximal ideal and

C*

(Again, this

is non-negative.

is in a sense a "simplest" possible case, such that jective as A-module, but Example 4.

Let

a

M of

Then

A

a.

is in-

A is not a field).

be any complete discrete valuation ring that

is not a field and let ideal

A

Let

sea m

be any generator of the maximal

be any integer

> 2

and let

A = a/Mm.

is injective as left A-module.

Let

(~d*)

be the skew-commutative, non-negative differen-

tial graded a-algebra constructed in Example 3 above. (Example 3), we have seen that the graded poincare duality (with

n = 4).

a-algebra

Then, C* obeys

Define

Then again it is easy to see that the (skew-commutative, nonm negative) d.g.a. c; over the ring A=a/M obeys Poincare duality.

It follows that all the hypotheses of Theorem 2 hold

for the d. g. a. negative and (since

C*

c;

A = a/Mm.

Since

n - l ( =d 3 ). Cn - l ( =c 3 ) ~Cn( =C 4 ) dm m· m m m m of Example 3 is non-negative and

definition zero), 2 holds.

over the ring

c;

is non-

is the zero map d\

c 3 ~ C4

is by

we have also that condition (2) of Theorem

It follows from Theorem 2 that the conditions (1), (2)

and (2') of Theorem 2 all hold.

In particular,

(condition (1»),

the (skew-commutative, non-negative) graded A-algebra

(Hi(C;»iE~

obeys Poincare duality over

conclusion (1) of Proposition 1) ule, all integers

i E:

~.

Hi(C*) m

Nevertheless,

A

- and therefore (by

is reflexive as A-modHiCC*) m

is not projec-

Duality

Poincar~

tive as A-module, for

705

i = 1,2,3.

In fact, by explicit computation, we have that

H2

(C~)

'"

~

H3

(C~)

'"

k,

Ell k,

m H4 (C*) '" A( = O/M ),

and

m

and of course reflexive as

Hi (C*) m = 0,

i < -lor

i> 5 .

Of course

is

D<

A( =O/Mm)-module, but is not projective as A-module.

Thus, under the hypotheses of Theorem 2, even if conditions Ci

is

projective and finitely generated as A-module, all integers

i,

(1), (2) and (2') all hold, and

the same need not be true for if and

A = O/Mm, m

0

is non-negative, if

all integers

Hi(C*),

i

(even

is a discrete valuation ring not a field,

is any fixed positive integer :: 2).

Theorem 3. t

where

C*

Let

A

be a commutative ring with identity and let

be a non-zero divisor in the ring

A.

Let

C*

be a differ-

ential graded A-algebra such that (1)

A

is t-adically complete, and

complete, all integers (2)

Hi(C*)

all integers

C

i

is t-adically

i,

has no non-zero t-divisible, t-torsion elements,

i.

(E.g., by Theorem 1 of Chapter 5, condition (2) holds if the ring

A

is Noetherian, and if

era ted as A/tA-module,all integers (3)

Hi (C*/tC*)

is finitely gen-

i.)

The endomorphism "multiplication by

t": Ci +C i

Chapter 7

706

is injective, for every integer

i.

Suppose also that (4)

The ring

A/tA

(This

is injective as (A/tA)-module.

is a very strong condition, see Examples 1 and 2, and Remark I, following Theorem 2.

This hypothesis in essence cannot be elim-

inated, however, see Remark 6 following the proof of this theorem), and that (5)

The graded (A/tA) -algebra

negative, skew-commutative and obeys n

be an integer

Hi (C* /tC*), i E z, duality.

Poincar~

is nonThen let

such that Axioms (EUl) and (pn2) hold for the

graded (A/tA)-algebra in (5). Assume also that (6)

The image of

Hi(C*/tc*)

{t-torsion elements in

Hi(C*)}

in

is finitely generated as (A/tA)-module, all integers

i.

Then the following eight conditions are equivalent: (1)

The graded A-algebra i E z,

Hi (C*) / (topological t-torsion),

obeys Poincare duality, for the same integer (2)

n.

Hn(C*)/(t-divisible elements) is a free A-module of

rank one. (2 ') that

a c A,

a' x

t-di visible for all

n x E H (C*) ,

implies

a = O. (3)

Hn(C*)/(topological t-torsion)

is a free A-module of

rank one. (3 ') Hn(C*)

a E A,

for all

a· x

a topological t-torsion element in

xcHn(C*),

implies that

a=O.

Poincare Duality (4) Let

(E;,d;)iE7

707

be the generalized Bockstein spectral

r>O sequence of the cochain complex of A-modules to the endomorphism, "multiplication by ter 1.

C*

with respect

t", as defined in Chap-

Then l - 0 dn r ,

If

(4')

i

all integers i

(Er,dr)iEz

r >

o.

is the generalized Bockstein as in

r>O condition (4), the

E~

(4")

Eg =E~

is a free (A/tA)-module of rank one.

When this is the case, every topological t-torsion element in

Hn(C*)

is t-divisible (and therefore also infinitely t-

divisible) . Notes:

Hypothesis (3) can be dropped, if instead we modify

1.

the hypothesis (5) by replacing

with

(the percohomology groups of the cochain complex

"H! (C* ,A/tA)", of A-modules

"Hi(C*/tC*)"

C*

with coefficients in the A-module

A/tA,

as

defined in the latter part of Chapter 5). 2. (1')

Hypothesis (1) can be replaced by either condition

or (1") (1')

A

t-torsion) (1")

is t-adically complete and

Hi(C*)/(tOPolOgical

is t-adically complete, all integers t

i.

is in the Jacobsen radical of the ring

A,

A

has

no non-zero infinitely t-divisible elements and Hi(C*)/(topological t-torsion) is finitely generated as A-module, all integers

i.

(Hypothesis

(1) implies (1') of this Note, by Theorem 1 of Chapter

4.

In

708

Chapter 7

fact by Theorem 6 of Chapter

4, hypothesis (1) even implies

Hi (C*)/)t-divisible elements) is t-adically complete for

that

i).

all integers Remarks:

1.

If

is t-adically complete, then

A

Hi(C*)/topo-

logical t-torsion) is finitely generated as A-module (and there-

is finitely generated as

for complete) iff

This is because the former,

® (A/tA),

A/tA- module.

is the latter.

The proof

A

is by the method of proof of conclusion (4) of Proposition 2 of Chapter 6. 2.

In all practical applications of this Theorem

that I know, hypothesis (1") of this Note holds - but there are such "practical" applications when hypothesis (1) of the Theorem fails (see the Corollarly below)). Proof:

Obviously, condition

(3)~condition

tion (3)

(3').

(since i f

(2'), and condition

(2)~(condition

Clearly also, condition (2) implies condi[Hn(C*)/(t-divisible elements)]

is a free

A-module of rank one, then the t-divisible part and topological t-torsion part of

Hn(C*)

Next, let us show that

clearly coincide.) (3)~(2).

M=Hn(C*)/(t-divisible elements). that

Suppose (3).

Let

Then condition (3) tells us

M/(topological t-torsion) is a free A-module of rank one.

Since a free A-module of rank one is a projective A-module, and since

M/(topological t-torsion) is a quotient module of

follows that (1)

A

is a direct summand of

M,

i.e.,

M,

it

that

n H (C*) / (t-divisible elements) "" A E9 N

as A-module, where

N

is some A-module.

By hypothesis (5) of

this Theorem, and conclusion (2) of Proposition 1, we have that

Poincare Duality

709

hypotheses (1) and (2) of Lemma 1 of Chapter (2) of this Theorem (for

i=n)

Note 1 to Lemma 1 of Chapter

6 hold.

Hypothesis

implies hypothesis (3') of

6.

Therefore all the hypotheses

of Lemma 1, as modified by Note 1, of Chapter

6, hold, and

therefore so do the conclusions of that Lemma, in particular conclusion (5). (2)

Hn(C*)

That is, @

(A/tA) "" Hn(C*/tC*)

as

(A/tA)-module.

A

By hypothesis (5) of this Theorem and Axiom (P.Ul), we have that n H (C* /tC*) "" A/tA

as

(A/tA) -module.

Tensoring equation (1) above over

A

with

A/tA,

and substi-

tuting equation (2) and our last observation, we see that (A/tA) Ell (N/tN) "" (A/tA) as

(A/tA) -module.

It follows that

stated that the A-module (1),

N

N

N/tN = 0,

is t-divisible.

or otherwise But by equation

is isomorphic to a sub-module of an A-module that has

no non-zero t-divisible elements.

Therefore

N = O.

Returning

to equation (1), we see that Hn(C*)/(t-divisible elements) "" A as A-module, which proves condition (2). Thus, conditions (2), (2'), (3) and (3') are all equivalent. (Notice also, in the proof that

(3)~(2)

just given, that we

have observed that, when these conditions hold, that {topological t-torsion elements in

Hn(C*)} = {t-divisible elements in

Hn(C*)}.

This proves the statement in the last paragraph of the Theorem.)

710

Chapter 7 Also, by hypothesis (5), and by conclusion (2) of Proposi-

tion 1, we have that cular for

i=n+l.

Hi (C*/tC*) = 0

for

Therefore

i

~n + 1,

and in parti-

Therefore, from the

general theory of spectral sequences, it follows that for all integers

r >0

(4') are equivalent.

E~=E~.

iff

Also, since

I.e., conditions (4) and

En = Hn(C*/tC*)

is a free

o

(A/tA)-module of rank one condtions (4') and (4") are obviously equivalent. Next, let us show that condition (4") implies condition (3).

In fact, by hypothesis (2) for

i = n + 1, and by the short

exact sequence (*) in the conclusion of Theorem 2 of Chapter 1, we have that

[Hn(C*)/(t-torsion)] 0 (A/tA)""E n . A

En"" A/tA.

(4"1 tells us that

But condition

00

Therefore

00

n [H (C*) / (t-torsion) ] 0 (A/tA) "" (A/tA) , A

or equivalently, if we let (30) R=Hn(C*)/(topological t-torsion) then

(3)

R 0 (A/tA) "" (A/tA) A

as

(A/tA)-module.

But, by hypotheses (I') or (1") of Note 2,

we have therefore that

R

is simply generated as A-module.

(The argument is as follows: that image of module.

u

in

choose any element

R! (A/tA)

as A-module). (4)

such

generates this latter (A/tA)-

Then, by the argument used in the proof of conclusion

(4) of Proposition 2 of Chapter R

u ER

R", A/J

That is

6, we see that

u

generates

Poincar~

as A-module, where

J

(3) and (4) imply that

Duality

711

is an ideal in the ring J + tA = tA,

i.e.

A.

Equations

that

Jc tAo

(5)

But considering the definition, equation (30)' of the A-module R,

we also have that

R

has no non-zero t-torsion; substituting

this into equation (4) yields (6)

xEA,

txEJ=xEJ.

Next, I claim that the ideal t-divisible.

In fact, let

exists

such that

ye J. (1)

yEA Thus,

J

J, x E J.

x=ty.

considered as A-module, is Then, by equation (5), there But then, by equation (6),

is t-divisible, as asserted.

(or (1') or (1") of Note 2) imply that

A

But hypothesis is t-adically

complete (or, respectively, t-adically complete, or without nonzero infinitely t-divisible elements). ring

A

fore

J = O.

Thus, in all cases, the

has no non-zero infinitely t-divisible elements.

There-

Returning to equations (3 ) and (4), it follows 0

that n H (C*) / (topological t-torsion) '" A as A-module, i.e., that condition (3) holds. (4") implies condition (3), as asserted. dition (1)

Thus, condition

Also, obviously, con-

(considering Axiom (pn.l») implies condition (3).

Therefore, (2)-=- (2' )-=- (3)-=- (3' ) ,

in which case the last

paragraph of the conclusions of the Theorem hold;

712

Chapter 7 (4)~

(4')<:=:> (4") ;

(4")=(3),

and

(1)=(3).

Therefore, to complete the proof of the Theorem, it suffices to show that condition (2) implies conditions (1) and (4).

But

the hypotheses of Theorem 4, as modified by Note 2, of Chapter 6

then hold.

of Chapter

Therefore, by Theorem 4 (as modified by Note 2)

6, we have that the conditions (1), (2), (3), (4) and

(1') stated in the conclusion of Theorem 4 (as modified by Note 2) of Chapter

6 are equivalent.

But ((1) (a)) of VI.4

is true by the hypotheses of this Theorem, and ((1) (b)) of VI.4 (as modified by Note 2 to VI.4) holds by condo Hence condo

(1) of VI.4

(2) of this Thm.

(as mod. by Note 2 of VI.4) holds, and

therefore so do conditions (2), (3), (4) and (1') of VI.4.

But

condition (4) of VI.4 is the same as condition (4) of this Theorem.

Thus, condition (2) of this Theorem implies condition (4). Therefore, to complete the proof of this Theorem, it suf-

fices to show that condition (2) implies condition (1). We have already established that condition (2) implies conditions (2'),(3),(3'),(4),(4') and (4"). holds, Le.,

d n- l = 0 r

'

all integers

r >

In particular,

o.

Thus, by hypothesis (5), the graded (A/tA)-algebra i E:l, . ~nteger

(4)

Hi(C*),

is non-negative and obeys Poincare duality (for the fixed ) n.

But

E~O' = Hi (C*).

Th us,

( EO' i d i0 ) iE:l'

is a differ-

ential graded A-algebra, and by hypothesis (5) of this Theorem and by condition (4) with

r = 0,

we have that the hypotheses

of Theorem 2 hold, and that condition (2) of Theorem 2 holds. Therefore, by Theorem 2, the condition (1) of Theorem 2 holds, i.e., the cohomology of the differential graded (A/tA)-algebra

713

-Poincare Ouality i

i

(EO,dO)iEZ

obeys Poincare duality for the same integer

cohomology of this cochain complex is graded

(A/tA)-algebra n.

induction on

we see that

(7)

r,

Poincar~

duality for

Proceeding in this way using Theorem 2 by

For every non-negative integer r,

graded (A/tA)-algebra the fixed integer

The

thus, the

obeys

the same integer

n.

i EZ,

the (non-negative)

obeys Poincare duality (for

n).

Next, hypothesis (6) of the Theorem tells us, after considering Cor. 1.1 of Chapter 1, that for each integer i, is finitely generated as

(A/tA)-module.

Boo(E~l

i

But

i

B (EO) = U B (EO). r>O r there exists a positive integer 00

Therefore, for each integer r = r (i),

r

depending on

i,

be any integer ':'r(i)

o 2. i

2. n.

i,

such that

Let

for all integers

i

such that

Then,

for all integers

i

(since

i>n+l

if

or i f

i2.- 1 ).

But then

all integers (8)

i.

Considering equation (7), if follows that

obeys Poincare duality (for the same fixed integer Now, let Then

i E z,

The (non-negative) graded (A/tAl-algebra n).

oi=Hi(C*)/(topological t-torsion).

i E Z,

is a graded A-algebra.

To prove condition (1)

(and therefore complete the proof of the Theorem), we must show that the graded A-algebra

oi,

i EZ,

obeys Poincare duality,

714

Chapter 7

for the integer

n

described in hypothesis (S).

dition (3) implies (PDl). n

In fact, con-

It remains to prove (P.Q2).

I.e., let

be the map described in the statement of

Axiom (P.D2); then we must show that First, notice that

Di


is an isomorphism.

has no non-zero t-divisible elements,

and also no non-zero t-torsion elements, and that we have an isomorphism of graded (A/tA)-modules, (9)

i D ,

(In fact, considering the definition of

Di

has no non-

zero topological t-torsion elements, and therefore also has no non-zero t-divisible elements.

On the other hand, considering

the short exact sequence (*) in Theorem 2 of Chapter I, and hypothesis (2) of this Theorem, we deduce equation (9)). an integer
i.

Then

is one-to-one:

In fact, let

n i,

v ED -

Then, for every

we have

tion (9), it follows that, if then

uUv = 0

that

u=O.

that

t

for all

-

n-i

v E Eoo

u E Di uUv

Considering egua-

= O.

u

is the image of

•

By equation (a), it follows

Then

• U

divides

I,

u

in

Considering equation (9), it is equivalent to say u.

zero t-divisible elements. u = t


be such that

We have seen, preceding equation (9), that

j

Fix

where

j ~ 0,

Therefore, i f and

u' E Di

0=
u

Di

~ 0,

has no non-

we can write i u' ii! t • D .

is such that Since

dn- i

has no

non-zero t-torsion elements (by the observation preceding equation (9)), it follows that ceding paragraph applied to

¢i (u') = O. u',

But then, by the pre-

it follows that

tlu'

in

Poincar~

a contradiction.

Duality

Therefore if

¢

i

715

(u) = 0

then

u = 0,

as

asserted.

pi

is onto:

Identify the A-modules

condition (3). Rom (0 A

n-i

,A).

Then

n-i

RomA(O

Also, identify

n ,D) oi

on

and

A

by means of

is identified to

~ (A/tA)

and

E!

by means

A

of equation (9).

And also identify

Rn(C*/tC*)

with

A/tA,

using «RU1) of) equation (8). Let

v E RomA (0

phism, call it

v,

n-i

,A).

Then

v

induces an (A/tA) -homomor-

on-i ~ (A/tA)

from

into

A/tA.

Considering

A

equation (9), we see that n-i RomA/tA(E", ,A/tA).

v

By (RO.2) of) equation (8), there exists an

such that

element such that

is an element of

v(w) =uJw,

v = cupping on the left with

WEE~-i).

all

there exists an element

u E oi

(Le. ,

u

But, by equation (9),

such that

u

is the image of

Then, from the commutative diagram:

u

qJ

oi

(10)

;>

Jp

.J

01/tOi

/I Ei '" we see that

Rom (0 n-i ,A) A

;>


Rom / tA (0 A

n-i

/to

n-i

, A/tA)

II

u .... ( cu1212 in g: on the> n-i RomA/tA(E", ,A/tA), left with u)
and

v

Rom / (0 n-i /to n-i ,A/tA) . A tA

both map into the same element in But since the element

is a

tEA

non-zero divisor, the kernel of the natural homomorphism P: Rom (0

A

(Proof: in

n-i , A) .... Rom / (0 n-i /to n-i , A/tA) A tA If

f:On-i .... A

is zero, all

t · Rom (0 n-i , A) . A

is an A-homomorphism that maps into zero

RomA(On-i/tOn-i,A/tA) n i xEO - ,

is

then the image of

Le.,

tlf(x)

for all

f(x)

in

xEA/tA.

