Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, Z~irich
92 Category Theory, Homology Theory and their Applications II
1969
Proceedings of the Conference held at the Seattle Research Center of the Battelle Memorial Institute, June 24 - July 19,1968 Volume Two
Springer-Verlag Berlin. Heidelberg. New York
All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer Verlag. 9 by Springer-Verlag Berlin- Heidelberg 1969 Library of Congress Catalog Card Number 75"75931 Printed in Germany. Title No. 3698
Preface
This is the second part of the Proceedings of the Conference on Category Theory, Homology Theory and their Applications,
held at the Seattle Research Center of the Battelle
Memorial Institute during the summer of 1968. The first part, comprising 12 papers, was published as Volume 86 in the Lecture Notes series. Following the Table of Contents,
there is appended a list of papers to be published in
subsequent volumes. It is again a pleasure to express to the administrative and clerical staff of the Seattle Research Center the appreciation of the contributors to this volume, and of the organizing committee of the conference,
for their invaluable assistance
in the prepa-
ration of the manuscripts.
Cornell University,
Ithaca, January,1969
Peter Hilton
T a b l e of C o n t e n t s
H. B. B r i n k m a n n
Relations
S. U. C h a s e
Galois objects
P. D e d e c k e r
T h r e e - d i m e n s i o n a l n o n - a b e l i a n c o h o m o l o g y for g r o u p s
R. R. D o u g l a s ,
for g r o u p s and for e x a c t c a t e g o r i e s and extensions of Hopf Algebras
H-spaces
C. E h r e s m a n n
C o n s t r u c t i o n de
K. W. G r u e n b e r g
C a t e g o r y of g r o u p e x t e n s i o n s
M. A. K n u s
Algebras graded by a group
F. W. L a w v e r e
Diagonal Arguments
S. M a c L a n e
Foundations
B. M i t c h e l l
On the d i m e n s i o n o f o b j e c t s a n d c a t e g o r i e s
F. U l m e r
.....
10 .
. . . . . . . . . . . . . . . . . . . . . . . . structures
libres
. . . . . . . . . . .
. . . . . . . . . . . . . .
. . . . . . . . . . . . . .
and Cartesian Closed Categories
for c a t e g o r i e s a n d sets
Hochschild dimension E. R o o s
I
32
P. J. H i l t o n
a n d F. S i g r i s t
J.
......
Kan extensions,
cotriples
74 105 117
. . .134
. . . . . . . . . .
146
III
. . . . . . . . . . . . . . . . .
Locally Noetherian categories
65
. . . . . . . . . . . . .
and A n d r 6 ~ o ~ o m o l o g y
....
165 197 278
Papers to appear in future volumes
J. F. Adams
Generalized cohomology
J. Beck
On H-spaces and infinite loop spaces
D. B. Epstein and M. Kneser
Functors between categories of vector spaces
D. B~ Epstein
Natural vector bundles
P. J. Freyd
New concepts in category theory
J. W~ Gray
Categorical fibrations and 2-categories
R. Hoobler
Non-abelian sheaf cohomology
M. Karoubi
Foncteurs d6riv~s et K-th~orie
F. E. J. Linton
Relative functorial semantics
E. G. Manes
Minimal subalgebras for dynamical triples
P. May
Categories of spectra and infinite loop spaces
P. Olum
Homology of squares and factorization of diagrams
-I-
RELATIONS
F O R GROUPS
A N D F O R EXACT C A T E G O R I E S
by Hans-Berndt
Relations
(Correspondences)
defining
connecting
spectral
sequences.
and L e i c h t
among
Puppe
answer
the q u e s t i o n
satisfying
Applied
as desired.
fractional
calculus.
A
category
note
is e x a c t
of a cokernel
unique
up to
(1)
The
former
categories
chasing
and
for
or d i f f e r e n t i a l s
MacLane,
a n y e x a c t (1) c a t e g o r y and thus b e i n g
p . 1 8 and A x i o m s
Puppe,
in
Hilton
for groups.
categories
(1)
category,
A construction
We give
was
out
mentioned in
we r e c o v e r
[i0!
a positive for any
we o b t a i n the
of r e l a t i o n s
[4] then d e s c r i b e d
for some m i n o r
the case
, if it has
and
changes
a calcustandard
for a b e l i a n
this case b y m e a n s
independently
functorial
an a b b r e v i a t e d
is i n c l u d e d
a zero o b j e c t
and
of a
also c o n s i d e r e d
after
choice.