A/tA Since

716

t

Chapter 7 is a non-zero divisor in

g (x)

=f

xE on-i.

Thus, since

t . g = f).

under

all

(x)/t,

we then can define

Then

i

¢ (u)

g E Hom (0 A

and

v

n-i

,A)

and

map into the same element

i

i t follows that

p,

A,

tl (¢ (u) - v)

in

Hom (0 A

n-i

,A).

We

have shown that Every element of

(11)

t,

Hom (0 A

n-i

to an element in the A-submodule To complete the proof that

cpi

,A) 1m ¢

is congruent, modulo i

.

is surjective, we divide

into cases. Hypothesis (1" ) of Note 2 holds.

Case l.

on-i

Then

is

finitely generated as A-module, and therefore so is Hom (0 A

n-i

,A) .

Since (by hypothesis (1") of Note 2)

Jacobsen radical of

A,

is in the

t

equation (11) and Nakayama's Lemma MornA (0

n-i

,A)

(applied to the finitely generated A-module Im(¢i) complete the proof. Case 2.

Hypothesis (l'l of the Theorem holds.

is t-adically complete.

Therefore so is

Then

Hom (0 n-i ,A) A

a closed A-submodule of the t-adically complete A-module By hypothesis (I'), we have also that plete.

i

Therefore Im(¢ ),

oi

A

(being n-i AD

).

is t-adically com-

being the image of a homomorphism of

t-adically complete A-modules, is also t-adically complete.

But

then equation (II), and the argument in the proof of conclusion (4) of Proposi tion 2 of Chapter

HomA(O

n-i

6,

shows that

1m ¢ i =

,A).

The proof of Note 1 is similar.

Q.E.D.

Remarks 1.

The hypothesis (6) in Theorem 3 is unnecessarily re-

strictive.

In fact, the proof only required that we assume that,

Poincar~

for each integer

i,

717

there exists an integer

every t-torsion element in some

Duality

Hi(C*)

tr-torsion element of

r,

such that

is congruent, modulo

Hi(C*).

t,

to

In fact, a more penetrating

analysis can be made to show that hypothesis (6) can, in fact, be entirely eliminated.

(One starts by showing, by careful con-

sideration of exact sequences, using the fact that jective as

and

A/tA-module, that

ive as (A/tA)-modules, all integers 2.

A/tA

i,s,r

is in-

are reflexwith

s

~

0,

r

~

0).

The portion of hypothesis (5) of Theorem 3, that the

graded A/tA-algebra weay.er hypotheses.

Hi (C*/tC*)

is positive, can be replaced by

(A similar, obvious, observation can be made

in certain other theorems in this chapter). 3. which

Another generalization of Theorem 3 is possible, in

hypothesis (4) is eliminated, and (4") is replaced by:

"E~"" A/ (topological t-torsion)". that the element

One then removes the hypothesis

tEA is a non-zero divisor, and that

A has no

non-zero t-divisible elements, and e.g., in condition (1), one replaces the phrase where

"graded A-algebra" by "graded A '-algebra",

A' = A/ (topological t-torsion).

in conditions (2), (2'), (3), (3')

Similar changes are made

(replacing

A

by

A').

The re-

sulting, more generally applicable, theorem, is in the same relation to Theorem 3, as Corollary 3.1' of Chapter 6 1 following the proof of Corollary 3.1 of Chapter Corollary 3.1 of Chapter

6.

(in Remark 6) is to

We leave it to the interested

reader to write down such a "Theorem 3'''. 4.

Notice that, under the hypotheses of Theorem 3, but

not necessarily assuming hypothesis (6), we still have that conditions (2), (2'), (3), (3'), (4), (4') and (4")

are equivalent, and

Chapter 7

718

that all of these imply that, for each integer i >0

(A/tA)-algebra:

obeys

Poincar~

r,

the graded

duality for the same

(It is easy to see that, conversely, this latter

integer n.

condition implies e.g. condition (4».

And, if hypothesis (6)

also holds, then the equivalent conditions (1), (2), (2'), (3), (3'), (4), (4'), (4") all imply that the graded (A/tA)-algebra i E 1l ,

obeys

duality.

Poincar~

(And, conversely, whether or

not condition (6) holds, this latter condition implies (1), (2), ( 2 ' ) , (3) , (3' ) , (4) , (4') and ( 4 ") diately.

Then (4)=:>(2),

just observed. (2)=:> (1) i E 1l,

.

Proof:

(4) follows imme-

(2'), (3), (3'), (4') and (4") as we have

Finally, the latter part of the proof that

when hypothesis (6) holds also shows that if

obeys Poincare duality for the integer

n,

then condi-

tion (1) holds). 5. that

Suppose that condition (1) of Theorem 3 holds, and

Hi(C*)/toP010gical t-torsion)

A-module, all integers of the commutative ring element (1' )

t.

i. A

Let

is finitely generated as

A[t- l ]

denote the localization

at the positive powers of the

Then it is trivial to see that

The graded l ®A[t- ]

A[t

-1

]-algebra:

[Hi(C*)/(tOP010gical t-torsion]

obeys Poincare duality.

A

Conversely, under the hypotheses of Theorem 3, if condition (1') holds, then so also do conditions (1), (2), (2'), (3), (3'), (4), (4'), and (4"). Proof:

Let

Hn(C*/tC*) one.

n u E H (C*)

be such that the image of

u

in

is a generator of this free (A/tA)-module of rank

Then condition (1') implies that the A-submodule of

Poincare Duality

719

n M = H (C*) I (topological t-torsion) generated by rank one.

The proof that

u

is free of

follows readily by the

(l')~(3)

methods of the earlier part of the proof of Theorem 3.

(And

resembles the proof of Theorem 4 of Chapter 6). 6.

Theorem 3 above has one very restrictive hypothesis-

namely that

AltA

is injective as

that rarely holds.

(A/tA)-module, a condition

Can this hypothesis be eliminated?

The

following counterexample answers this in the negative. Example 1.

Let

A be a regular local ring of dimension 2, and

let

{s,t}

be a set of uniformizing parameters for the ring

Let

C*

A.

be the (non-negative) differential graded A-algebra

l such that C = A, C = free A-module of rank 1 with basis {x} , 2 3 C = free A-module of rank two with basis {h,k}, C = free

°

A-module of rank one with basis one with basis Make

C*

{u},

and

ci=O

{z} ,

c

4

= free A-module of rank

all integers

iI0,1,2,3,4.

into a

(non-negative) graded

O-algebra by requiring

1 E cO

be a two-sided multiplicative unit,

that

x

2

=h

2

=k

2

=z

2

= xh = hx -= xk = kx =

hz = zh = kz = zk = 0, xz=-zx=kh= hk=u. Then

C*

is a skew-commutative, non-negative graded A-algebra,

and we see that the graded A-algebra C* obeys Poincare duality. i i i+1 Define d :C + C , all integers i, to be the unique homomorphism of A-modules such that dO=d 3 =0, d

l

(x) =

st • h,

720

Chapter 7 d

2

(h) = 0,

2 d (k)=st

and

(Thus, necessarily or c*,

i

~

z.

o

d

i

= 0

for all integers

i

such that

i < 0

Then one readily verifies that the graded A-algebra

3).

just defined,

together with the coboundaries

is a non-negative, skew-commutative differential graded A-algebra. (The computation is similar to that in Example 3 following Theorem 2).

0 = A/tA

Let graded

k =AA t, s} • A.

and

Then

C* /tC*

is the

O-algebra: cO /tC O = 0,

l Cl/tC "" 0,

c 2 /tC 3

2

"" 0 Ell 0,

3

C /tC "" 0, 4 4 c /tC "" O.

and

The coboundaries in the d.g.a. (since i

~

2,3).

d

l

modulo

C*/tC*

t,

over

0

are zero, and

are all zero d

i

= 0,

Therefore

(1)

As graded O-algebra, Hi (C*/tC*),

iE~)

C*/tC*

(and therefore also

is isomorphic to the graded O-algebra con-

structed explicitly in Example 3 following Theorem 2. by Example 3 following Theorem 2, the graded O-algebra

Therefore, c*/tc*--

and by equation (1) therefore also the graded O-algebra Hi (C*/tC*), i E 2'-obeys Poincare duality, with the fourth the

Poincar~

highest non-vanishing group. orem 3 holds.

721

Duality

Thus, the hypothesis (5) of The-

The reader will immediately verify that the other

hypotheses of Theorem 3, except the hypothesis that injective as

AltA-module, all hold

AltA

(since all the groups in

question are finitely generated over a Noetherian ring

0».

is

(A

or

(Of course, hypothesis (1) must be replaced by hypothesis

(1") of Note 2 to Theorem 6, in the case that cally complete.

A

But the reader can insist that

complete - or even that

A

is not t-adiA

be t-adically

be a complete local ring, if desired-

and then hypothesis (1), and hypotheses (1') and (1") of Note 2 of Theorem 3 all hold). However, explicit computation shows that HO (C*) = A,

(2)

Hl (C*) = 0, 2

H- (C*) "" A/st • A,

H3 (C*) "" A/st • A, and

H4 (C*) "" A.

Therefore, ditions Chapter

H4 (C*)

is a free A-module of rank one, so that con-

(2), (2'), (3), (3') of Theorem 3 hold.

6, it follows that conditions (4) and (4') of Theorem

3 also hold, and therefore so also does (4") lent to (4». (3)

By Theorem 4 of

(which is equiva-

But from equation (2) we deduce that

HO (C*) I (topological t-torsion) = A, H1(C*)/{topological t-torsion) == 0, H2 (C*)/(topological t-torsion)

"".!!.. , sA

. ) A H3 (C*)/(topological t-tors~on "" sA '

and

H4 (C*) I (topological t-torsion) "" A.

722

Chapter 7

Thus, the (non-negative, skew commutative) differential graded A-algebra

Hi(C*)/(topological t-torsion),

obey Poincare duality over

A

iEZ,

does not

(in fact, the second and third

groups are not even reflexive; also the first group is zero, but the third isn't and the highest non-vanishing group is the fourth).

Therefore, condition (1) fails.

(Notice, amusingly

enough, that (I') of Remark 5 holds, and that all modules are finitely generated over

A).

we delete the hypothesis, even if

n = 4,

A

Hi (C*/tC*)

t

i

c =0

(and

"A/tA

is injective as

A/tA-module",

is Noetherian (even a complete local ring of

dimension two with negative,

Thus, Theorem 6 becomes false if

one of the parameters), for

i

~ 5),

are finitely generated

and

C*

is non-

i

C , Hi (C*) ,

A-,A-,

and

and

(A/tA)-modules,

respectively. It is of interest to see where the proof of Theorem 6 would break down in the above Example.

It is easy to verify in

the above example, that the differential graded algebra over (from the Bockstein spectral sequence of respect to the endomorphism "multiplication by in Chapter 1), is isomorphic, as d.g.a. over

t", 0,

C*

with

as defined

to the one

constructed explicitly in Example 3 following Theorem 2.

There-

fore, by Example 3 following Theorem 2, the cohomology of that d.g.a., which is over and

0,

i E , l

i EZ,

does not obey Poincare duality

and in fact explicitly

o

El = 0,

1 El = 0,

2~

El ~ jk,

3 El"" k

This is the point at which the proof of Theorem 6

would break down (precisely when one would have to use Theorem 2) •

It is also not difficult to see that in Example 1,

Poincare Duali ty all integers

723

(In fact, this is because in

i.

Hi(C*),

every

2 precise t -torsion element is a precise t-torsion element. Proposition 3 of Chapter 1 then implies the indicated result.) (Another computation of

can be obtained using the short

exact sequence (*) of Theorem 2 of Chapter 1 - the third group in

these sequences vanish, since

as A-module and

A

Hi(C*)

is finitely generated

is Noetherian, and therefore

Hi(C*)

has

no non-zero t-divisible elements.) Notice that Example 1 above has two other curious properties: Namely, the d.g.a. does the d.g.a.

C*

over

C*/tC*

A

over

obeys Poincare duality, as

A/tA=O.

The coboundaries in the d.g.a. i zero, so that Hi (C*/tC*) =c ,

A more novel detail:

C*/tC*

over

all integers

of course, the coboundaries

A/tA(=O) i.

are all

(Although,

in

are not zero, but are the ones constructed in Example 3 following Theorem 2.

x,h,k,s

1 -

(Thus,

dO (xl

= sh,

are the images of

x,h,k,s

we have that Remark 6'.

2 dO(k)=sz,

2 dOth) = 0,

for

in

C*/tC*.)

where

Again, since

j ~ 1.

Of course, it is easy to make trifling modifications

in Exmaple 1 above so as to obtain other counterexamples in which less coincidences occur. d.g.a. over

A

E.g., if

such that

A-module, all integers all integers

i,

duality over

A

duality over

A/tA,

i,

such that

Hi(D*)

D*

is any non-negative

is free of finite rank as

such that

Di

Hi (D*),

i E Z,

and such that

Hi (D*/tD*)

is flat as A-module, obeys Poincare obeys Poincare

C* 0 D*, where C* is as in A Example 1 above, is another counterexample to Theorem 6 with the hypothesis,

"A/tA

then d.g.a.

is injective as

A/tA-module" deleted.

E.g.,

724 let

Chapter 7 0* = the singular cochains of

the ring

A,

where

m

JPm«([;)

with coefficients in Then

is any positive integer.

C*

~

A

0*

is another counterexample to Theorem 3 with the hypothesis "AltA

is injective as

Remark 7.

AltA-module" deleted.

Suppose that all the hypotheses of Theorem 3 hold

except possibly for hypothesis (1)

(and also, that hypothesis

(I') and hypothesis (1") of Note 2 to Theorem 3 also need not hold).

Then, if in conditions (1), (2), (2'), (3), and (3') we re-

place each occurrence of the A-modules, t-torsion)",

"Hi(C*)/(toPological

"Hn(C*)/(topological t-torsion),"

and

"Hn(C*)1

t-divisible elements)", by their t-adic completions, i.e., by I\t the (A )-modules, (t-torsion) ]I\t"

and

i . I\t "[H (C*)/(t-torsl0n)] ," "Hn(C*)l\t"

n "[H (C*)I

respectively - and leave con-

ditions (4), (4') and (4") as is (and also in condition (1) write,

"graded A"t-algebra" and "Poincare duality over

Al\t"

in lieu of "graded A-algebra" and Poincar€! duality over

A"),

then the resulting Theorem is a true statement,

(one that does

not require any of the hypotheses (1), (I') or (I")).

The proof

is very simple. Proof:

Since

C*

obeys the hypotheses of Theorem 2 except

hypothesis (1) we see that the cochain complex of (C*)l\t

obeys all the hypotheses of Theorem 1.

Hons (1), (2), complex of and

C*

Therefore condi-

( 2 ' ) , (3) , (3' ) , (4) , (4 ') and (4"), for the cochain

Al\t-modules (C*)l\t

are all equivalent.

But

(C*)l\t

have the same generalized Bockstein spectral sequence

(since they have the same (C*)l\t

Al\t-modules

so that

(4) , (4') and (4") for

are,respectively,the same as (4), (4'), and (4") for

The short exact sequence (1.8) of [P.P.WCJ

C*.

(i.e., conclusion

725

Poincare Duality (1) of Theorem 1 of Chapter

2), applied to

C*, tells us that

the natural map is a monomorphism: (1)

Hi (C*)/\t ... lim Hi (C*/tjc*) ~

j~O

and similarly for

(C*)/\t,

,

so that we have the increasing

sequence of submodules: (2)

Hi (C*)/\tf.+H i ((c*)/\t)/\t~lj,m Hi (C*/tjc*). j~O

But the exact sequence (I.8) of [p.P.WCJ Theorem 1 of Chapter

2), applied to

(i.e., conclusion (1) of C*,

and hypothesis (2)

of Theorem 3 above imply that the monomorphism (1) is an isomorphism.

Therefore, so are both monomorphisms in the sequence (2),

i.e., the natural map is an isomorphism:

But, by Theorem 1 of Chapter 4 of

A/\t-modules

(c*)/\t,

applied to the cochain complex

we have that the natural map is an iso-

morphism:

all integers

i.

Equations (3) and (4) imply that we have a

natural isomorphism of (5)

A/\t-modules

Hi (c*)/\t"" Hi ((c*)/\t))/ (t-divisible elements),

all integers

i.

The isomorphisms (5) imply that each of the

A/\t-modules: Hi«C*)/\t)/(topological t-torsion),

726

Chapter 7

each integer

i,

Hn«c*)At)/(topological t-torsion) and Hn«C*)At)/(t-divisible elements) are respectively isomorphic to [Hi(C*)/(t-torsion)]At, [Hn(C*}/(t-torsion) ]At and

Hn(C*}At

respectively.

Substituting the latter three expressions into

conditions (1), (2), (2'), (3) and (3') for the former three expressions whenever they occur, the new conditions become the modified forms of the (five) conditions (1), (2), (2'), (3), (3') described in this Remark. the cochain complex of

This observation, and Theorem 3 for

(~t)-modules

(C*)At,

and our previous

observation that conditions (4), (4'), and (4") for the AAt-module (C*)At

are identical to the corresponding conditions for the

cochain complex of A-modules

C*,

completes the proof of Remark

7 •

Corollary 3.1:

Let

0

be a discrete valuation ring and let

be an element of the maximal ideal of

O.

Let

C*

be a differ-

ential graded O-algebra such that Either (1')

0

is a complete discrete valuation ring, and

Hi(C*)/(topolOgical t-torsion) O-module, all integers or

(1")

is a complete

i.

Hi(C*)/(tOPOlogical t-torsion) is finitely generated as O-module, all integers

t~O

i.

727

Poincare Duality Assume also that (2)

Hi(C*)

has no non-zero t-divisible, t-torsion elements,

all integers (3)

Multiplication by gers

(4)

i. i i t: C ... C

is injective, all inte-

i.

The graded

(O/tO)-algebra

Hi (C*/tC*),

iEZ,

is

non-negative, skew-commutative and obeys Poincare duality for some integer Then

Hi (C*/tC*)

all integers free

i,

and

n.

is finitely generated as

(O/tO)-module,

[Hi(C*)/(topological t-torsion)] is a

a-module of finite rank , all integers

i.

Also the following several conditions are equivalent:

(1)

The graded a-algebra i Ez ,

(1') Let

Hi(C*)/(topological t-torsion),

obeys Poincare duality for the same integer K

be the quotient field of

K-algebra

O.

Then the graded

[Hi(C*)/(tOPOlogical t-torsion)] @K,

a

iEz,

obeys Poincare duality. (2)

(Repeat condition (2) of Theorem 3, replacing by

(2')

(3)

(3')

(4)

(4')

"An

"0".)

(Repeat condition (4) of Theorem 3, replacing by

"A"

"0".)

(Repeat condition (3') of Theorem 3, replacing by

"A"

"0".)

(Repeat condition (3) of Theorem 3, replacing by

"A"

"0".)

(Repeat condition (2') of Theorem 3, replacing by

n.

"An

"0".)

(Repeat condition (4') of Theorem 3, replacing

"A"

728

Chapter 7 by

"0".)

(Repeat condition (4") of Theorem 3, replacing

(4")

"A"

"0".)

by

Also, when these equivalent conditions hold, we have that Hi (C*/tC*) i,

is finitely generated as

(O/tO)-module, all integers

and also that Hi(C*)/(topological t-torsion)

is a free O-module of finite rank, all integers

i.

And, when these equivalent conditions hold, we have also that every topological t-torsion element in

Hn(C*)

is t-divis-

ible. Note:

Hypothesis (3) above can be dropped, if in hypothesis (4)

we replace

"Hi(C*/tC*)"

changes in conditions (4), Proof:

H~(C*,O/tO)'"

by

and make similar

(4'), and (4").

Hypothesis (1')

(respectively:

(1"»

of the Corollary

implies hypothesis (1')

(respectively:

(1"»

of Note 2 to Theo-

rem 3.

Hypotheses (2), (3) and (4) of the Corollary imply, re-

spectively, hypothesis (2), (3) and (5) of the Theorem. thesis (4) of Theorem 3 is true (since (O/tO)-module, whenever tEO,

t"F 0,

3 holds.

0

And hypothesis (6) of Theorem

By hypothesis (4) of the Corollary and conclu-

sion (1), Proposition 1, we know that (O/tO)-module.

Therefore

(O/tO)-module.

Since

module of module.

is injective as

is a discrete valuation ring and

as is easy to see).

(Proof:

O/tO

Hypo-

Hi (C*/tC*)

Hi (C*/tC*)

(O/tO)

Hi (C*/tC*)

is a reflexive

is finitely generated as

is a Noetherian ring, every sub-

is also finitely generated as (O/tO)-

This proves hypothesis (6) of Theorem 3).

Therefore

729

Poincare Duality

the hypotheses of Theorem 3 hold, and therefore we have that conditions (1), (2), (2'), (3), (3'), (4), (4')

and (4") are equiva-

lent. Also, we have observed that

Hi(C*/tC*)

is finitely gener-

ated as (O/tO)-module (whether or not conditions (1), ••• ,(4") hold).

Therefore, since

quotient

of

logical t-torsion).

O/tO

is Noetherian, so is the sub-

Hi (C* /tC*) .

i

Let

i

D = H (C*)/(topo-

Then by the proof of Theorem 3, we have

that (1)

Also, by either hypothesis (1') or hypothesis (1"), Di,

finite subset of the image of

S

(O/tO)-module.

then

generates

Di

S

is a

as O-module iff

i (O/tO) generates D @ (O/tO) as 0 0 (The proof of this assertion was given in the in

i D

S

if

@

proof of conclusion (4) of Proposition 2 of Chapter

6).

But

we have just seen that

is finitely generated as (O/tO)-

module.

(1), Di is finitely generated as

Hence by equation

O-module.

Since

Di

has no non-zero t-torsion, and

discrete valuation ring, it follows that

Di

0

is a

is free of finite

rank as O-module. Finally, the equivalence of conditions (1) and (I') follows from Remark 5 above, and the fact, observed just above, that condition (1) imples that

Hi(C*) / (topological t-torsion) is

finitely generated as O-module. follows: group

Obviously

(l)~(l').

(The argument proceeds as By Theorem 4 of Chapter

6, the

M=Hn(C*)/(t-divisible elements) is simply generated as

A-module, and condition (2) holds iff the annihilator ideal of

Chapter 7

730

M is

{O}.

But, if condition (I') holds, then

[Hn(C*)/(topological t-torsion)] is a free A-module of rank one, and therefore its annihilator ideal, and a fortiori that of

M,

are zero.

tion (I') implies condition (2). equivalent to (I'».

Thus

Therefore condi-

(l)~(l')~(2),

(1)

is

This proves the Corollary.

The proof of the Corollary in the situation of the Note to the Corollary is similar. Example 1.

Let

p

Q.E.D.

be any rational prime and let

ring of p-adic integers,

and let

t = p.

calization of the ring of integers ated by

p.

Let

X

1L

Let

1L (p)

A =1L, p

the

be the lo-

at the prime ideal gener-

be any topological space.

Then we claim

that the following several conditions are equivalent (where the homology and cOhomology groups are all the usual singular homology and cohomology). (1)

Hi (X,1L/p1L)

is finitely generated, all integers

i,

(I' ) Hi (X,1L/P1L) is finitely generated, all integers i, i (2) H (X,1L ) is finitely generated over 1Lp' all intep A

i,

gers A

(2' ) H. (X,1L ) 1

gers (3)

P

is finitely generated over

Hi(X,~)

and «3) (b»

«3) (b»

The p-torsion in integers

below hold:

is a finite dimensional rational vector

space, all integers and

all inte-

i,

Both conditions «3) (a»

«3) (a»

1Lp'

i.

i,

Hi (X,1L)

is a finite group, all

Poincare Duality (3')

731

Both conditions « 3' ) (a)) and «(3') (b) ) «3') (a))

Hi (X,(I)

is a finite dimensional rational

vector space, all integers «3') (b))

The p-torsion in

i

H (X,Z (p))

(4')

Hi(X,z (p))

is finitely generated as

p}

Js

1

p}

EZ

such that

s

is prime

i.

(l)~

(1'),

group in (1) is the

«3)

(2)-=;' (2'),

«3)(b))~«(3')(b)),

and

(4)~(4')

(Z/~)-dual

(a))~

( (3') (a)),

(5)~(5').

of that in (I').

in (2') is the direct sum of the

zp)'

is prime

The usual universal coefficients theorem of topology

shows that

A

s

is a finitely generated abelian group, all

integers

Exti

such that

i.

H. (X,Z ) / { x : s· x = 0,

to

Proof:

]sEZ

is a finitely generated abelian group, all

integers (5' )

z(p)-module,

i.

i H (X,Z)/{x: s· x=O,

to

Z (p)-module,

i.

all integers (5)

is a finite

i.

is finitely generated as

all integers

i.

Hi (X,z)

group, all integers (4)

below hold:

2 p -dual

(In fact the And the group

of that in (2) and

(the group in (2) with a dimension shift of minus one), p

from which the equivalence of (2) and (2') follows.

(3) (a))-=- ((3') (a)), ... ,

(4)~ (4')

are proved similarly). Also,

by the usual universal coefficients theorem for homology,

i.. p

H. (X,Z )""'H. (X,Z( )) (S9 1 P 1 P Z

Since

~p

is faithfully flat as

(p)

z(p)-module, conditions (4') and (2') are equivalent.

From the

homology (or cohomology) sequence of the short exact sequence:

o .... zp

~zp .... (Z/pZ) .... 0,

we see immediately that (2') implies

732

Chapter 7

(1')

(or that (2) implies (1».

Thus, to establish the equiva-

lence of (1), (1' ) , (2) , (2' ) , (4) and (4') it is necessary and sufficient to prove that

(1)~(2).

We use Theorem 1 of Chapter 5. chains of Hi(C*)

X

with coefficients in

=Hi(X,~) p

and

Let A =?L

C* = the singular coand let

p

Hi(C*/tC*)R>Hi(X,Z/p;l),

t = p.

Then

all integers

i.

Therefore if condition (2) holds then all the hypotheses of Theorem 1 of Chapter 5 hold, and therefore the conclusion, that is condition (1), holds. We leave it to the reader to struggle out, using universal coefficients, that

(4')~(5').

(4'), and the fact that HOnz (Hi (X,Z) ,Z (p»

obviously

(5')~(3'),

One uses both condition

H.(X,;l)GZ( )R>H.(X,Z( Z

l

e ExtZ (H i - l

HOnz (p) (Hi (X,Z (p»'z (p»

(Hint:

pl. l

P

»,

(X,Z) ,Z (p) ) R> H (X,;l (p) )

i R> H (X,Z (p) ),

all integers

and that i).

Since

and fairly easily (using the homology se~

quence of the short exact sequence:

o .. Z

universal coefficients observation:

Hi (X,;l)

z .... Z/p:l ~~

.... 0,

'" Hi (X,[) ,

and the all

Z

integers

i),

(3')~(l'),

it follows that (3') and (5') are

also equivalent to each of the conditions (1), (I'), (2), (2'), (4),

(4').

Q.E.D.

Remark:

The use of Theorem 1 of Chapter 5 in the above example

can be avoided because of the simplicity of the ring a complete discrete valuation ring). A

However,

is a regular local ring of dimension 2 and

A

(namely,

even if,e.g., t

is part of a

set of uniformizing parameters, then Theorem 1 of Chapter 5 is not obvious.

Such applications,

(for very general

A), occur in

[P.AS::J.

The following amusing example from topology is of course

Poincare Duality

733

very familiar. Example 2.

Let

"'z

A

the localization of the ring of in-

(2)'

tegers at the prime ideal generated by 2, a discrete valuation ring.

Let

t'" 2 \; A.

fold and let in

A.

Let

be any connected topological mani-

C* = the singular cochains of

X

with coefficients

Then it is well known that all the hypotheses of Corol(The outstanding observation being hypotheses

lary 3.1 hold.

(4), that the graded i ( = H (X, (Z/2Z)),

n

X

= dimension

proof) .

X,

z/2z-algebra

i E Z)

Hi (C*/tC*),

i E z,

obeys Poincare duality for the integer

which is proved by the familiar topological

(The other hypotheses of Corollary 3.1 are satisfied,

since by Example 1 above (just the equivalence of (1), (I'), (2), ( 4 ) and ( 4 I as

)

)

we know that

Z (2) - module, all integers

Hi(X,z/~)

i

is finitely generated

H (X'Z(2» i

(condition (4»

is finitely generated, all integers

(1», and this latter holds since the graded i H (X,z/2z), i H (X,Z/2Z)

i\;Z,

since i

(condition

(z/2Z)-algebra

obeys Poincare duality, and therefore

is a reflexive (Z/2Z)-module, and therefore finitely

generated).

Therefore, by Corollary 3.1, the following condi-

tions are equivalent (noting that, by definition of singular cohomology, (1)

Hi (C*) = Hi (X,Z (2) ), the usual singular cohomology):

The graded Z(2)-al g ebra

i H (X'Z(2»/(2-torsiOn),

obeys Poincare duality for the integer (I' )

The graded rational algebra

H

i

(X,[),

iE;z>:,

n. i E;z>:,

obeys

poincare duality. (2)

n H (X,;z>: (2»

(4)

n l ~ d - : En-l~En r r r

is a free Z (2)-module of rank one. . th e zero map, a 11 1n . t egers 1S

r> O.

734

Chapter 7 (4' )

E~

(4")

is of dimension one as Cz/ZZ)-vector space.

Of course, the equivalence of these conditions is essentially well known, and the topological

~anifold

X

is called orientable

iff each of these equivalent conditions hold. Notice that by Theorem 4 of Chapter 6,

Hn(X,~)

is a cyclic group.

we have that

Therefore another condition equiva-

lent to (1), (2), (4), (4') and (4") is

(2.0)

The annihilator ideal of

Hn(X,Z (2))

is zero.

It is well known that, when the above equivalent conditions

X

fail, i.e., when

is not orientable, that in fact

Hence, using the results of Chapter 1, in this case (4.1)

n-l dO

is surjective,

dn-l=O, r

all integers

r > 1.

and ( 4. 1 ") E~ = 0 •

Conditions (2.1),

(4.1) and (4.1") are therefore equivalent, in

this case, to the negation of each of conditions (1), (1'), (2), (4),

(4') and (4").

the condition:

(Condition (4.1) is therefore equivalent to

IId~-l 'I 0").

Of course, all of this is well-known by other, more topological methods.

(Except for the equivalence of condition (1) with

the other conditions, as (I') is the condition usually stated). Of course, there are other, less well kno\'ln,

(and less

trivial) applications of Corollary 3.1 and of Theorem 3, in parti-

Poincar~

cular to algebraic geometry.

Duality

735

(E.g., see the examples earlier

in this chapter after the statement of Axioms (P.0l) and (P.U2)).

This Page Intentionally Left Blank

CHAPTER 8 FINITE GENERATION OF THE COHOMOLOGY OF COCHAIN COMPLEXES OF I-ADICALLY COMPLETE LEFT A-MODULES FOR A FINITELY GENERATED IDEAL I

Throughout most of this manuscript, we have dealt with a simply generated ideal,

t-A.

Certain results can be general-

ized to finitely generated ideals.

Some such theorems, analo-

gous to those in Chapter 4 , will appear in a later paper.

But

we generalize most of the theorems of Chapter 5, including Theorems 1 and 5, here. Definition 1.

Let

First, we pose some definitions. A

sequence (t , ... ,t ) l r ring

A

be a ring with identity. of elements of

if it is of length

r,

A

Then a finite

is an r-sequence for the

and if the following two condi-

tions hold. (1)

The right ideal generated by

sided ideal,

tl, ... ,t i

is a two-

1 < i < r.

(It is equivalent to say that

Ati c tlA + '"

+ tiA,

1 2. i 2. r), and (2)

If

ti

is the image of

AI (tlA + •.. + ti_1A) , then the function, "left multiplication by

is one-to-one,

t i "..

1 < i < r. 737

ti

in the quotient ring

738

Chapter 8

Definition 2.

Let

left A-module. of

A

be a ring with identity and let

Then a finite sequence

(tl, .•. ,t r )

A is an r-sequence for the left A-module

length

r,

(1) A-module

M

M

be a

of elements

if it is of

and if the following two conditions hold: tIM + ••• + tiM,

The abelian subgroup, M

is a left A-submodule,

(It is equivalent to say that

of the left

1 < i < r. AtiMc tIM + .. , + tiM,

i .::. i.::. r), and (2)

For each integer

morphism, "multiplication by M/tlM + ... + ti_lM, Clearly, if tive integer and A,

A

A

(tl, .•. ,t r )

~O,

ring for

A, A"

and

r

a non-nega-

a finite sequence of elements of

is an r-sequence for the ring (tl, ... ,t ) r

A

in the

is an r-sequence for the

in the sense of Definition 2.

Also clearly, if ger

M - = i l

of the left A-module

t.1. "

is any ring with identity,

sense of Definition 1 iff left A-module

we have that the endo-

is injective.

(tl, ... ,t r )

then

1 .s.. i .s.. r,

i,

A

(tl, ... ,t ) r

is a ring with identity, a sequence of

r

r

an inte-

elements of the

then if condition (1) in the definition of "r-sequence holds, then for every left A-module

in the definition of

M,

condition (1)

"r-sequence for the left A-module

M"

holds. If A,

A

is any ring with identity and

then let

(S)

tity

A,

any sequence of and if

is any subset of

denote the two-sided ideal generated by

Then it is easy to see that, if tl, ... ,t r

S

M

r

r

is any integer

~

S.

0,

elements of the ring with iden-

is any left A-module such that condition (1)

in the definition of "r-sequence for the left A-module

M"

Finitely Generated Ideal

739

I

(that is, condition (1) of Definition 2) holds, then the left Mi = M/ (tlM + ... + tiM)

A-module

quotient ring Example 1.

is a left module over the

A/({tl, ... ,t i }).

In the case that the ring

A

is commutative, Defi-

nitions 1 and 2 agree with the familiar ones posed by AuslanderBuchsbaum ([A.B.)). Example 2. Then

A

Let

be a Noetherian, commutative local ring.

is regular iff there exists an n-sequence

for the ring that

A

In particular,

A

tl,···,t n

Example 3.

Let

where

n

(tl, ... ,t ) n is the Krull dimension of A, such

generate the maximal ideal of A

be a ring with identity and let

be a finite sequence of elements of left A,

A-module.

A.

If

(tl, ... ,t r )

A.

Let

(tl, ... ,t r )

M be any flat

is an r-sequence for the ring

then

(tl, .•. ,t r ) is likewise an r-sequence for the flat left A-module M. (This Example is a special case of Lemma 2

below. ) We are now ready to state and prove a generalization of Theorem 1 of Chapter 5. (This generalization is not as strong as Corollary 1.1 below. ) Theorem 1.

Let

left ideal in

A

A

be a ring with identity and let

such that

A

that there exists an integer such that

tl, ... ,t r

we have a

(~-indexed)

generate

I

be a

is I-adically complete.

r,

and elements I

t l , ... , tr E I

as left ideal.

cacha in complex

C*

Suppose

Suppose that

of left A-modules,

such that (1)

ci

is I-adically complete for all integers

i.

Sup-

Chapter 8

740

pose also that the ring

(2)

A

is left Noetherian (*), and that i (tl, ... ,t ) is an r-sequence for c , iE7, r

The sequence

(E.g., this is the case if (tl, ••• ,t r ) is an r-sequence for the ring A, and if c i is a flat A-module, all integers i.) Suppose also that

i C

module integers Let

n

+ t ,C i

The left submodule

(2' )

of the left A-

J

1::, j

is closed for the I-adic topology,

::. r-l,

all

i.

be a fixed integer. n H «A/I)0 C*)

(3)

Then if

is finitely generated as left A-module,

A

for the fixed integer

(4) Proof:

HnCC*) For

n,

then

is finitely generated as left A-module.

r = 1,

this is Theorem 1 of Chapter 5.