[I0].
p.9-19]
version
readily
Standard or in
No addition
of
[1].
now.
if e v e r y m o r p h i s m
The d e c o m p o s i t i o n
[9~ 1.7-15,
is q u a s i e x a c t
of Buchsbatun.
of g r o u p s
b y a kernel.
in M i t c h e l l
terminology
shown
can be c a r r i e d
of g r o u p s
the p r o b l e m
a calculus
for the p u r p o s e s
It w a s also
exact
to the c a t e g o r y
admits
can exist.
below
small
(w
[3].
followed
found
KI-3]).
described
Hilton
suited
of r e l a t i o n s
and c o l o c a l l y
is, e x c e p t
isomorphism
m a y be
Lambek,
[I0].
are such that
sition
categories
b y Riguet,
Applied
For exact
[6] and C a l e n k o
The p r e s e n t
I.
studied
axioms
relations
w a s g i v e n b y Puppe
made
operations
such c a l c u l u s
to a l o c a l l y
categories
The c h a n g e s
for d i a g r a m
cohomology
The c o n s t r u c t i o n
of h o m o m o r p h i c
by Leicht
tool
order
whether
[10~4.18,
that at m o s t one
lus of r e l a t i o n s category
certain
(Puppe
to the problem.
category.
a useful
others.
in the b e g i n n i n g p.17]
higher
They have been
posed
of r e l a t i o n s
4.15,
morphisms,
provide
Brinkmann
is a compo-
turns
properties
out to be of e x a c t
[2].
is r e q u i r e d
as in the exact
_
Exact additive
categories
group
we o b t a i n we w o u l d
obtain
of r e l a t i o n s
compatible
: K
)K
which Z
consists
K
Beispiel
w i t h one object. a
of a c a t e g o r y
A, p.23].
Adjoining
(natural)
as its m u l t i p l i c a t i v e
satisfying
we w r i t e
[101w
does n o t e v e n admit
w i t h composition)
(conversion)
simplicity
[1Osl.3,
category
(Puppe
is a c a t e g o r y
a field h a v i n g
(i.e.
For
of i n t e g e r s
an exact
2. A c a t e g o r y K
Z
n e e d not be a b e l i a n
a zero o b j e c t
addition,
K, a
(natural)
order
relation
order preserving
A~ = A
and
for o b j e c t s of d a t a
since
else
group.
and a c o n t r a v a r i a n t
for the triple
The
f~
= f
(K,<,~).
Details
, groups
RG
( on
functor
for m o r p h i s m s . m a y be
found
in
p.3-4].
Examples:
Relations
2.2.
K
Let
for sets
be a c a t e g o r y
RS,
pointed
sets
RS,
f E K
of relations.
w i l l be c a l l e d
, abelian
a map,
groups
RADb.
iff
(2) f~f
~ 1
f f@ > i
The a r r a n g e m e n t inition.
The m a p s
(Identities
Lemma
f < g
g
< ff~g
MK
hence
restrict
Examples: (abelian
of
are maps).
2.3
Proof:
of the p i c t u r e
to
and
~ fg#g
K
refers
form a s u b c a t e g o r y Obviously
f,g
( f
~ MK
MRS =
for
(sets),
groups).
(2) fg m e a n s first
f
then
imply
g
involved
in the def-
contains
all o b j e c t s
of
f = g.
I < f f@,
no further
f ~ MK_, f#
MRS,
MK_, w h i c h
of d u a l i t y
we h a v e
using
so far c a r r i e s
MK_, since
to the two types
= S,
f~
< g
structure
and
g g
< I.
than b e i n g
a category
(@
does not
n e e d not be a map).
(pointed
sets),
MRG = G
(groups),
M R A b = Ab
-3-
3.
Let
out
that
[10]!
K
be a c a t e g o r y MK_
4.8,
precise
(locally
p.15]).
We w i l l
form of these
especially MK
is a
nice
[10! 4.15,
K
of relations.
Examples:
The axioms
for
K3b,
(E.g.
however,
replace
expressed and
K3
by:
Also p.
15-16]
satisfied
for
that
K1-3
RAb.
K
such
u = m ee m
that
u
= u
(subquotient)
(Axioms
to b o t h e r
K,
KI-3
the reader
the e m b e d d i n g
of
MK
[10] w i l l be c a l l e d
for
in this case m a y be w e a k e n e d u
on
is up to i s o m o r p h i s m
of
Except
are i m p o s e d
category
details
and that
axioms
axioms
exact
it turns out
satisfying
Every
in the form
= u
K3b
2
(symmetric
such that
m,e
of
[IO]!
with
the
into
K
by
a pseudoexact
satisfied
recover
uniqueness
idampotent)
are m a p s
is
determined
t h e y are also
so as to still
it turns
may be
and
mm
= I
exists
a
e e = 1.
4. We can n o w state
Theorem
C
pseudoexact
sets.
the
following
result:
4.1 Let
be a l o c a l l y
category
and c o l o c a l l y
of relations
K
small
such that
The s m a l l n e s s c o n d i t i o n
of the t h e o r e m
The c o n s t r u c t i o n
is b e l i e v e d
pendent
below
exact
Then
category.
there
C = MK.
is o n l y u s e d
to insure
to g i v e an h o n e s t
that
mathematical
K
has hom
object
inde-
of this assumption.
Let
E_xx d e n o t e
categories
and
both with
their
w
are
small)
not go into e n o u g h
[10! 4.8-4.12, p.17].
If further
and c o l o c a l l y
axioms.
category
RG.
of relations.
Psex
4.1 r e s u l t s
K Ex~----~--~MPsex
the
(illegitimate)
(illegitimate)
appropriate
functors.
in that we d e f i n e d
is an e q u i v a l e n c e
To m o t i v a t e category
the
category
We c o n s i d e r
of l o c a l l y
a functor
Ex
categories
to the r e s u l t s K
~Psex
of
such that
(M is the e x t e n s i o n
to be given, a diagram
and c o l o c a l l y
of p s e u d o e x a c t
With regard
of c a t e g o r i e s
the c o n s t r u c t i o n
of relations.
category
w e assume
that
of
K
small
exact
of r e l a t i o n s
[10] m e n t i o n e d
in
the pair "maps"
to a functor).
is a p s e u d o e x a c t
-4-
f
-
g
(4.2)
in
MK
and analyze when
(4.3)
f@g
< g t f t@
(4.4)
f@g = g'f'@
and
We have Lemma 4.5
f~g < g'f'e,
iff the diagram commutes in
Lemma 4.6
f~g = g'f'#,
iff the decomposition in
MK (fg' = gf').
MK_ of the diagram to
(4.7)
yields in
(I) a pushout,
(2) a pullback and (3) and (4) bicartesian
(pullback and pushout)
MK . A proof is given in [2], the hypotheses may be weakened to the form mentioned
in w
Examples
(See [2; Anhang]).
A square as 4.2 in If
C
MK_
such that
fSg = g'f'~
will be called fully commutative.
is abelian, Hilton [4; Theorem 3.3, p.258] described fully commutative squares
by the existence of a completion to a commutative diagram
(4.8)
such that the inner square is bicartesian~
This is readily seen to be equivalent to
-5-
the c h a r a c t e r i z a t i o n
5_:.
We are
morphism cwordC
thus A
given
led to the
f )B
in
of c w o r d s
Every
on
object
Every map For
every
If
a
C
We c a n h e n c e
of
A---~fB map
~
nor
having
).
concatenation
of
from
A
This
Let
R
the
will
will
by
9 f#
a l s o be d e n o t e d
from if
C
~
.
A,
is a c w o r d
from
B
to
~ | ~
is a c w o r d
finite
diagrams
C
from
(A
(5.3)
(f | g,fg)
) A,A)
f
abbreviates
r extends
~ R
for
all
to an a n t i a u t o m o r p h i s m
fl
between
A.
cwords
defined
by
A, f
)
*
mI (5.4)
(m I ~ m 2,m I
(5.5)
(m ~ e ~ , e 'e ~ m')
(3) The
terminology
| m~)
is due
~ R
g "
,
*
;
m2
, if
is a p u l l b a c k
if
is b i c a r t e s i a n
~ R
to Freyd.
to
f#
We h a v e
~ . <
for all
A
B.
f2
~ R
and
as e.g.