Theorem is true for

r-l;

to prove it for

the two-sided ideal in the ring

A

r.

Suppose the

Let

generated by

I r l

be

tl, ••. ,t _ • r l

Then by condition (1) of Definition 2, the left A-module i i i C /(tlC + ••. + tr_lC) each integer

i.

is also a left

(A/lr_l)-module, for

But then the hypotheses of Theorem 1 of Chap-

ter 5 apply, to the ring and the cochain complex

A' = A/I

r-l'

the element

t

= tr E A',

«A/lr_l)QC*) (=C*/(tlC*+ ••• +tr_lC*).) A

Therefore, by Theorem 1 of Chapter 5, we have that

is finitely generated as left (A/lr_l)-module.

But then, apply-

ing the inductive assumption to the cochain complex ring

A,

the ideal

I - , r l

and the sequence

C*,

the

(tl, ••. ,t _ ), it r l

follows that

e*) erated.

In the sense that every left ideal in A is finitely gen-

Finitely Generated Ideal

741

I

is finitely generated as left A-module. Remark:

Theorem 1 as stated is perhaps too restrictive in its

hypotheses.

In particular, the topological hypothesis (2')

would be a bit annoying in applications.

We shall see (e.g.,

in Corollary 1.1, following Lemma 3) that Theorem 1 can be improved as follows: 1.

If, in Theorem 1, we delete the technical hypothesis

(2'), but insert instead the technical condition (TC) of Lemma (tl, ••• ,t ) is an r-sequence r then the Theorem remains valid.

1.1.1, and assume also that in the ring 2.

A,

If, in Theorem 1, we delete the technical hypothesis

(2'), and instead assume that the elements the center of the ring

A,

Proof of assertion 1:

If

the ring

A,

tl, ... ,t

n

are in

then again Theorem 1 remains valid. is an r-sequence for

then since we have hypothesis

(2) of Theorem 1,

it follows from Corollary 2.1 that the natural map is an isomorphism:

Therefore Corollary 1.1 completes the proof of assertion 1. Proof of assertion 2:

The notations being as in Corollary 1.2,

is an r-sequence in the ring then by hypothesis (2) of Theorem 1 (Tl, ... ,T ) r m for the Z(Tl, ... ,Tr)-module C , all integers

Z(T , ... ,T ), r l is an r-sequence m.

Therefore

by Corollary 2.1 the natural map is an isomorphism of left Amodules:

742

Chapter 8 Hn'T T) (Z\Tl,···,T )/(Tl,···,T ),C*} "'iZ ' l ' ... ' r r r

Therefore Corollary 1.2 implies assertion 2. Corollaries 1.3 and 1.3' below are additional refinements of Theorem 1,

(which do not involve "percohomology groups", but

involve higher Tor's, with coefficients in

A/ (tlA + ... + trA)

in Corollary 1.3', and with coefficients, over in

Z\T , ... ,Tr)/(T , ..• ,T } r l l

in Corollary 1.3}, in the case

is not an r-sequence for

that m

Z(T , ... ,T ), r l

all integers

(and, in fact, when both conditions (2) and (2') are removed).

{In Corollary 1.3, one assumes that the elements the ring

A

tl, .•. ,t

r

of

are in the center of the ring; in Corollary 1.3',

one assumes instead that the technical condition {TC} of Lemma 1.1.1 holds, and that ring

(tl, ••• ,t ) r

is an r-sequence in the

A.} Corollaries 1.1 and 1.2 {following Lemma 3} are other gen-

eralizations of Theorem 1.

Both involve percohomology groups

(in Corollary 1.1, the group lary 1.2, the group

H~(T I.l..

H~(A/(tlA+ •.• +trA),C*}; in Corol-

1'···'

T) (Z(Tl, .•. ,T )/(Tl, ... ,T ),C*}, r

and very mild hypotheses on the ring (TC) of Lemma 1.1.1 holds, ~ that center of the ring

A.}

r

A

r

that either condition

tl, ••• ,t

are in the r (And, in all of the Corollaries 1.1,

1.2, 1.3 and 1.3', neither hypothesis (2) nor hypothesis (2') of Theorem 1 is required, vide infra.) Lemma 2. ~o,

Then

Let

and let

A

be a ring with identity, let

(tl, ... ,t r )

r

be an integer

be an r-sequence for the ring

A.

Finitely Generated Ideal 1.

If

M

is any left A-module, we have that

all integers 2.

If

conditions, (2a)

M

743

I

i,j,

with

02.i2.r,

j.:.i+l.

is any left A-module, then the following three

(2a),

(2b) and (2c), are equivalent:

(t , ... , t ) l r

is an r-sequence for the left A-module

M; (2b)

+tiA),M) =0, all integers

i,

(2c)

all integers

j':'l,

12. i 2. r; +tiA»,M) =0,

all integers

i,

l
The proof is by induction on

tion is trivial.

Suppose that

known for the integer

r-l.

r >1

r.

For

r = 0,

the asser-

and that the Lemma is

Then since

(tl, ... ,t _ ) r l

is an

(r-l)-sequence, by the inductive assumption we have conclusion 1 of the Lemma for

02.i2.r-l,

j.:.i+l,

and also the equiva-

lence of (2a), (2b) and (2c) for the integer

r-l.

conditions

", respectively.)

Since

"(2a)r_l"' "(2b) (tl, ..• ,t ) r

r-l

"and "(2c)

r-l

(Call those

is an r-sequence in the ring

A,

have the short exact sequence of (right) A-modules:

o -+-

(left Iilul tiplica tion by [Aj (tlA + ... + tr_lA) 1 [Aj (tlA + '"

+

tr_1~1 -+-

trJ

--------------------------~~.>

[Aj (t A + ... + trA) 1 l

which yields the long exact sequence of Tor's:

(1) + tr_lA»

,M)->

-+-

0,

we

Chapter 8

744

By conclusion 1 of the Lemma for the integer A

Tor j ((AI (tlA + ... + tr_lA»

,M)

=

0

for

j

~r.

r-l, we know that Therefore from

the long exact sequence (1), we deduce that A

Tor.((A/(tlA+ ... +t A»,M) =0

j~r+l.

for

r

J

This completes the inductive verification of conclusion 1. Next, let us prove the equivalence of (2a), (2b) and (2c). By the inductive assumption, we can assume that (2b)r_l

both hold--i.e., that

sequence for

M,

(t , ... ,t _ ) r l l

(2a)r_1 and is an (r-l)-

and that

A

Tor.((A/(tIA+ ... +t.A»,M) =0, J

1

j~l,

all integers

i,j

with

l
We must show that (3)

(left multiplication by

t ): r

MI (tl M + ... + tr_IM) ...

MI (tIM + ... + tr_lM) is injective if and only if (4)

all integers

and that condition (4) for the integer all integers

j

~

1.

equation (2b) r-l' for

j

~

2,

j = I

j

~l;

implies (4) for

But from the long exact sequence (1) and we see that

and that

A

Tor j ( (AI (tlA + ... + trA) ) ,M) = 0

A

TorI ((AI (tlA + .•• + trA»

only if condition (3) holds.

,M) = 0

i f and

This completes the inductive proof

of the equivalence of (2a), (2b) and (2c). Corollary 2.1.

Let

A

be a ring with identity, let

r

be an

Finitely Generated Ideal ~O,

integer ring

A.

Let

and let C*

(tl, ... ,t r )

745

I

be an r-sequence for the

be a (z-indexed) cochain complex of left A-

modules. If (tl, ... ,t ) is an r-sequence for the left A-module r m c , all integers m, then the natural map is an isomorphism of A-modules

all integers Proof:

i.

Ey hypothesis,

m C

obeys condition (2a) of Lemma 2, and

therefore obeys condition (2b) of Lemma 2, for all integers 1,:, i .:. r,

and in particular for

i = r,

all integers

m.

i,

The

Corollary then follows from the definition of percohomology (Chapter 5, Lemma 2.1.1). Lemma 3. ¢:B ->-A

Let

and

B

be rings with identity and let

be a homomorphism of rings with identity.

integer ring

A

B,

~O,

and let

(ul' ... ,u r )

Let

r

be an

be an r-sequence in the

such that, if

ti = ¢ (u ), i is an r-sequence in the ring A. Let

C*

be any cochain com-

plex, indexed by all the integers, of left A-modules. Then if i we regard C as a left B-module by means of the homomorphism of rings,

¢:B->-A,

all integers

i,

then the natural mappings

are isomorphisms:

all integers

n E;z.

(where the cohomology groups are the per-

cohomology groups as defined in Chapter 5). Proof: The proof is by induction on the integer r

=

0

the assertion is trivial.

Suppose

r >1

r >

o.

For

and that the

746

Chapter 8

assertion is established for the integer (u ' ... ,u - ) r l l

and

rings

A

B

and

(tl, ... ,t r _ ) l

r-l.

Then

are (r-l)-sequences in the

respectively, so that by the inductive assump-

tion we have an isomorphism analogous to the conclusion (1) of the Lemma with r-l replacing

r.

From the short exact sequence:

(left multiplication by

and the analogous short exact sequence with "B"

"t.

and

replacing

II

~

"U

i

",

"A"

u ) r >

replacing

we have the long exact se-

quences of percohomology: n-l

(2)

~ HB((B/(UIB+ n

H~((B/(uIB+

+ ur_lB))

,C*)~

d

n

+ UrB)) ,C*)....:;:........ ••• and n-l (3)

d

t

'>

H~((A/(tlA+ ... +tr_lA)),C*)--'£"'" H~((A/(tlA+ ... +tr_lA)),C*)

H~( (A/(tlA + ... + trA)) ,C*) d The ring homomorphism into the sequence (3).

----;>

n

>

induces a mapping from the sequence (2)

cP

The inductive assumption and the Five

Lemma complete the proof. Remarks:

1.

Of course, if we delete either the hypothesis that

"(ul' ... ,u ) is an r-sequence in r an r-sequence in

A",

very easy to construct.

B",

or that

then Lemma 3 fails.

"(tl, ... ,t r )

is

Counterexamples are

Finitely Generated Ideal

747

I

Before stating and proving Corollary 1.1 below, we require a Lenuna. Lenuna 1.1.1. teger A.

>0

Let

Let

A

and let I

tl, ••• ,t r • (TC)

be a ring with identity. Let (t , ••• , t ) r l

be an in-

be an r-sequence for the ring

be the right (= two-sided) ideal generated by Suppose that

A

is I-adically complete, and that

For every free left A-module

we have that the are closed in

~I

i < i < r - 1.

for the I-adic topology,

Then for every free left A-module is an r-sequence for

F,

AI AI tl F + ••• + ti F

left A-submodules:

Note.

r

F,

we have that

(tl, ••• ,t ) r

FAI.

The technical condition (TC) is equivalent to the condi-

tion that (TC')

The I-adic topology on the ideal

ti(A/(tlA+ ••• +ti_1A»

AI (tlA + ••• + ti_1A)

A _l i

tuple: tuple:

A _ , i l

is I-adically Hausdorff.

(Xj)jES (tixj}jES

1 < i < r - 1.

By induction on

(TC ' ) for the integer

i

in

If

i, S

we can assume that is any infinite set, then

is equivalent to asserting that, a

A~_l

in

Ai _ l =

is the induced topology from the

I-adic topology of Proof of the Note.

in the ring

S

Ai _ l

converges I-adically to zero iff the converges I-adically to zero.

this latter is equivalent to asserting that I-adically closed in

(Al~i)AI; i.e.,

(tlA + ••• + tiA) «A (S) )1\1) Proof of Lemma 1.1.1.

t. ({A~S»AI) ~

~-l

And is

that

is I-adically closed in

The proof is by induction on

(A (S) )1\1. r.

748

Chapter 8

By (TC),

tl«A(S))AI)

is I-adically closed in

(A(S))AI.

Hence

(A(S))AI/tl«A(S))AI) is I-adically Hausdorff, and is therefore isomorphic to «A/tlA) (S))AI

But then the ring

the (r-l) -sequence (t 2 + tl A, ••• tr + tl A),

A/tlA,

and

obey the hypotheses

of the Lemma.

Q.E.D.

A more general, and possibly more elegant, statement than Theorem 1 is the following Corollary,

(which makes use of per-

cohomology as defined in the latter part of Chapter 5). Corollary 1.1.

Let

A

be a ring with identity, let (tl, ... ,t ) r

non-negative integer and let the ring let

A.

Suppose that the ring

1= tl A + ..• + trA

A

r

be a

be an r-sequence in

is left Noetherian, and

be the ideal generated by

{t ,·,·, t } . r l

(By the Remark following Definition 1, the right ideal and the two-sided ideal generated by Suppose that the ring

{tl, ... ,t } r A

coincide.)

is I-adically complete, and that

the technical condition (TC) of Lemma 1.1.1 holds. to Lemma 1.1.1,

(TC) of Lemma 1.1.1 is equivalent to (TC ' ) of

the Note to Lemma 1.1.1). eliminated if the elements ring

Ai

(By the Note

(The technical condition (TC) can be tl, ••• ,t

r

are in the center of the

see part (2) of the Remark following Proposition 4.)

(Also, if the ring

A

course holds, since

A

is commutative, then condition (TC) of is by hypothesis Noetherian.

the Krull intersection theorem to prove

(TC ' )

(One uses

of the Note to

Lemma 1.1.1)). Let such that n

C* C

be a i

(Z-indexed) cochain complex of left A-modules

is I-adically complete for all integers

be a fixed integer. (1)

H~(A/I,C*)

i.

Let

Then if the percohomology group is finitely generated as left (A/I)-module,

Finitely Generated Ideal for the fixed integer (2)

Hn(C*)

fixed integer proof:

n.

Choose a

Cl'-indexed) cochain complex

i H (¢*)

and such that

'C*

of left

¢ *: 'C* ->- C* of cochain complexes of left i 'C is free as left A-module, all integers

A-modules such that

i H (¢*)

Then since

then

is finitely generated as left A-module for the

A-modules and a map

i,

n,

749

I

is an isomorphism, all integers

is an isomorphism, all integers

i,

i.

from

the universal coefficients spectral sequence:

and the similar spectral sequence for (1)

H!(A/I,¢*)

ci

Since image of

¢*

(~-indexed)

(2)

Since

'C

it follows that

is an isomorphism for all integers

i.

is I-adically complete for all integers under the functor

i,

the

"I-adic completion" is a map of

cochain complexes of left A-modules

(¢*)I\:

i

'C*,

('c*)1\

-+C*,

where

"1\"

= "1\1".

is free as left A-module, all integers

i,

and

since we have the condition (TC) of Lemma 1.1.1, by the conclusions of Lemma 1.1.1, we have that the left A-module is such that

(tl, ... ,t r )

(ICi)1\

is an r-sequence, all integers

i.

Therefore by Corollary 2.1 we have that H! (A/I, 'C*) "" Hi ((A/I)

@

'C*),

A

H! (A/I,

('C*)I\) ""

H! ((A/I)

@ ('C*)I\),

all integers

A

Since also

(A/I)

@

A (3)

[(,c*)I\]

""(A/I)

@

('C*),

it follows that

A

H!(A/I,'C*)""H!(A/I,('C*)I\),

all integers

i.

i.

Chapter 8

750

Combining equations (1) and (3), we see that the map (4)

Hi (A/I, (¢*)"): Hi (A/I, ('C*)") ... Hi (A/I,C*)

is an isomorphism, all integers

ci

i.

i (IC )"

But since

and

are I-adically complete, it follows from Proposition 4

Hi(~*")

below, and equation (4), that

is an isomorphism of

left A-modules,

all integers

i.

Therefore to prove the conclusion, equation

(2), of the Corollary it suffices to prove the corresponding assertion for the cochain complex conclusions of Lemma 1.1.1,

i

( I C )" , i

tl('C )"+ ... +tj(IC )" integers

i.

(tl, ••• ,t ) r

i

for the left A-module

('C*)".

of

But by (TC) and the

is an r-sequence

and the submodule i (IC )"

is closed,

l~j~r-l,

Therefore the hypotheses of Theorem 1 hold for the

cochain complex of left A-modules rem 1, we have that

Hn((,C*)")

('C*").

Therefore, by Theo-

is finitely generated as left

A-module, which completes the proof. Remark 1.

all

Q.E.D.

In the Note to Lemma 1.1.1, we have observed that the

technical condition (TC) of Lemma 1.1.1 is equivalent to condition (TC') of the Note to Lemma 1.1.1.

Another way of writing

(TC') is as: (TC") n > 0

For every integer

such that if

A _ i l

A/ (tlA + ••• + ti_lA),

m

n

~-

~-

n

there exists an integer

is the quotient ring

then

I t.A. 1;:) (I A. 1) ~

m.:. 0,

(t.A. 1)' ~

~-

A _ = i l

Finitely Generated Ideal

751

I

i < i < r-l.

Another equivalent way of writing this condition is that, (TC'" ) n>O

For every integer

m> 0

such that, for all integers

in the ring

right

(=

that

A

l~i~r-l,

A.

proposition 4. tl, ... ,t r

i,

there exists an integer

Let

A

be a ring with identity and let

be an r-sequence of elements of two-sided) ideal generated by is I-adically complete.

technical condition to Lemma 1.1.1,

(TC)

A.

Let

{t , ... , t }. l r

I

be the Suppose

Suppose also that the

of Lemma 1.1.1 holds.

(By the Note

(TC) of Lemma 1.1.1 is then equivalent to

(TC') of the Note to Lemma 1.1.1).

(The technical condition (TC) can be eliminated if the elements (Le., i f t.t.=t.t., of the ring A commute r l J J l l,2,i,2,j ,2,r); see part (1) of the Rc:aark following this Proposi-

tl, ... ,t

tion) . Let

C*

and

D*

A-modules such that all integers

n,

be (z-indexed) cochain complexes of left n C

and

and let

plexes of left A-modules.

n D

are I-adically complete for

¢*: C* .... D*

be a map of cochain com-

Then the following two conditions are

equivalent (1)

Hn (¢*): Hn(C*) .... Hn(D*)

modules, all integers (2)

is an isomorphism of left A-

n.