1A
(5.2)
of c w o r d C ,
fn -, f
relation
identity
We h a v e
, f~l
.....
category
) B
. (
.
every
fn
*
~
illegitimate
it is an
to
of all
.
fn-I
the b i n a r y
B
, then
(
by
For
B,
fn- 1
. < be
to
and
Here
be d e n o t e d
f ~
The p o s s i b l y
) .
f2 ) A,
A
of
be a c a t e g o r y .
A to A a n d
category
as c o m p o s i t i o n .
C
from
.....
) .....
B(
B
.
C
follows(3):
is a c w o r d to
Let
fn-1
~
fn (5.1)
with
f#
as
from
is an o b j e c t
A( A I
cwordC.
defined
is a c w o r d
f~
cwordC
fl (5.1)*
a symbol
f2
A
Duality
C
f )B,
~
identify
construction:
is a c w o r d
A
fl (5.1)
is t h e n
is a c w o r d
neither
following we a d j o i n
C
A
b y 4.6
in
in
C
,
C
of
if C.
-6-
Let R u R*
N
u R ~ u R ~*
Obviously equipped there
be the n a t u r a l
~ with
,
where
Theorem
e.g.
R*
with
an i n v o l u t i v e
automorphism
by
relation
has
is c o m p a t i b l e
is an o b v i o u s
be d e n o t e d
equivalence
$
on c w o r d C
the o b v i o u s
and we o b t a i n
) KC
~ ,
functor
C
.
The
(locally
and c o l o c a l l y
generated
meaning
(a,~)
a quotient
which
image
~ R*~>
category
is the i d e n t i t y
of
f ~ C
under
by
KC:
(a*,~*)
= cwordC/~
on objects.
this
~ R.
Also
functor w i l l
f.
5.6 Let
relation
C <
be a
m a y be i n t r o d u c e d
in
KC
small)
by means
exact
of
category 9
f#g
( glf t#
Then a natural for c o m m u t a t i v e
order dia-
grams
in
C
and
furthermore
such that
In
Theorem
KC
[I] 5.6
is a p s e u d o e x a c t
and
category
of r e l a t i o n s
.
is p r o v e d
C
be a l o c a l l y
has a u n i q u e
is a s u b o b j e c t (4) and by
with
in the f o l l o w i n g
form:
5.7 Let
in
MKC = C
KC
% ~ ~', iff there
and c o l o c a l l y
representative e
m
-x
is a q u o t i e n t
is a c o m m u t a t i v e
small r
exact f
category. e
) .~
o b j e c t (4) in
C
Then every morphism in c w o r d C
. We i n t r o d u c e
such that a relation
m "("
diagram ft
in
C
(4)i.e.
for the r e p r e s e n t a t i v e s .
selected
from its class
Then
KC
with
<
and
~
is a p s e u d o e x a c t
category
of
-7-
relations
such that
MKC = C
The p r o o f g i v e n briefly some
described
standard
a relation
common
Lemma
among
diagram
and
Finally
cwords.
[I])
in
C
cwords
We then
[8] (5)
C
. If
the
The
cword
to this
standard
to
For the u n i q u e -
form and i n t r o d u c e
the s t a n d a r d
by shortening
idea m a y be
is e q u i v a l e n t
in the theorem.
show that
obtained
we use
Let
form is the s h o r t e s t
the same c w o r d
admit
a
following
be an e x a c t
category.
(1) is a p u l l b a c k
and
Let
(2) is a pushout,
then
(3)
(4) is a pushout.
the e q u i v a l e n c e
in d e f i n i n g
is u s e d
[2],
of M a c L a n e
as d e s c r i b e d
of shortening
To do this,
(Proof:
is a p u l l b a c k
>. ~
and that a n y two cwords
shortening.
5.8
the m e t h o d s
It is not h a r d to see that e v e r y
<.
the p r o c e s s
be a c o m m u t a t i v e
used
.<
"shorter"
form p o s s i b l e
[I] u s e s
as follows:
form
ness we a n a l y z e
in
KC
to define
relation
. This o b v i o u s l y
(!).
generated
suffices
The r e s t of the p r o o f
by
"shorter"
to p r o v e (Axioms
is
~ , the r e l a t i o n
the u n i q u e n e s s KI-3
etc.)