H~{A/I,¢*): H~{A/I,C*) ""H~{A/I,D*)

is an isomorphism

752

Chapter 8

of left (A/I)-modules, all integers (l)~

proof:

(2) follows from the universal coefficients spec-

tral sequence; to show that (1).

n.

The proof is by induction on Choose

such that

'C* 'C

n

and and

'0* 'On

Assume (2); we must prove

(2)~(1).

r.

cochain complexes of left A-modules are free left A-modules, and such that

we have 'C*..,.C*

p*:

and 1*:

'0*""0*

homomorphism of cochain complexes of left A-modules such that Hn(P*)

and

n H (1*)

are isomorphisms for all integers

n,

and

such that there exists '¢ *:

'C*..,.' 0*

a homomorphism of (z-indexed) cochain complexes of left A-modules such that the diagram '!J2*

'C*

!J2*

>0*

('C*)A

and

('O)A

C*

tions of

'C*

Let and

'0*.

1

Then since

complete for all integers

n,

'C*

(1)

1

1

p*

is commutative.

) '0*

t 1

(' C* {'

C*

*

n C

be the I-adic compleand

On

are I-adically

we obtain a commutative diagram: '~*

;;. '0*

1

(' ¢*)A ;;.(' o*{' !J2*

1 0*

Finitely Generated Ideal

753

I

The composite of the left (respectively: right) column of the commutative diagram (1) the image, call it n H

is

p*

(respectively:

T*).

Therefore

Hn(l), of the diagram (1) under the functor

has the property that the composite of the left (respec-

tively: right) column is an isomorphism. the commutative diagram Hn (¢*)

Hn(l), we deduce that, to show that

is an isomorphism, it suffices to show that

is an isomorphism.

n C

and

On

free left A-modules, all integers theses of this Proposition, A,

are I-adic completions of n.

But then, by the hypo-

(tl, ... ,t ) is an r-sequence in the r

and by the technical condition (TC)

the hypotheses

of Lemma 1.1.1 hold.

of Lemma 1.1.1, we have that

c

quence for

n

Hn(,¢*A)

Therefore it suffices to prove the Proposi-

tion in the case that

ring

By diagram chasing in

and for

on,

Corollary 2.1 we have that

we have that

Therefore by the conclusion

(tl, ••• ,t ) r all integers

is also an r-sen.

Therefore by

n H~ (A/I, C*) "" H ( (A/I) 0 C*) ,

and

A

similarly for For

0*.

r == I,

the Proposition follows from Proposition 3

(following Prop. 2, several Examples, and Cor. 2.1) of Chap. 5. If r .:: 2, and we have proved conclusion (1)

for r-l, then consider

the map ( (A/t A)0 C*) l

-+

A

The cochain complexes

((A/t Al0 0*) • l A

(A/tlA) 0 C*, A

(A/tlA) H¢* integer tlC

n

r-l.

(A/tlA) 00*

and the map

A

obey the hypotheses of this Proposition for the (Notice that, by the technical condition (TC),

cn ,

is I-adically closed in

and therefore

(A/tlA) 9 C

n

A

is J-adically complete where

J

is the image of

I

in

A/tlA,

Chapter 8

754 all integers

n.

And similarly for

On,

all integers n.)

Therefore, by the inductive assumption applied to the ring (t +t A), ... ,(t +t A) and the ideal 2 l r l n is an isomorphism for we have that H ((A/tIA) 6Sl ¢*)

(A/tIA),

the sequence

J = I/tIA,

A

all integers

n.

But then by Proposition 3 of Chapter 5

equivalently, by the case A,

the ring sor in

and the ideal

of this Proposition) applied to since

tlA,

n C

jective in

Dn,

and in

all integers

n,

is an isomorphism for all integers

Remark:

is a non-zero divi-

tl

and the endomorphism "multiplication by

A

n H (¢*)

r = I

(or,

tl "

is in-

i t follows that

n.

Concerning the technical condition,

Q.E.D.

(TC), in Corollary

1.1 and in Proposition 4, (1)

(Le.,

If in Proposition 4 the elements l~i~j~r),

titj=tjti'

t

i

Ti -+t i ,

l~i~r,

power series in

where r

Z(Tl, ... ,T > r

Z-indexed cochain complex

7.

A

by

1"'"

is the ring of formal Then every

If

I' =

then for every left A-module Z{T , ... ,Tr)-modules. l C*

Consider

such that

Z(Tl, ... ,Tr)-module.

(T , ... ,Tr)Z(Tl, ... ,T ), l r have that M'I", M'I' as

"'~{T

(T , ... , Tr> -+ A I

(Proof:

variables over the integers.

left A-module becomes a

Lemma 3 that

;l,

commute

then this hypothesis can be

eliminated, and the proposition remains true. the homomorphism of rings

1
,

M,

we

Also, for every

of left A-modules, we have by

H~((A/(tlA+ ... +trA)),C*) T 'j(Z(Tl,···,T )/(Tl,···,T ),C*). r r r

Z(Tl, .•. ,T r ),

tl, ... ,tr

if necessary, we can assume that

by

Tl, ... ,T

Therefore replacing r

A =Z{T , ..• ,T ). r l

trivially, the condition (TC) holds for the ring completing the proof).

and

I

by

I'

But then, A=Z(Tl, ... ,T ), r

Finitely Generated Ideal (2)

, 1 ~i ~r, are i then the technical hypothesis,

If in Corollary 1.1 the elements

in the center of the ring

755

I

A,

t

(TC), of the Corollary can be deleted, leaving a true Corollary. (Proof:

¢:;Y(Tl, ... ,T ) -+A r

Let

rings with identity such that

"H!(T

i,

1 < i < r.

+ '"

Then re-

+ trA) ,0*)"

with

T) (Z(Tl,···,T )/(Tl,···,T ),0*)", r r r all cochain complexes 0* of left A-modules) (L

(all integers

¢ (T i) = t , i

"H~ (N(tlA

placing the left A-modules, the left A-modules,

be the unique homomorphism of

1"'"

in the proof of Corollary 1.1, we see that the proof of Corollary 1.1 goes through with this substitution throughout. 3

But Lemma

¢:Z(T , ... ,T ) -+A then l r completes the proof of Corollary 1.1 (in the case that we delete applied to the homomorphism of rings

the technical hypothesis (TC) in that Corollary, but assume instead that

tl, ... ,t

are in the center of the ring

r

A). Q.E.D.

The proof of the part (2) of the last Remark also proves Corollary 1.2.

Let

A

be a ring with identity and let

I

be

a finitely generated (left, right or two-sided) ideal in the ring

A.

Suppose that the ring

that there exist an integer I

that generate the ideal

two-sided ideal) such that ring

A. Let

r I

A

~

0,

with identity such that

and elements

t

l

, ... , tr

in

(as left ideal, right ideal, or

tl, ... ,t

Suppose also that the ring ¢ :;y (T l' ... , T r) ... A

as I-adically complete, and

r

are in the center of the A

is left Noetherian.

be the unique homomorphi sm

¢ (T ) = t , i i

1 < i < r.

Then let

0

f rings C*

be

a fixed z-indexed cochain complex of left A-modules, such that c i is I-adically complete, for all integers i, and let n be any fixed integer.

Then if

756

Chapter 8 (1)

Hn'T T) (;Z\Tl,···,T )/(Tl,···,T ),C*) ;Z'l""'r r r

is finitely generated as left (A/I)-module, then (2)

Hn(C*l

Remarks:

1.

Corollary 1.2 is an improvement on Corollary 1.1

(in the case that A),

is finitely generated as left A-module.

tl, ... ,t

r

belong to the center of the ring

since the (strong) hypothesis, that

r-sequence for the ring 2. "tl, ... ,tr

A",

"(t , ... , trl l

is an

has been deleted.

The full strength of the hypothesis, that belong to the center of the ring

A",

appears neces-

sary for the statement of Corollary 1.2 (since otherwise

~/T ~IZ,

1"'"

T) (;Z\Tl,···,T )/(Tl,···,T ),C*) r

r

r

does not in general

possess a natural structure as left A-module) • 3.

In Corollary 1.2, one can replace

";Z\T ,· .. ,T )" r l "cjl:;Z(T , ... ,T ) ->- A" r l

throughout with

"A(T , ... ,T )", replacing r l by the unique homomorphism: "A(T , ... ,T ) +A" I r identity that induces the identity on

A

(of rings with

and is) such that

T.

maps into

l

t , 1 < i < r. By Lemma 3, this will not affect the i percohomology group appearing in equation (1) of Corollary 1.2. (I.e., one can replace the percohomology group in equation (1)

of Corollary 1.2 by the canonically isomorphic group, HAn(T

1"'"

4.

T) (A(Tl,···,T )/(Tl,···,T ),C*». r r r However, very definitely, under the hypotheses

of Corollary 1.2, one

~

not replace the percohomology group

in equation (1) of the corollary with

"H~(A/I,C*)'" (unless

(t , ... ,t ) r l

is an r-sequence in the ring

A),

since

Finitely Generated Ideal when

(tl, ... ,t ) r

757

I

is not an r-sequence in the ring

H~(A/I,C*)

percohomology group

A,

the

is very different from

~(T T \ (Z(Tl,···,T )/(Tl,···,T ),C*). Z I'···' n' n n 5.

Can the technical condition (TC) in Proposition

4 and in Corollary 1.1 be deleted in all cases (even if tl, ... ,t

r

do not commute

in the center of the ring

elementwis~

A)?

or, respectively, are not

In the Remark following Proposi-

tion 4, we showed that this was the case, if in, e.g., Propositl, ... ,t

tion 4,

commute.

r

Studying that proof, we see

that we are led to study the "free ring" exists an r-sequence

t

tity with generators

tl,.··,t r ,

l

R

such that there

, ... , tr E R -- i. e., the ring with idenXijk

(l~k~i<j~rl,

Xkjkal£l···as£s all integers

s

~

1,

with the relations: l
each integer

s > 1.

It is not difficult to see that, if tity, and

(ul' ... ,u ) r

A

is any ring with iden-

any r-sequence in the ring

there exists a homomorphism of rings with identity that

cP (ti'

=ui '

1 < i < r.

A,

then

cp:R + A

such

It seems possible (but the negation

758

Chapter 8

also seems quite possible) ideal generated by

I = tl R +

that, if

(tl, ..• ,t ), r

+t R r

is the

then

(t , .•• ,t ) is an r-sequence for the ring R/\I. (It r l is certainly true, by the construction, that the first condition (1)

of Definition 1 of "r-sequence" holds), and that (2)

Condition (TC) of Lemma 1.1.1 holds for

A=R/\J..

If indeed (1) and (2) above hold, then the proof of the first part of the Remark following Proposition 4, would enable one to remove the technical condition (TC) from the statement of Proposition 4. It is not difficult to show that

is the ring

of non-commuting polynomials in denumerably many variables over the corresponding ring for the integer inductive effort to study that,

tlRAI

Therefore, an

(1) and (2) should focus

(left multiplication by

whether

r - 1.

on proving

t

) is injective in l has the induced topology in RAI. (*)

R/\I,

and

A similar effort to remove the hypothesis (TC) altogether from Corollary 1.1, would lead one to study a more complicated set of rings than

R.

If

A

is as in the hypotheses of Corol-

1ary 1.1 (deleting the hypothesis, .. (TC) "), then one needs a ring

5

together with a sequence

the completion by

(ul' •.. ,u r )

t

of

5

(ul' ••• ,u ) r

in

5

such that

with respect to the ideal generated

is such that

(ul' ... ,u r )

is an r-sequence, is

such that the technical condition (TC) holds, and is such that we have an epimorphism 1 < i < r.

4>:5

/\

.. A

Also, one needs that

of rings such that 5/\


removing (TC) in Proposition 4, one needs only such a homomor-

(*) Prop. 4 holds, even without (TC},for

any two-sided ideal I.

Finitely Generated Ideal phism

rjJ,

759

I

and does not need a "left Noetherian" assumption.)

The reason

rjJ

must be an epimorphism for affecting

an improve-

ment in Corollary 1.1, is that one needs to know that finite generation of

as left

sA-module

as left A-module.

is equivalent to finite generation of

«1) and (2) are always isomorphic canoni-

A

cally as left S -modules by Lemma 3. cp:SA -+A

If the homomorphism

is surjective, then "finite generation as left SA_ mod -

ule" becomes equivalent to "finite generation as left A-module".). (Of course, trivially, if

A is a ring, complete with re-

spect to the right ideal generated by an r-sequence such that exists a left-Noetherian ring

S,

tl, ... ,t , r

complete with re-

spect to the right ideal generated by an r-sequence S

such that

obeys the technical condition (TC), and such that

there exists an epimorphism that

cp (u ) = t

i

ul, ... ,u ' r

i ,

1

~

i

~

r,

rjJ:S -+A

of rings with identity such

then the ring

A

also obeys the

hypothesis (TC)). 6.

Of course, the hypotheses in Theorem 1, in Corol-

lary 1.1 and in Corollary 1.2, that "the ring

A

is left

Noetherian" can be eliminated, if we were to modify the statement of the Theorem and Corollaries appropriately.

However,

such a modification is rather bulky and awkward in the case when the ideal

I

is not simply generated.

some finite sequence

(tl, ... ,t ) r

(One has to fix

of elements of

I

that gen-

Chapter 8

760

erate

I.)

The length of the necessary hypotheses to eliminate

the indicated quoted hypothesis increases with the length

r

of the sequence. A strengthening of Proposition 4, analogous to the strengthening, "Corollary 2.1" of "Corollary 1.1", can be stated, in which

"Hn(T T ) (;l.(Tl,···,T )1 (Tl,···,T ) ,C*)" replaces Z 1'···' r r r "H~(A/I,C*)"; (or, one can work with "Z[Tl, ... ,Trl" in lieu

of

"Z(Tl, ... ,T )".) One has to assume that tl, ... ,t r r mute, but one removes the very strong hypothesis that is an r-sequence for the ring

A".

com-

We state this

result as a Corollary to proposition 4. Corollary 4.1.

Let

A

be a ring with identity and let

a two-sided ideal in the ring exists an integer

r> 0

A.

I

be

Suppose also that there

and elements

t , ... , tr that generate l the ideal I as right ideal such that the elements tl, ... ,t r COITUllute in the ring A, (Le., such that to' to =to· to, J

1

l~i,j~r).

Let

¢:Z[Tl, ... ,Trl +A

J

1

be the unique homomorphism

¢(T ) =t , l
of rings with identity such U,at Let

C*

and

0*

lexes of left A-modules, such that complete, all integers

n.

Cn

and

on

are I-adically

Let

p*: C* .... 0* be an arbitrary map of cochain complexes of left A-modules. Then the following two conditions are equivalent

integers

n.

Finitely Generated Ideal (2)

H~[T fL.

1"'"

761

I

T ](;?[Tl,···,T ]/(Tl,···,T ),p*): r

r

r

->Hn[T T 1 (;? [Tl,···,T l/ (Tl,···,T ) ,0*) ;? l " " ' r r r is an isomorphism, all integers

n.

Proof: By means of the ring homomorphism q,:;?[Tl, ... ,Tr]-+A, n C and on are ;?[Tl, ... ,Tr]-modules, all integers n. Since n C and On are I-adically complete, Cn and on are also ;Z(Tl, ... ,Tr)-modules.

If

I'

Tl,···,T r

in ;Z(Tl, ... ,T r ),

I

and

by

I'

tl, ... ,t r

is the ideal generated by then replacing

by

Tl, ... ,T

;Z(T , ... ,T ), r l if necessary, we can

r

by

, 1 < i < r. But i then the hypotheses of Proposition 4 all hold (the technical

assume that

A =;Z(T l ,··· ,T r ),

I = I' ,

A

Ti = t

condition (TC) not being needed by the first part of the Remark following Proposition 4).

Therefore, by Proposition 4, condi-

tion (1) of this Corollary is (2' )

~(T

-7.

an(T -7

1"'"

1"'"

eq~ivalent

to:

T > (;7 ( T l , ... , T ) / (T l , ... , T ), p *) : r r r T) (;Z(Tl,···,T )/('l'l,···,T ),C*) r

r

r

n --+H71 (T T) (;Z(Tl,···,T )/(Tl,···,T ),0*) 1"'" r r r is an isomorphism, all integers

n.

But by Lemma 3, equations (2) and (2') are equivalent.

Q.E.D.

An immediate consequence of Corollary 1.2, in which statements about ordinary cohomology replace percohomology, is Corollary 1.3.

Let

an ideal in the ring

A A.

be a ring with identity and let Suppose that the ring

A

I

be

is I-adically

Chapter 8

762

complete, and that we have an integer tl, ••. ,t r

in

I

r >0

that generate the ideal

and elements I

such that

t , .•. ,t are in the center of the ring A. Let 1 r CP:Il[T1, ... ,T r ] +A be the unique homomorphism of rings with rp (T i) = t , i is left Noetherian.

identi ty such that ring

A Let

C*

plex of left

7 [T , i-or:!. 1

••• , T]

i,jEIl,

i>O.

Let

n

(1)

Hn+i(D~)

then (2)

r (Il [ Tl ' ... , Tr

be any fixed integer. l.

for

i, let

be the cochain com-

D'l'

l.

(A/I)-modules, such that

oj - 'T

]/ (

Tl' ••• , Tr ) ' Cj ) ,

Then if

is finitely generated as left (A/I)-module,

0':' i.:. r,

Hn(C*)

the fixed integer Proof:

is I-adically complete for all inte-

Then for each integer

i.

Suppose also that the

be a fixed (Il-indexed) cochain complex of left

A-modules, such that gers

1 < i < r.

is finitely generated as left A-module, for n.

We have the second spectral sequence of percohomology, Il[Tl,· .. ,T r ] p Tor-q (Il [TI, ..• ,T ]/ (Tl, ••• ,T ),C ) r r q ==I> HfJ+[T T ](.z[Tl,···,T ]/(Tl, .. ·,T ),C*), .l 1"'" r r r

for which

By definition of

all

p,qEIl,

Also, by Lemma 2,

IEP,q 1

01, q

~

=0

P 'EP,q Dq' 2

we have that 'EP,q 1

= HP(D*) q ,

O. if

q~r+l

(or

q

<

-1) .