(The u n i q u e n e s s
is then
straight-
forward.
The r e l a t i o n By lemma
5.8 t h e y agree
for g r o u p s
in 5.4,5.5
for exact
in the c o n s t r u c t i o n
It is clear 5.2,
5.3
this
free c a t e g o r y ,
reasons
defined
is the
that
pertaining
categories.
slightly
from
The c h a n g e
the r e l a t i o n
is m a d e
used
to include
in [i].
relations
(see w
the q u o t i e n t
(illegitimate)
differs
of cwordC
free
b y the n a t u r a l
-category
c w o r d C m a y thus be a v o i d e d
on
C
. KC
relation
generated
then is a q u o t i e n t
and it a p p e a r s
only
by of
for t e c h n i c a l
to the p r o o f o f 5.7.
(5) That
these methods
could be applied
here occured
to m e w h i l e
reading
MacDonald
[7].
_
6. S t a r t i n g
m etm'e ~
from
we m a y
(6.1)
9
by means
of a p u s h o u t
bicartesian. -4
form the f o l l o w i n g
r
(1) and a p u l l b a c k
It follows
)>.<(
>>- <
<->
>>- ~
<-<(
> -~(
could have been used
).
squares
.<
.>
other groups
(I) and
commutative) -
for the c a t e g o r y
exist
).
to r e d u c e
(Lambek
forms can in g e n e r a l and
-r
shows
(-
that both
>>->
> .~
squares
are
.,
.,
fully c o m m u t a t i v e ~.~
w
5.8 then
(2).
in
>.,
4.5 and 4.6 are true
<.
)
that a n y of the forms
.~
.>
9
diagram
of r e l a t i o n s
every relation
[5! P r o p o s i t i o n s
be c o n s t r u c t e d
(2) in 6.1
in s t a r t i n g
formed
as p u s h o u t
5.7.
for g r o u p s in
1,2,
for groups,
theorem
RG
(6)
to the
p. 47,48]).
however,
and e n o u g h form
None
since
of the
5.8 fails
n e e d not be b i c a r t e s i a n
for
(or fully
A pushout
o
for groups, commutative
where
m
(pushout
in 5.6 is
RG
is n o t n o r m a l ~ e = cokm,
pullback
be a p u l l b a c k ~ m = kere
and h e n c e
!). K G
cannot
with order
be
fully
relation
as
.
Certainly
we c o u l d h a v e
directly
using
position
is a s s o c i a t i v e .
.<
<.
~.~
-7
Diagram
.<
<.
(6)4.5
does
not use a x i o m s
assumptions
defined
~-~
this.
weaker
cannot
.>
).
KI-3.
mentioned
relations >..
lemmas was
Then
similar
(for e x a c t
As m e n t i o n e d in w
Example.
for g r o u p s
and e x a c t
the p r o b l e m
arises
categories whether
to 5.8 h a v e
to be u s e d
categories)
used by Calenko
before,
4.6 m a y be p r o v e d
the com-
in a s s e r t i n g [3],
under
the
-g-
<
(-
). ~
.
Added in Proof
as suggested in [10! 4.8, p. 18] was used by Leicht [6].
(December 5,1968): Details are given in the "Anhang" of [2].
REFERENCES [I]
H.-B. Brinkmann,
"Relations for Exact Categories", to appear
[2]
H.-B. Brinkmann und D. Puppe,
(Preprint 1968).
"Exakte und Abelsche Kategorien, Korrespondenzen",
Springer Lecture Notes in Mathematics, to appear. [3]
M.S. Calenko,
"Relations for Quasiexact Categories", Mat. Sbornik
73(115)! 564-584 (1967). [4]
P. Hilton,
(1958).
J.B. Leicht, Manuscript
[7]
[8]
254-271, Springer 1966.
J. Lambek, "Goursat~s Theorem and the Zassenhaus Lemma", Canadian J. M~th., 10! 45-56
[6]
In Russian.
"Correspondences and Exact Squares", Proc. Conf. Cat. Alg. 1965,
(La Jolla, California)! [5]
(N.S.)
"Remarks on the Axiomatic Theory of Additive Relations", Unpublished
(1964).
J.L. MacDonald,
"Coherence of Adjoints, Associativities and Identities", Arch. Math.,
19! 398-401
(1968).
S. MacLane,
"Natural Associativity and Commutativity", Rice Univ. Studies, 49 (4)!
28-46 (1963). [9]
B. Mitchell,
"Theory of Categories", Academic Press 1965.
[10] D. Puppe, "Korrespondenzen in Abelschen Kategorien", Math. Ann., 1481 1-30 (1962).
-I0-
GALOIS OBJECTS AND EXTENSIONS OF HOPF ALGEBRAS
By Stephen U. Chase*
Let
R
a commutative,
be a commutative
cocommutative
coalgebras
and terminology isomorphism
of
projective
(a)
topology
[21]).
X (A)
U
algebra
Our principal
sheaves
to be defined relative
on the category
(when dis-
result is a natural
(A*,U)
(i)
later,
(b)
to a suitably
of commutative
S
R-algebras,
by the linear dual
A*
its multiplicative
group of invertible
of
(c)
R-
elements.
"Zariski"
they arise from certain rings of fractions
* The author wishes to acknowledge the GP-7945, Battelle Memorial Institute, and the Alfred E0 Sloan Foundation. results presented in this manuscript sequent issue of the Springer Lecture
A*
A, and
to each commutative
are essentially
of
chosen Grothendieck
assigns
in our topology
classes
is the category
is the sheaf which
coverings i.e.,
R-module
is the group of isomorphism
is the sheaf represented (d)
be
-
Galois A-objects, of abelian
A
and Hopf algebras we shall use the notation
X(A) ~ Ext I where:
and
Hopf R-algebra with antipode
which is a finitely generated cussing
ring with unit,
of
The
coverings; R.
support of N.S.F~ Cornell University, Detailed proofs of the will appear in a subNotes, ETH Series.
-II
The i s o m o r p h i s m
(i) a s s u m e s
cal f o r m in the f o l l o w i n g
special
the c h a r a c t e r i s t i c
of a f i e l d
i t i v e n th r o o t of
i,
in K,
F
and
H
and
respectively, any f i n i t e
with
abelian
U
the
left-hand
the c o m p a c t
k
receives
contains
F = {i}
and
is,
theory p.
its
in e s s e n c e ,
(see, e.g.,
H
F-module
a primitive
(2) b e c o m e s
~
kS/k
of k.
to
a primof
1
and
K/k,
Then,
for
-
, U(K))
continuous
(2)
homomorphisms
to the d i s c r e t e elements
str~cture
n th r o o t of
from
group of
J, K, and
f r o m t h a t of
i, t h e n
U(K)
U n-
K = k
and
-
~ Ex~(Hom~(J,Un), the w e l l - k n o w n
[i, pp.
19-22]
Kummer
or
(3)
U(k)) isomorphism
[19, C h a p i t r e
X,
of f i e l d
w
163] ). The
for f i n i t e mula p.
group
n
classi-
be p r i m e
with
of
closure
g r o u p of i n v e r t i b l e
Homc(H,J) which
groups
(Hom~J,Un)
side d e n o t e s
n
of n th roots
of e x p o n e n t
~ Ex~F
is the m u l t i p l i c a t i v e
If
J
Let
K = k(~)
a separable
topological
H o m z ( J , U n)
k,
be the g r o u p
n
group
Homc(~,J) where
case.
be the G a l o i s ks
a somewhat more
isomorphism
commutative
(i) m a y be v i e w e d
group
for a b e l i a n v a r i e t i e s
184]
and
[14, C h a p t e r
Cartier-Shatz formulates
formula
and g e n e r a l i z e s
of the W e i l - B a r s o t t i
([18, C h a p i t r e
III,
of
schemes,
18]);
as an a n a l o g u e , for-
VII,
Theorem
6,
it is a l s o
related
to the
413].
It re-
[17, P r o p o s i t i o n
i, p.
some of the w o r k
of H. H a s s e
[12],
-12-
P. W o l f
[23], D. K. H a r r i s o n
[i0], M. O r z e c h
[8] and o t h e r s
on G a l o i s
relies
on c o a l g e b r a i c
heavily
t i o n to the m e t h o d Galois principal
homogeneous
cases
a naive
definition
antipode. tative
and
gives
(unadorned
8
map
A
phism
S
7S:
S | S
w i l l be c a l l e d
both
mention.
some rela-
[6].
notions
of the being
[9].