Therefore,

Finitely Generated Ideal

763

I

from the general theory of spectral sequences, since the ring A

is left Noetherian, to prove that the n'th group of the

abutment is finitely generated as left A-module, it suffices to prove that 'E~,q

is finitely generated as left A-module for

D~q~r.

p+q=n,

of the Corollary.

But the latter is exactly hypothesis (1) Therefore the n'th group of the abutment is

finitely generated as left A-module, i.e.

H~[T

(1' )

fL.

1"'"

T 1 (nTl,···,T l/(Tl,···,T ),C*) n n n

is finitely generated as left A-module.

But, by Lemma 3, the

percohomology group in equation (1') is canonically isomorphic ~(Tl,

to the corresponding percohomology group with replacing 1.2

Z[TI, ... ,Tnl.

... ,Tn)

Therefore equation (1) of Corollary

holds, and therefore by Corollary 1.2 we have that equaQ.E.D.

tion (2) of Corollary 1.2 holds. Remarks 1. with

In Corollary 1.3, one can replace

";r(TI, •.• ,T n )",

"A[Tl, •.• ,Tnl"

or

~[Tl'

... ,Tnl

"A(TI, ... ,T n )"

according to one's preference. The Corollary remains valid, R i since by Lemma 3 Tor (R/ (TIR + ... + T R),C ), all integers q

q, i,

q

~

is independent up to canonical isomorphisms of

0,

R=Z[TI,.··,T n ], 2.

r

If

Z(TI, .. ·,T n ), (tl, ... ,t ) r

A[TI, ... ,TnJ

or

A(TI, ... ,T n ).

is an r-sequence in the ring

A,

in Corollary 1.3 one can define .

A

~

l

.

D~=Tor.(A/I,CJ),

all integers

i,j

with

(since then, again by Lemma 3, this definition of

i>O

D~

~

coincides,

up to canonical isomorphism, with the one given in the Corollary) .

Of course, if

(tl, •.. ,t ) r

is not an r-sequence, then

764

Chapter 8

one cannot do this . Of course, there is a statement analogous to Corol-

3.

lary 1.3 in the situation of Corollary 1.1, with percohomology over

A

replacing percohomology over

Corollary 1.3'.

Let

A

respect to an ideal

~

[T , ... ,Trl. l

be a ring with identity, complete with

I,

such that

I

is the right ideal gener-

ated by an r-sequence (t , ... ,t ) for the ring l r ger >0.

Suppose that the ring

the technical condition Let

C*

c

such that let

n

Namely,

A

A,

r

an inte-

is left Noetherian, and that

(TC) of Corollary 1.1 holds.

be any fixed cochain complex of left A-modules i

is I-adically complete, all integers

be a fixed integer.

i,

Define cochain complexes

and D~ 1

of

left (A/I)-modules by requiring that .

A

1

1

.

D~ = Tor. (A/I}:]) ,

all integers

Then if, for the fixed integer (1)

0:" i

:.. r, (2)

H

n +i (D~)

i, j

wi th

i > O.

n,

is finitely generated as left (A/I)-module,

1

then Hn(C*)

is finitely generated as left A-module.

(The proof is similar to that of Corollary 1.3.) An amusing application of Corollary 1.3 is given by the following four examples. Example

1.

Let

identity, let

t l , ... ,t r integer

A

I

be a Noetherian commutative ring with

be an ideal in the ring

A

and let

be a set of generators for the ideal ~O).

scheme over

Let X

X

I,

(r

an

be an arbitrary locally Noetherian pre-

(i.e., a prescheme that is locally isomorphic to

spectra of Noetherian rings).

Then we have the completion

Finitely Generated Ideal

Ox

of the structure sheaf

765

I

with respect to the ideal

I,

which is a sheaf concentrated on the closed subset: Y = xspe~ (A) Spec (A/I)

(~I) -modules over we would say,

of

Let

F

be any coherent sheaf of

(In the language of "formal schemes",

X.

"let

X.

F

be any coherent sheaf of modules over the

structure sheaf of the formal prescheme Let

M

i> 0,

u

such that

I

U

'

M = O.

Then, for each in-

consider the assignment: A

U'V'v>Tor (M,f(U,F», i

(1 )

where

Y) ").

be any finitely generated A-module such that there

exists an integer teger

(y,O~II

U

runs through the affine open subsets of

X.

Then the

presheaf (1), restricted to the open base consisting of affine open subsets of

X,

Ox-modules over

X -- in fact, is a coherent sheaf of

Ox

is a sheaf, and is a coherent sheaf of

x S (A/lu)-modules over the prescheme, Spec (A) pec

(Y, Ox

x S ec (A/l u ) ). (Notice that we are speaking here of a Spec (A) p .

coherent sheaf over the structure sheaf of a prescheme, rather than of a coherent sheaf over the "completed up" structure sheaf of a formal prescheme). Proof: ideal

Let I.

tl, ... ,t r

be a finite set of generators for the

By induction on

j,

o~ j

~

r,

I claim the assertion

is true if there exists an integer

B>0

such that

B t j +1 • F = •••

then

tk • F = 0,

B

+ tr • F = O.

and therefore

IrB. F = 0,

If

j=O,

whence

modules over the sheaf of rings

F

B

is a coherent sheaf of and therefore is a

Chapter 8

766

°-

coherent sheaf of

modules over

x

Then the assertion

X.

follows from the elementary facts that, if Noetherian algebra over U

A

(in the application,

any affine open subset of

generated A-module and the application,

N

X),

i,

N=r(U,F»,

M

M

is a finitely

A

then

Tori(M,N)

isafinitely

i > 0 -- this is proved by in-

using the long exact sequence of Tor's associated

to any short exact sequence some integer

then if

R=r(U,Ox),

is a finitely generated R-module (in

generated R-module, all integers duction on

R is a commutative,

s

~

0,

0 .... M' .... As .... M .... 0

of A-modules for

(such a short exact sequence exists since

is finitely generated as A-module)--and the fact that

A A Tori(M,Nf)""Tori(M,N)f' ules

M,

all integers

all R-modules i >0

all fER, all A-mod-

(this latter fact not even re-

quiring any Noetherian assumptions on following from the fact that

N,

Tor~1

A

and

R,

but just

commutes with direct limits

over directed sets.» Assume the assertion is established from some

o 2. j

< r - 1.

To prove the assertion for

B'

that, if

B'

tj+2 • F = ... = tr

• F = 0,

j + 1.

j,

We must show

then the pre sheaf (1) on the

open base consisting of affine opens is a coherent sheaf on that base over the sheaf of rings lU·M=O, of

X,

I

u

A • Tor. (M, r (U,F» 1

= 0,

Since all affine open subsets

U

and therefore clearly the presheaf (1) is a presheaf of

modules over

O~l /lu • O~I "" 0X/lu • OX.

Therefore it suffices

to prove that the presheaf (1) on the indicated open base, (which is therefore a presheaf of base on

Xl,

Ox - modules over that open

is both a sheaf and coherent over

show that, for every affine open subset

U

of

OX. X,

I.e., to if

Finitely Generated Ideal R=r(U,Ox)' (2)

every

and

N=r(U,F),

A

then

is finitely generated as R-module and for

Tori (M,N)

fER,

767

I

the natural map is an isomorphism

(3)

for all integers

i > O.

ated module over

RAI,

B'

(Notice that

N

is

a finitely gener-

and that we know that

B'

tj+2 ' N = ... tr ON = 0;

and that by the inductive assumption, we

know that equations (2) and (3) are true if there exists an integer

C>0

C

such that

tj+l

tj+l-torsion element in

N.

°

Let

N = 0) •

Then since

ated module over the Noetherian ring

N o N

RAI

ing Example 4 below for the proof that

be the set of

is a finitely gener(see Remark 2 follow-

R0

I

is Noetherian) it

is likewise finitely generated follows that the submodule N o AI as R - module. Since NO is a tj+l-torsion module, there exists an integer j

C> 0

+l~k~r,

such that where

C

t j + l No = O.

D=sup(B',C).

But then Therefore by the

inductive assumption, equations (2) and (3) hold if we replace N

with

No'

Throwing the short exact sequence

o -+ No

-+

N -+ N/N

0

-+

0

of A-modules through the exact connected sequence of functors: A

(Tori(M, »i>O' hold for N

No'

and using the fact that equations (2) and (3) we see that, to prove equations (2) and (3) for

it suffices to prove them for

N/N . o

I.e., we're reduced to

proving equations (2) and (3) in the case that generated

If I

But then

module with no non-zero u

NI = N/tj+IN

N

is a finitely

tj+l-torsiono 1\1

is a finitely generated R

-module,

768

Chapter 8

and

where

D

= sup (u, B')

,

so that by

the inductive assumption, equations (2) and (3) hold with replacing

N.

Nl

The long exact sequence of Tor's associated to

the short exact sequence of A-modules

is the long exact sequence: U

d i +1

A

A

t'+ l

A

di

--"'> Tor. (M,N)~>Tor. (M,N) ...... Tor. (M,N )--=-;>

l

1 1 1

Since

1

U

' M = 0,

we have that

t

U j+l

endomorphism, "multiplication by A

Tori (M,N)

t

. M = 0,

u " j+l

i > O.

is zero, all integers

and therefore the

of the A-module Therefore the above

long exact sequence of Tor's yields the short exact sequences

i ,:,1,

all integers A

Tori_l (M,N) i,.:,l,

from which it follows immediately that

is finitely generated as R-module, all integers

and also (by induction on

that equation (3) holds.

i)

This completes the inductive proof, that the presheaf (1) on affine opens is a coherent sheaf of

0x/1u. Ox - modules over

Y . Example I

2.

Let

B

be any commutative ring with identity, let

be any ideal in the ring

O~ = l,im(0wl1jOw)'

B,

let

W= Spec (B) ,

let

a sheaf concentrated on the closed subset

j,:, 0

of

(I = 0)

W

(let

Y

denote this closed subset).

any quasicoherent sheaf of (either over

X,

and let

F

OW-modules, or .

~1

= G· = Itm(G/1J j":'O

• G).

Then

Let

G

be

~1-mOdUleS)

F is concen-

Finitely Generated Ideal trated on the closed subset

Y

of

Hi(W,F)=Hi(Y,FIY)=o,

(1)

X,

769

I

and

all integers

i>l.

(This is of course very well known, and fairly trivial.

We in-

clude a proof for completeness.) proof:

Let

m

be any finite set of affine open subsets of

such that the union is (2)

W.

W

Then we show first that

~i(m,W,F)=O,

i>l.

v

v '

In fact, C* (m ,W,F) = Ij,m C* (ur,W,G/IJG). j.:=:O

Also, the mappings in the inverse system of which the inverse limit is taken are epimorphisms. of Chapter (3)

3,

we have the short exact sequence

0+ [lj,m l Iii-l(m ,W,G/IjG)] +i'li(UI ,W,F) j.:=:.O V·

+

all integers sheaf of

Therefore, by Corollary 1.1

.

[l.im Hl(UI ,W,G/IJG)] j.:=:.O i> 0.

+

0,

But the sheaf

is a quasicoherent

Ow-modules over the affine scheme

collection of affine open subsets of W.

G/IjG

X

W,

and

i >

O'

Also, from equation (4), when

jr

(W,G/IjG),

i = 0,

is a

such that the union is

Therefore

=

UI

the map

°

i = 0.

Chapter 8

770

is an epimorphism, all integers

j::. 0,

so that

(5)

Equation (4)

for

i> 1

and equation (5) plugged into the short

exact sequence:; (3) imply equation (2). Next, noting that the set of all finite collections

X

open subsets of sets of

X

of

ill

the elements of which are affine open sub-

such that the union of

ill

X,

is

is cofinal in

the eech sense, and passing to the direct limit over such ill, we obtain that (6)

lii(W,F) = 0,

Replacing

all integers

W by any affine open subset in

larly, for every affine open subset (7)

i> 1.

vi H (V,F) = 0,

V

Finally, let

X

of

we have simi-

W,

that

i> 1.

be the set of all affine open subsets

the topological space topology of

all integers

V

W,

X.

Then

V

V

of

is an open base for the

closed under finite intersections.

By Theorem

5.9.2., pp. 227-228, of [G], it follows that

(8)

i

vi

H (W,F)"OH (W,F),

all integers

i.

Equations (6) and (8) imply that H

Since also

i

(W,F) F

=

0,

i > O.

is concentrated on the closed subset

Y

of

W,

Finitely Generated Ideal

771

I

These two equations imply equation (1). ExampLe 3. let

X

Let

A

Q.E.D.

be a commutative ring with identity and

be a locally Noetherian prescheme over

be any finitely generated ideal in the ring let

tl, ... ,t

ideal

I.

F

A.

Let

Let

r>

°

I

and

be a sequence of elements that generate the

r

Let

F

be any coherent sheaf of

the topological space let

Spec (A) .

X

°

AI

X - modules over

(in the language of "formal schemes",

be a coherent sheaf of modules over the structure sheaf

of the formal scheme:

I

AI Y), (Y,Ox

Let

U

Y

is the closed subset

be a fixed open subset of

Let such that

CP:il [T l , ... ,Trl -+A cP

each integer

where

where

m ~ 0,

of

X).

Y.

be the unique ring homomorphism

1 < i < r.

(T ) = t , i i

"I = 0"

Then by Example 1 above, for

the assignment:

W runs through the set of all affine open subsets of

X,

is a coherent sheaf of modules over the structure sheaf of the (Where

prescheme

l~i~r,

"A",

"Ti"

"I",

M=,l[Tl, ... ,Trl/(Tl, ... ,Tr)'

ger

m,:,O,

replaces

let

".". [T l' ... , mlr 1" fL.

"ti'"

replaces

"(Tl, ... ,T r )"

and

u=l).

replaces

For each inte-

TO~[Tl,···,Trl(il[Tl, •.. ,Trl/(Tl, ... ,Tr),F))

denote this coherent sheaf of

(Ox/I)-modules over

Suppose also that the topological space pact and that the ring fixed integer such that

A

is Noetherian.

Y

Y.

is quasicom-

Then, if

n

is any

772

Chapter 8 is finitely generated as

(A/I)-module,

02i2r,

then Hn(Y,U,F)

(3)

is finitely generated as

fixed integer Proof:

A-module for the

n.

First, I claim that there exists a covering

of

ill

Y

(in the sense of [P.P.W.CJ, Chapter I, pg. 129), the elements of which are affine, such that

is finite, and such that

ill

[P.P.W~J

is a refinement (in the sense of covering

{Y,U}.

In fact, since

Y

pg. 128, line 4) of the

is a quasi-compact and

locally Noetherian prescheme, so is the open subset Therefore there exists a finite collection subsets of ill',

Y

such that

Y

ill'.

intersections of elements of ill"

ill'

Then

U

Let

Y.

of affine open

of

Y

can be written

ill" = {all finite non-empty

ill'}.

ill'" is a covering of

Chapter I, pg. 127),

Let

ill III = all elements of

ill'"

Y

(in the sense of [P.P.W.CJ,

is finite and every element of

is an open subset of a Noetherian affine scheme. induction on the finite set E E ill III

[P.P.W~.l,

illE

ill '"

U['''

By decreasing

ordered by inclusion, for

we define a finite covering (in the sense of

Chapter I)

Having defined fine

of

that cannot be written as unions of strictly smaller ele-

ments.

each

U

is the union of the elements of

and such that the open subset

as a union of elements of

ill

illE

ill , E

of

E

for all

by affine open subsets of E'Eill lll

to by any finite covering of

such that E

E~

E',

E. de-

that is a refine-

ment (in the sense of [P.P.W.C.], Chapter I, pg. 128, line 4) of H smaller elements of

cannot be written as a union of strictly ill

O

'

where

illO ={F

n E:

F E ill E'})'

for

Finitely Generated Ideal all

I) ill

ill = U ill . EEill E is a finite set, and is a covering ([P.P.W,CJ, Chapter

E' E ill '"

Then

ill of

773

I

Y

E~

such that

E'.

Then define

the elements of which are affine opens, and such that

is a refinement ([P.P.WCJ, Chapter I, pg. 128) of the covering

{x,u}. paper,

(Note:

This construction is similar to one used in the

"On a Conjecture of Andre Weil", [CAW,], Amer. J. of Math.,

Chapter I, in proving that combinatorial cohomology of sheaves forms a system of derived functors). Next, let (4)

E* = C* (ill, (Y,U) ,F),

as defined in [P.P.WC.l, pg. 128, lines 22-26.

Then

Ei

is a

(finite) product of I-adically complete A-module3, and is therefore

I-adically complete, iE;y.

=

Also, for all integers

i,

j

~

0,

:7[T, Tor. 1 ... ,Tl( r J' [ T , ... , T l/( T , •.. , T ) , E j) . l r r l l

(This is so, since the covering the functor

ill

j G'VU> C (ill , (Y, U) , G) ,

of abelian groups on

Y

is finite, and therefore (from the category of sheaves

into the category of abelian groups)

is the finite direct product of functors of the type: G'VU>f (Wi'G),

i E S,

where

affine open subset of

Y.

is a finite set, and

S

Y,

"Cj(ill,(y,U),

)",

"f(W , )" i

replacing the func-

w i ·lS an a ff'lne open su b se t a f

h were

and this latter assertion follows from Example 1.)

then, since the elements of is quasicoherent over

is an

Equation (5) therefore follows from

the corresponding equations with tor

Wi

ill

(O~I) iY,

But

are affine open, and since and by Example 1 the sheaves

F

Chapter 8

774

TO?[Tl,···,Tnl(;l[Tl, ... ,T 1/(Tl, ... ,T ),F) r

1

0x/I. Ox'

are coherent over

r

we have, by ([P.P.W.c.l, I. 6, Theoj~O,

rem 7, pg. 152), for all integers

(7)

that

Hj (y,U,TO~ [T l ,·· ..,Trl (;l [T ,.· .,Trl/(T , •.. ,T ) ,F) l r l

""Hj(TO~[Tl,···,Trl(;l[Tl, ... ,T 1/(Tl, ... ,T ),E*)). r

1

(Here we are using the fact, for

W Em,

since

r

W is affine

open, we have that j H (W,F) = 0,

all integers

Hj (W,Tor/ [T l ,·· .,Trl

= 0,

all integers

j

j

~ 1,

(d'

[T , ... ,Trl/(T , ... ,T ) ,F)) r l l

~

1.)

(The second of these assertions follows from Example 1, since by Example 1

Tori;l[Tl,···,Trl (;l[Tl, ... ,Trl/(Tl, ... ,Tr),F)

coherent over the structure sheaf of the prescheme and

W is affine open in

Y.

A,

E*

the ideal

(in lieu of

I, the integer

of elements in the ring lary 1.3 above.

I

(Y, (Ox/I)1 Y)

The first of these assertions

follows from Example 2 above) of A-modules

is

But then the cochain complex C* r >0

in Corollary 1.3), the ring and the sequence

t

1

,··· , tr

obey all the hypotheses of Corol-

The conclusion of Corollary 1.3,

(that equa-

tion (1) of Corollary 1.3 implies equation (2) of Corollary 1.3), and equations (6) and (7) above show that, equation (2) implies equation (3), completing the proof of this Example. (in the notations of Corollary 1.3, To r~

D~= 1

[T l' ••. , T r 1 (z [T l' ... , Tr 1I (T l' ..• , Tr) , E *) )

Finitely Generated Ideal 4. let

Let

A

775

I

be a Noetherian commutative ring with identity and

be a Noetherian prescheme over Spec (A).

X

any ideal in the ring

A

Y = X Specx (A) Spec (All)

Let

I

be

such that the prescheme is proper over

(The word

Spec (All).

"proper" implies "of finite presentation").

The (I = 0) . palr

is a sheaf concentrated on the closed subset: of

Let

X.

(Y,OAxI

ly )

Y

denote this closed subset.

. t'lng conS1S

0

f th ioglca ' 1 space e topo

F

Y

and

O~I

is a formal prescheme.)

be any coherent sheaf of

(OA I) _ modules over the

this "sheaf of topological rings" Let

(Then the

topological space

X

X

(or equivalently over

Y).

(Then

F

is

a coherent sheaf of modules over the structure sheaf of the formal pre scheme

AI

(y,Ox

IY)).

Then Hn(Y,F)

is finitely generated as

all integers Proof:

Taking

U

(AAI)-module,

n > O.

to be the empty set in Example 2 above, it

suffices to prove that each of the sheaves: F. =TO?,[Tl,···,TrJ (;r[Tl, ... ,T J/(Tl, ... ,T ),F), 1

for

r

1

O.s. i .s. r,

r

have finitely generated cohomology groups as

(A/I)=modules over the topological space replacing

Y.

"A", "Ti"

But, by Example 1 replacing

we have that the sheaves

lit.

II

1

F.

1

are

776

Chapter 8

coherent sheaves of over Spec (A/I), H

j

y.

Since

Y

is proper

it follows that is finitely generated as

(Y ,F. ) ~

all integers

0y-modules over

j,:::,O,

all integers

i

with

(A/I) -module, O.:':i~.r,

as re-

quired.

Q.E.D.

Remarks. 1.

Of course, a special case of Example 4 is the one

in which

is proper over Spec (A),

X

which is of course very

well known. 2.

In the course of proving Example 1, we used the

fact that, if and if

I

B

is a Noetherian commutative ring with identity

is any ideal in the ring

Noetherian.

B,

then

BAI

This is of course very well known.

be deduced very easily as a Corollary of

is also

But it can also

Lemma 1.1.1 at the be-

ginning of Chapter 5. Proof:

A consequence of that Lemma is that, if

B

is a

Noetherian commutative ring with identity, then so in

B(T),

the ring of formal power series in one variable over then by induction on the integer But if

I

is any ideal in

by r-elements, then

3.

BAI

B,

and if

so is I

But

B(T ,··· ,T r ). l

is generated as ideal

is isomorphic to a quotient ring of

The proof of the Lemma 1.1.1 of Chapter 5 proves

slightly more (by induction on (1)

r,:::, 0,

B.

Lemma.

Let

B

r):

be a (not necessarily commutative)

ring with identity that is left Noetherian, and let

I

ideal in B

and a set

tl, ... ,t

r

such that there exists an integer of elements of

I

that generate

r >0 I

be an

as ideal and

Finitely Generated Ideal such that the right ideal generated by

777

I

tl, ... ,t

i

is a two-

(t ,··· ,t ) r l obeys condition (1) of Definition 1 at the beginning of this sided ideal,

chapter).

(Le., such that the sequence

Suppose that the ring

BII

Then

1
B

is left Noetherian iff

B

is I-adically complete. is left Noetherian.

That Lemma has the immediate Corollary

(2)

Corollary.

Let

B

be a (not necessarily commutative) ring

with identity and let

I

be an ideal in the ring

left Noetherian, and if there exists an integer sequence I

tl, ... ,t r

of elements of

B

B. r > 0

If

B

is

and a

that generate the ideal

and such that the right ideal generated by

two-sided ideal, all integers the sequence

(t , ... ,t ) r l

i,

1 < i
{tl, ... ,t } is a i (L e., and such that

obeys condition (1) of Definition 1

at the beginning of this chapter), then the I-adic completion BtU

of the ring

I

is also a left Noetherian ring.

By the Lemma applied to the ring it suffices to prove that

(BAllI

BAI BAI)

But this letter ring is isomorphic to Noetherian since Remark.

B

and the ideal

(Proof: I . BAI,

is left Noetherian.

B/IB,

which is left

is left Noetherian) .

In the course of proving Example 3, we needed to know

that, in the notations of Example 3, i f F is a coherent sheaf AI and if W is an affine open subset of Ox - modules over X, of

X, (1)

then Hi (W,F) = 0,

i > 1.

Of course, this is actually covered in Example 2. in Example 2, equation (1) is proved, not for

F,

However,

but for

therefore to make use of Example 2, we must know that

F = FAI.

778

Chapter 8

Since

F

is coherent over

°X

AI

it is equivalent to show

'

that: 1.

If

M

is a finitely generated module over the

Noetherian commutative ring

B,

and if

I-adic topology for some ideal ly complete. AI

B=

r (W, OX

),

Ai

where

W

I = ideal in

since

B,

F

is complete for the then

M=

M

is I-adical-

r (W, F) ,

generated by the ideal

B

is any affine open subset of

rank (the set of such Y

in

(In the appl ica tion, take

is the quotient of a free

of

I

B

AI

(OX

) Iw

X

I

in

such that

Flw

sheaf of modules of finite

W's forms an open base for the topology

is coherent over

0X

AI

)).

Observation 1., which is of course well-known, follows from Lemma 5 below.

(Note:

The proof of Lemma 5 is much easier in

the commutative case). Lemma 5. let

I

Let

A be a left-Noetherian ring with identity and

be an ideal in the ring

I-adically complete. and a sequence

A

such that the ring

A

is

Suppose that there exists an integer

tl, ... ,t r

of elements in the ring

A

r >0

such

that, (1)

For each integer

ideal generated by equivalent to say,

i,

1,::, i::r,

we have that the right

{tl, ... ,t } is a two-sided ideal. i "and such that the sequence:

(It is

obeys condition (1) of Definition 1 at the beginning of this chapter". ) Suppose also that (2)

There exists an integer

N> 0

1.::. i.::. r,

such that and

Finitely Generated Ideal (3)

There exists an integer

B> 0

779

I

such that

t. 'A.t~ct.A, J

1

Then, if have that Notes.

M

1.

12. i 2. r, right

M

J

is any finitely generated left A-module, we

is I-adically complete.

If the ring

A

is such that, for every integer

the left ti -adic topology on

ti-adic topology on

A,

A

i,

coincides with the

then trivially conditions (2)

and (3) above both hold. 2. ring

A,

Of course, if

tl, ... ,t r

are in the center of the

then condition (I), (2) and (3) above hold. 3.

If the left ideal generated by

the right ideal generated by

t.

1

coincides with

then conditions

I-

~i'

(I), (2) and (3) hold. Proof: r - 1.

By induction, assume the Lemma is true for the integer Observe that every finitely generated left A-module

has the property that the natural map: (since

M

M .... rJ'I

is surjective

is a quotient of a free left A-module of finite rank,

and is therefore a quotient of a complete left A-module). fore

M

diction.

n IhM = {O} • Let us reason by contrah>O Suppose the Lemma-is false for some finitely generated M.

Then let

that the Lemma is false for

Q

be a left A-submodule of M such

M/Q,

mal (among left submodules of

M)

A

Q

is left Noetherian, such a M/Q

and such that

Q

is maxi-

with this property. exists.)

(Since

Then replacing

M

if necessary, it follows that there exists a finitely

generated left A-module M,

There-

is complete iff

left A-module

with

M

M,

such that the Lemma is false for

but true for every proper quotient (left A-module) of

M.

Chapter 8

780

Then every proper quotient left A-module of cally complete.

n

D=

(1)

M

is I-adi-

If we let IjM,

then since

M

is not complete,

D t- 0,

j~O

and

MID'" " I .

Therefore every proper quotient module of

is a quotient module of

MID.

Therefore

D

must be a smallest

(in the sense of inclusion) non-zero left A-submodule of In particular, if D = Ad.

And

of

ID

M.

d

d

is any non-zero element of

D,

M.

then

is an element of every non-zero left A-submodule

is a left submodule of

D.

Therefore, since

a smallest non-zero left submodule of 10 =

M

M,

either

10 = 0

0

is

or

o. I claim that it is impossible that then there exists an element

10 = 0,

Then

At"d

At I d = O.

at I d let

t" E I

is a non-zero left A-submodule of In particular, there exists

t = at ".

=

(In fact, if

ID=O.

a EA

such that 0,

t"d t- O.

and therefore

such that

d

Then

t EI

and

td = d, whence (1 - t) • d = O.

But the element verse

u = 1

(1 - t)

of the ring

+ t + t 2 + ...

Therefore

d=u' (l-t) 'd=u'O=O,

A has the two-sided in-

Finitely Generated Ideal a contradiction, since of

d

781

I

was chosen to be a non-zero element

D. Therefore ID={O}.

(2)

I _ = (tlA + ... + tr_lA) r l

The ideal

theses of this Lemma, and the ring plete.

obeys all the hypo-

A is (Ir_l)-adically com-

Therefore by the inductive assumption every finitely

generated left A-module, and in particular cally complete.

n

j.:::.O r-l I f I;_lM,/-O

M = 0.

for all

j.:::.O

then

I;_lM:::lD,

n I j IM:::l 0 '/- 0, a contradiction. j.:::. ran integer jo .:::. such that that

°

tion on

n,

(4) ... t

in

:l::,i , ... ,i ::.r}, l n

Let For

S

n =

To prove for I

=

Therefore there exists

Lm

i =r

n

={t . . . . t.

°

A,

n.

11

1n

all integers :l
n> 0.


n-

all integers

the assertion is immediate. For i = r.

Assume true for

n - 1

n = 1, where

it is n > 2.

The induction assumption can be written:

and hypothesis (1) of the Lemma for

mES

n-l

be wri tten:

implies, by induc-

is the right ideal generated by

condi tion (1) with

n-l

so

that

In

(Proof: n> 0.

j.::.O,

°

Hypothesis (1) of the Lemma for

11

all

jo Ir_loM={O}.

(3)

{to

is (Ir_l)-adi-

Therefore

j

I

M,

I = tlA +

0

••

+ trA.

'T'herefore

i =r

can

782

Chapter 8

In = In-I. I

=1 L m A)' I

=

\mES _ n l

!

mES

m ( t lA + . . . + t A) r n-l

L

=

L m (A mES _ n l m t. A= l

mES _ n l l
Lm

I) =

I =

mES _ n l [,\' m'A , mES

completing the in-

n

duction. ) For any integer quences teger,

let

n::' 0,

(s l' ... ,s v' k , ... ,kv) l

be the set of all se-

n

where

v

is a non-negative in-

,sv,k l ,··· ,k v - l ~ 1,

s 2""

sl + ... + Sv + kl + ... + kv = n. (1)

T

of the Lemma for

i = r)

Then using the fact (by hypothesis that

I = I _ + trA, r l

equation (4)

can be rewritten:

L t ( sl' .•. ,s ,k , ... , k )ET v l v n r

(4' )

all integers

n>O.

two-sided ideal in module

M),

Let A

K={a,=A:

a'M=O},

Then

K

is a

(the annihilator ideal of the left A-

and by equation (3) we have

(5)

If

n> 1

and

(sl"'"

kl + ... +kv~jo' sl kl s2 k2 tr I r - l tr I r _ l Therefore,

if

Sv,k l , ... ,kv) E Tn"

then

Sv kv ••. tr I r - l

C

kl+· .. +k v Ir- l

C

jo I r _l

C

K.

Un=~sl,· .. ,sv,kl,· .. ,kv): kl + ... +kv<jo}'

then from equation (4'), we have:

(6)

and if

Finitely Generated Ideal

783

I

If

k

If also

(s 1"" ,sv,k l ,··· ,k v ) E Un'

Therefore

v-l < jo'

Also since

then

kl + ... + kv < jo'

Le.

sl + ... +sv+kl + .,. +kv=n

it follows that

sl + '"

+sv >n- jo'

and

(8)

kl + '"

Since also

it follows that there must exist an integer

1,:: i.:: v,

v-> 0.

i,

+kv < jo'

sl""'sv':: O, such that

and such that n-j s. > _ _ 0

(v

v

1

~

°

since

n.::l).

By equation (7)

so that equation (8) yields S.>~-l. 1 Jo

That is, if

n>l

then for every

there exists an integer

i

(Sl, ... ,sv,kl, •.. ,k ) EU v n

such that

1 < i
and such that

(9)

But then

sl kl s. 1 k. 1 s. k. s. 1 k. 1 s k 1- ) t 1 (I 1 • t 1+ I 1+ t VI v ) (t r I r-l'" t r 1- I r-l r r-l r r-l •.. r r-l

Chapter 8

784

s.

I c

{ t

r-

It ~A,

if

r

i >

s. r

~A

if

'

1)

c

I

r-

s. It ~ r

i = 1

substituting by equation (9) this inclusion into equation (6), and replacing

II

nil

by

"n]'

"

o '

we see that for

n >1

we have

that

nj (10)

I

0

By hypothesis integer

integer

+I

r-

1 . t n. A

r

+ t n A. r

(3) of this Lemma, with

B>0

all integers

c K

j = r,

there exists an

such that

i,

r - 1

l
we have that

Ir - l •

t~ =

But by hypothesis (1) for the tlA + ... + tr_lA = I - . r l

(tlA + ..• + tr_1A) •

Therefore

t~

B B (tlAt r + ... + tr_1At r ) ctA+ ••. +tA r

r

that is, (11) If

B = 0,

then replace

we can assume that

B

by

I

r-l

In fact, for

• tB • I r k = 0,

in equation (11).

Therefore

B > 1.

I claim, by induction on (12)

+1

k,

that for

k

~

0,

we have

kj B 0

c K + tkA. r

the assertion is trivial.

Assume that

Finitely Generated Ideal Equation (12) Equation I

(12) • t

r-l

is true for the integer for the integer

B • I r

'"'

I

• tB •I r

r-l

where

k > O.

To prove

In fact,

k + l.

(k+l)j B 0

k

I

ki-'0 B

• I

jo R

which by the inductive assumption,

which by Equation (10) c K+ t

for

n = B

B k (K + I t B . A + t • A) r-l r r r I

t B • A + t k + B • A, r-l r r

which by E qua tion (11)

C

K + t k • (t A) • A + t k + B . A r r r

completing the inductive verification of Equation But then,

k~O,

for I

we have that

(B + Bj k) j 0

which by Equation (10) with

0

n = B + Bjok

B+Bj k B+Bj k c K+ I • tr o . A + tr o. A r l =K+I

C

K

+

I

Bj k B+Bj k t B • (t 0 A)+t 0 A r-l r r r B

t r-l r

Bj k I

0

which by Equation (12)

+t

B+Bj k °A

r

'

(12)

.

786

Chapter 8 B+Bj k C K + (K + tkA) + t r

That is, for (13)

I

k

~

0

0

Recall also that (14)

we have that

0,

Bj +Bj 2 k

A

0

r

C K + tkA.

K

r

is the annihilator ideal of

M,

so that

K' M= {oL

From Equations (13) and (14) we deduce that, for every integer k

~

we have that

0,

.

1

.2

BJ 0 +BJ k 0

'McK'M+t

k k k 'A'M={O}+tM=tM. r r r

That is, for every integer (15)

k~O,

Bj +Bj 2k o . MC tkM. I 0 r

Therefore

n InMC n tkM. n>O Since also

k>O r

t~MC IkM,

all integers

n tkMc n InM. k>O r n>O Therefore

n tkM = n InM k>O r n>O But, by Equation (1) (1)

n InM n>O

= D.

k

~

0,

Finitely Generated Ideal

787

I

Therefore (16)

Next, recall that by hypothesis (3) of this Lemma applied to the integer N>0

i = r,

we have that there exists an integer

such that

(17)

t N • Ac At r

r

I claim, by induction on the integer

i.::. 0,

that

(18) In fact, for i

i = 0

the assertion is trivial, and for

the assertion is equation (17).

= 1

tion is established for the integer i + 1.

assertion for the integer

Assume that the asseri

where

i.::. OJ

to prove the

In fact, we have that

t N (i+1) A = tNtNiAC tNAt i cAt t i = At i + l r r r r r r r r'

completing the induc-

tive proof of Equation (18). Now let V={xEM:

V = (the tr -torsion part of

There exists an integer

Then I claim that

V

h >0

and

a EA

exists an integer Equation (19)

whence

then h > 0

a· x

~

V.

t

i = h)

h r

M.

h tr • x = O}.

In fact,

We must show that, if

In fact, since

such that

(for the integer

M.

i. e., let

such that

is a left A-submodule of

is obviously an abelian subgroup of x EV

M),

• x =

o.

x E V,

there

But then by

we have that

V

788

Chapter 8

as required. Therefore Since

M

Since

is a left A-submodule of

M,

as asserted.

is a finitely generated left A-module over the left

Noetherian ring m> 0

V

A,

and elements

xl' ... ' xm (; V

it follows that there exists an integer xl' ... ' xm E M

such that

there exists an integer

But then, for every integer h Ax. CA. t J r

1

j,

x. = J

.s. j .s. m,

h >0

such that

we have that

o.

Therefore hN tr

hN hN hN V = t r (Ax 1 + ••. + AX ) = t r • Ax 1 + ... + t r • Ax m = 0 m

We have shown that (19)

There exists an integer

We will now use Equations

(16) and

By Equation (19), since elements in

M,

V

C >0

such that

tCV ={o}. r

(19) to complete the proof.

is the set of all t -torsion r

it follows that, for the fixed integer

C 2. 0,

we have that (20)

If

xEM

and

th.x=O r

where

h2.C,

then

t C • x = O. r

By Equation (16), since the element

dE 0

is on the left side

of Equation (16), for every integer

n> 0

there exists an ele-

ment

Y EM n

such that

Finitely Generated Ideal d = t~ • Y

(21) Define

Z

n

C

n = t r Yn+C •

(22) 0= d - d

Since

789

I

t

n Z =d r n '

Then

all integers

And

n >0 •

= t n + l + C Y + + - t n + C Yn+C -- tn+c(t r Yn+l+c - y ) n+C r n I C

n::.- 0,

n + C >C

so by Equation (20)

(with

00

h = n + C,

But then t

Z

r n+l

-

Z

n

= tC t t C(t ) = O. - tC r rYn+C+l r Yn + C = r rYn+C+I - Yn + C

I. e.,

(23)

trzn+l = zn'

By Equation (22) for (24)

Zo

all integers n = 0,

n> O.

we have that

= d.

But the Equations (23) imply that k

zl E n t M, k>O r and by Equation (16) this implies that

Since

D = Ad

therefore there exists an element

a EA

such that

Chapter 8

790

Then d (1 - tra)· d

= t r z 1 = t r ad

= O.

Bu t the element

(1 - ta) E A r

admi ts the two-

u = 1 + (\a)+ (t a)2+ (\a)3+ ... + . . . r

sided inverse: A.

'

in the ring

Therefore d

= 1 • d = u • (1 -

a contradiction, since in the left A-module

t a)' d

d

r

= u • 0 = 0,

was chosen to be a non-zero element

M.

This contradiction completes the proof. Corollary 5.1. let

Let

A

be a ring with identity as in Lemma 5,

M be a finitely generated left A-module and let

submodule of logy of Proof:

M.

N

The quotient module A.

M

MIN

for the I-adic topo-

is a finitely generated module

N

Let

is closed in A

is I-adically com-

M.

be a ring as in Lemma 5 and let

any left ideal in the ring

A.

Then

Apply corollary 5.1 with

Example.

MIN

Therefore by Lemma 5

Therefore

Corollary 5.2.

Proof:

is closed in

be a

M.

over the ring plete.

Then

N

is closed in

J

M= A

and

J

be

A.

N = J.

Under the hypotheses of Lemma 5, each of the ideals

(tlA + .,. + tiA)

is a left ideal,

Corollary 5.2 is closed in

1

~

i

~

r,

and therefore by

A.

Theorem 1, Corollary 1.1, Corollary 1.2, Corollary 1.3 and Corollary 1.3' are generalizations of Theoremsl and 5 of Chapter 5.

Are there similar generalizations of Corollary 1.1

Finitely Generated Ideal of Chapter 57

791

I

The answer is, "yes", as is shown by the following

sequence of Theorems.

(In the situation of Theorem 1, Corollary

1.1, ... ,Corollary 1.3', there is a corresponding statement about inverse limits of cohomology.

We will label these "Theorem 6,"

"Corollary 6.1", ... , "Corollary 6.3'"

respectively.)

For the sake of completeness, in our sequence of Examples we also note the following well-known fact. Example 5.

The hypotheses and notations being as in Example 4,

suppose also that

X

a coherent sheaf

G

of

A

and that there exists

Ox - modules over

/\

as sheaves of h> 0

is proper over

Ox - modules over

X

such that

G/\ "" F

Then for every integer

Y.

the natural map is an isomorphism: Hh (X,G) Q A"I ~ Hh(y ,F) A

First, since

Proof:

H

n

(X,G) QA

/\I

""H

n

A where

1T:XX (A/\I) A

in which Let

A

~X

is the projection, we reduce to the case

is I-adically complete.

tl, ... ,t r

be a set of generators for the ideal

if there exists an integer assertion is true.

n >0

(Since then

the ideal generated by

t 2 ,··· ,tr

tl-torsion subsheaf" of

G,

X.

Then

T

n

then the

tlG = 0,

in the ring

where A,

Next, let

and thereT

be "the

i.e. , the subsheaf such that r (U,G),

all open sub-

is a coherent subsheaf of

Ox-modules

of G over X, and there exists an integer n>O such that (Proof:

Let

Ul' ... 'U

h

I.

First notice that,

GAl = d'Ir - I,

r(U,T) = the tl-torsion in the A-module of

r.

such that

fore the inductive assumption applies).

U

),

A

The proof is by induction on the integer

sets

/\ I

(Xx (A

n

tlT=O.

be a finite set of affine open subsets

792

of

Chapter 8 X

the union of which is

X.

f(U.,T) = U J i>O f(U.,G) .... r(u.,G)], the union of

Ker[(multiplication by

Then

J

J

an increasing sequence of submodules of the finitely generated module

f(Uj,G)

r(Uj,T)

over the Noetherian ring

is finitely generated as

f(Uj,Ox),

whence

r(u.,Ox)-module and there n. J tlJf(U.,T) = 0, 1 < j < h.

n.

such that J J Therefore if n=sup(nl, ... ,n ), tn.T=O, and h n T = Ker «multiplication by t ): G .... G), and is therefore a co-

exists an integer

OX-modules over

herent sheaf of

Xl.

Therefore the assertion

is true for the coherent subsheaf of 0x~modules A

GO~>GO

assignment:

OX-modules over A

Ox-modules over subset and

B

U

AI

of

T

of

G.

from the category of coherent sheaves of

X

into the category of coherent sheaves of

Y

is exact.

X,

The

if

= f (U, OX) ,

B

is flat over

(Since, for every affine open

B,

A

then

AI

r(u,G )""r(U,G )0B O O B

since the ring

B

is Noetherian.)

Therefore we have the commutative diagram with exact rows:

o .. T

.... G

O .... } .... of sheaves of

....

G/T--> 0

!. . (G~T)A_>

0

Ox-modules over

X.

This yields a commutative

diagram with exact rows: di - l . . . di ~Hl(X,T)_Hl(X,G)~Hl(X,G/T) _~ ... i l d -

••• ~H

.

1

JT" )->H .!

(Y,

1

(Y,F)-'>H

. J(G/T)

1

(Y,

A

di )----» •••

Since the assertion is by the inductive assumption true for by the Five Lemma, to prove the assertion for to prove it for

G/T.

Therefore, replacing

G, G

by

T,

it suffices G/T

if

,

Finitely Generated Ideal

793

I

necessary, we can assume that the endomorphism "multiplication by

t

1

of

"

is injective.

G

But then the exact sequence I.8.1 of Chapter

2 applies,

and yields the short exact sequence:

(*)

0

-+

n H (X,G) Atl-'p

[a~

n H (X,G/tiG)]

lim (precise ti-torsion in [ i>O all integers Hn+I(X,G) ring

A,

Hn(X,G) Since

n > o.

G

n l H + (X,G) )]-> 0,

is proper over

Hn (X,G)

X,

and

are finitely generated modules over the Noetherian and, since and

A

Hn+l(X,G)

tl E I,

are therefore I-adically complete. Hn(X,G)

and

Hn+l(X,G)

are tl-adi-

Therefore

n

H (X,G) Hn+l(X,G)

is I-adically complete, by Lemma 5

a fortiori

cally complete.

and

Since

-+

A

tl

n = H (X, G)

has no infinitely tl-divisible elements.

This

latter observation implies that the third group in the short exact sequence (*) (*)

is zero.

Therefore the short exact sequence

implies that the natural map is an isomorphism: (1)

Hn(X,G)

~ lim Hn(X,G/tliG),

all integers

n> O.

i>O Since the functor: Ox-modules over Y

X

Go~>~

from coherent sheaves of

into coherent sheaves of

is exact, and since (multiplication by

tive, it follows that (multiplication by tive 2

A (F", G ).

t t

l

0y-modules over l

):

): F

G -+ G -+

F

is injec-

is injec-

Therefore the exact sequence (L8.1) of Chapter

again applies, and in the same way that we derived equation

Chapter 8

794

(1), we prove that the natural map is an isomorphism ~

n

i

n

all integers

H (Y,F) ':; lim H (Y,F/tlF),

(2)

i~O

But the coherent sheaf

n > O.

is such that

G/tiG

so that the inductive assumption applies, and since (since the functor: sheaves of

Ox-modules over

0y-modules over

Y

X

Govv>~

from coherent

into coherent sheaves of

is exact), we have that the natural map is

an isomorphism:

Equations (1), (Remark:

(2)

and

(3)

complete the proof.

A proof of the result stated in Example 5 above can

be given which is quite different from the one given above (and perhaps even easier).

This alternative, more fundamental, proof

depends on some theorems of mine that yield yet another new, relevant, spectral sequence about cohomology of completions.

I

will come back to this in a later ms.). Theorem 6.

The hypotheses and notations being as in Theorem 1,

assume in addition that: For each integer

h, there exists a sequence

hl, ... ,h

r

of

integers such that hI':: h, ... , h r ':: h and such that hI hr (2 strong) (t , ... ,t ) is an r-sequence for the left A-module l l c i , all integers i E'l!, and such that

(2~tronq) i

2.

j

For each integer

2. r-I,

the left A-submodule:

ci

closed in Let

i E'l'

n

and every integer j, h .. hI i 1 of tl C + .•. + t . JC J

for the I-adic topology. be any fixed integer.

Then if

is

Finitely Generated Ideal Hn(C*/IC*)

n

the natural map is an isomorphism:

and also for the fixed integer

Let

795

is finitely generated as left (A/I)-module,

then for the fixed integer

Proof:

I

Sl={h : l

n

(hl, ... ,h ) r

there exists a sequence

in hypotheses (2 strong)

and

(2~trong)}.

as

Then by Corollary 1.1

of Chapter 5' we have that the natural map is an isomorphism:

The proof is by induction on applied to the ring hI tl ), and the ideal

where where

By the inductive assumption

A' =A/(the two-sided ideal generated by I' = image of

sided ideal generated by A'),

r.

ti, ... '

I t~

in

A'

(I'

= images of

is the two-

t , ... , tr 2

in

we have that

I

is (the two-sided ideal generated by hl,h J is the two-sided ideal in A generated by

These two equations imply the conclusions of Theorem 6. Remarks:

In connection with hypothesis (2 t ) of s rong Theorem 6, it is easy to see that, if r is any integer and if

1.

(tl, ... ,t r )

with identity (tl, ... ,t r )

A,

is a sequence of and if

r

t1, ... ,t r

0,

elements in any ring

M is any left A-module, and if

is an r-sequence for the left A-module

if the elements

~

commute elementwise

M,

then,

(i.e., if

Chapter 8

796

then: positive integers are any sequence of r nr , ... ,t ) obeys condition (1) of Definition 2 (t such that r l nl nr ) , ... , t (t for the left A-module M, then the sequence nl, ... ,n

If

nl

r

r

l

obeys both conditions of Definition 2 for the left A-module i.e., is an r-sequence for the left A-module tl, ... ,t

M.

(Proof:

M, Since

commute elementwise, we have the unique homomorphism

r

of rings with identity: l
'Z' [T , ... ,Trl +A l

Therefore

M

is a

that maps

T. l

'Z' [T , ... ,Trl-module. l

into Since

we are assuming that condition (1) of Definition 2 holds for nl nr nl nr the sequence (t , .. . ,t ), it follows that ( t l , ... ,t ) r l n r n is an r-sequence for the left A-module is an r-sequence for the replacing

A

if necessary, that the ring

with

iff

'Z'[Tl, ... ,Trl-module

(T

M.

and

'Z' [T , ... ,Trl l

1

l

r , ••• ,Tr )

Therefore T. , l

it suffices to prove the assertion in the case A

is commutative.

But then, by induction on n

M

r,

if we know that

l

,t , •.. ,t ) is an r-sequence for M, then it would follow 2 r nl n2 nr that (t ,t , ... ,t ) is an r-sequence for M. (Since, bel r 2 (t

l

n

cause

(t

l

l

,t , ... ,t ) r 2

is an r-sequence for

M,

we have by n

is an (r-l)-sequence for M/t l M. l n2 nr Whence, by the inductive assumption (t , ••• , tr ) is an (r-l) 2 n Definition 2 that

sequence for

M/t

sequence for

M).

l

But by Definition 2, since the ring A nl n2 nr is commutative, it follows that (t is an r' t2 , ... , t r ) l l

case that the ring n,::: 1,

we have that

M.

Therefore it suffices to verify that, in the A

is commutative, that for each integer is an r-sequence for

And this is readily verified by induction on

n).

M.

Finitely Generated Ideal Notice also that, if

r

797

I

is any integer

>

0

and if

is an r-sequence for the left A-module AtiMC tiM,

1 < i
(which of course is the case if

M,

then if

Ati C tiA,

then of course condition (1) of Definition 2 holds. 2. tl, ... ,t

As noted in Remark 1 above if the elements

in the ring

r

A

commute elementwise, and if say

then condition (2 t ) of Theorem 6 fols rong lows from condition (2) of Theorem 1.

i

C ,

and if, e. g. ,

each integer

i).

is an r-sequence for hl hr then, (t l ,.·· ,t r )

At. c t .A, J

J

is also an r-sequence for

ci

(More precisely, if

,

all integers

However, condition

hl, ... ,hr2.l,

(2~trong)

is of course

a messy condition. Suppose that all the hypotheses of Theorem 6 hold,

3.

except possibly hypotheses (2

strong

) and (2

I

strong

are not assuming hypothesis (2 ') of Theorem 1).

)

•

Then if merely

is an r-sequence for the left A-module integers

i

(Also, we

ci

,

all

(i.e., if hypothesis (2) of Theorem 1 holds),

if the technical condition

and

(TC t ) of Corollary 6.1 below s rong

holds, then the conclusionsof Theorem 6 remain valid.

(The

proof of this is analogous to that of the analogous Remark following the proof of Theorem 1.)

E.g., this is the case if

is an r-sequence for the ring A

is commutative.

sequence in

A,

and if

one can eliminate Corollary 6.1. lary 1.1.

Alternatively, if tl, ••• t

r

A

and if the ring

(t1, ••• ,t r )

is an r-

are in the center of A, then

(TC t ), see Remark 1 following Cor. 6.1. s rong

Let the hypotheses and notations be as in Corol-

Suppose also that

Chapter 8

798

(TCstrong)

For each integer

h .:.1,

there exists a se-

of integers such that hI hr (t , •.. ,t ) and such that the sequence:

quence

l

the ring

A,

and s.t. the left submodule

closed in the A-module li,

A-modules

r

F/\

is an r-sequence in hI /\ h j /\ is tl F + ••• + tj F

for the I-adic topology, all free

1 .::. j .::. r - 1.

(If the elements

tl, ••• ,t

are in the center of A then one can

r

eliminate (TCstrong)' see Remark 1 following the proof of Corollary 6.1). Then for every integer

n E 'l',

the natural map is an iso-

morphism:

and liml [h>O Proof.

Let

H~ (A/Ih , C*)]

'C*

"" 0,

all integers

be as in the Proof of Corollary 1.1.

have observed in Corollary 1.1 that gers

n.

n.

Hn(C*)~Hn(,C*/\)

Then we all inte-

Also, the proof of Corollary 1.1 shows (using

(TCstrong) in lieu of (TC)) that

H~(

h (AI (tllA

h

h

+•.. + trrA)) ,C*) "" Hn ((AI (tllA + •••

all integers

n.

But, by condition (TC t ) and the conclusions of Lemma 1.1.1, s rong the cochain complex

'C*

(2'strong) of Theorem 6.

/\

obeys conditions (2 t ) and s rong Therefore the two conclusions of Theo-

Finitely Generated Ideal rem 6 applied to the cochain complex Remarks:

1.

Suppose that

I

'C*A

(tl, ••• ,t ) r

799

complete

is an r-sequence for

the left Noetherian ring with identity

A,

plete for the I-adic topology where

tlA + ••• + trA.

1=

the proof.

and that

A

is comThen by

the same observation that we have made within the statement of corollary 1.1, it follows that if the ring

A

is commutative,

condition (TCstrong) holds, and therefore we have the conclusions of Corollary 6.1. On the other hand, if all of the hypotheses of Corollary 6.1 hold, except possibly (TC s t rong ), and if an r-sequence for the ring

A,

and if also

(tl, ••• ,t ) r

are

tl, ••• ,t

are in r the center of A, then by the argument in part (2) of the Remark

following Proposition 4, we again have that the conclusions of the Corollary still hold. Corollary 6.2. lary 1.2.

Let the hypotheses and notations be as in Corol-

Then for every integer

n

the natural map is an iso-

morphism:

And for every integer

(Here

"(T l' ... , T r )h"

(Tl, ••• ,T ) r

n

we have that

denotes the h'th power of the ideal

generated by

Tl, ••• ,T

r

in the ring

1(T , ••• ,T l r

»).

The proof of Corollary 6.2 is much the same as that of Corollary 6.1. Remark:

As observed in the (analogous) Remark following Corol-

800

Chapter 8

lary 1.2, we can replace

"7(T , ... ,T r )" l

"A(T , ... ,T r )", l

1J'[Tl, ... ,Trl"

or

by anyone of:

IA[Tl, ... ,Trl"

in the

statement of Corollary 6.2. Corollary 6.3. lary 1.3.

Let the hypotheses and notations be as in Corol-

Then all of the hypotheses, and therefore also the

conclusions, of Corollary 6.2 hold. CorolJary 6.3'. lary 1.3'.

Let the hypotheses and notations be as in Corol-

Assume also that the hypothesis,

Corollary 6.1 holds.

(TCstrong)' of

Then all of the hypotheses, and therefore

also the conclusions, of Corollary 6.1 hold. Of course, Corollaries 6.3 and 6.3' follow immediately from Corollaries 1.3 and 1.3', respectively.

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