Hopf
We use Let
R-algebra with S
is a c o m m u -
is an R - a l g e b r a
alone,
The p r o o f
for o u r p u r p o s e s .
homomor-
of a r i g h t A - c o m o d u l e
For brevity we S
H. E p p
analogue
in a t o p o s
the s t r u c t u r e
> S | A =
in
(S,e), w h e r e
> S | A
| ).
explicit
7s(X | y) S
~: S
means
needs
torseurs
is s u f f i c i e n t
(S,a) by the s y m b o l a
of g e o m e t r y ,
is a p a i r
to
and bears
and R o s e n b e r g
be a c o m m u t a t i v e
An A-object
phism which
pair
spaces
which
and
R-algebra
techniques,
16],
theory.
form a ring-theoretic
of J. G i r a u d ' s
be as above,
and K u m m e r
u s e d by C h a s e
objects
special
R
algebras
[15,
shall denote
writing
We d e f i n e
by the f o r m u l a
(x | l)as(y)
the
a = aS when
the
the a l g e b r a h o m o m o r -
(x,y in S)
if the a G a l o i s A - o b j e ~ct c t IZ
(4)
f l1 l7 o7 wing zo•
conditions
hold S 7S: Before
is a f a i t h f u l l y S | S
considering
a notational
dealing with
coalgebras
C
with
(5b)
> S | A is an i s o m o r p h i s m .
we introduce
algebra
(5a)
flat R-module.
several
examples
device which
and c o m o d u l e s
comultiplication
of G a l o i s
is q u i t e [21].
~C: C
For
objects,
useful when x
in a co-
> C | C, w e w r i t e
-13-
to d e n o t e
Z(x)X(l ) | x(2 ) denote
M
(A C | l)~c(X)
is an R - m o d u l e
=
~c(X),
(I|
etc.
homomorphism,
(x~)f(x(1),...,X(n)) In s i m i l a r
fashion,
is an A - o b j e c t ,
if
we w r i t e
= f((x~)X(1)
A
we w r i t e
Z(x)X(1) to d e n o t e
and so on.
for
formula
Ys(X | y) = Suppose generated a finite finite only
Hopf
Hopf
if
we h a v e
now that
algebra.
arising
ing the u s u a l an e l e m e n t
> S
8S: A* | S gives 8S
to
gives
S to
8 s(u | x) = u(x) establish
side,
for
the f o r m u l a e
case
A
u in A*, below-
will
be c a l l e d
is l i k e w i s e
if
S
~ HomR(A*
a
if and
is an A - o b j e c t ,
on
A
| S,S)
Then
side.
structure
The
x in S.
~S: S
be-
> S | A,
to a m a p fact
yields
structure.
(6)
and the s e c o n d
corresponds
in the r i g h t - h a n d
-
and a f i n i t e l y
-
isomorphism.
a left A * - m o d u l e
then b e c o m e s
R-algebra
case,
from our h y p o t h e s e s
a right A-comodule S
> S | A
(i | AA)es(X),
(x,y in S)
In this
of the l e f t - h a n d
=
S
and
as(X),
and is c o c o m m u t a t i v e
HomR(A*,S))
adjointness
algebra
to d e n o t e
A* = HomR(A,R) 413]
>
C
X(n ))
(eS | 1)~s(X)
in w h i c h
[17, p.
f: C | 1 7 4
Hopf
| x(2 )
S | S
isomorphisms
H O m R (S,S 8 A) ~ HomR(S, the first
Then
is c o m m u t a t i v e .
the n a t u r a l
|174
is a Hopf
A
to
-
8 Y(2)
R-module,
R-algebra
A
YS:
(~)xY(I)
projective
If
is a c o m m u t a t i v e
7(x)X(l ) G x(2 ) ~ x(3 ) The
7(x)x(l ) | x(2 ) 8 x(3 )
that
easily
eS that
We shall w r i t e
Routine
computations
- 14-
u(x)
=(~)x(1)(u,x(2) }
u(xy) for
x,y in S
and
d u a l i t y pairing.
as
tion that that
u in A*
structure
,
where
S
to
is a group,
with
A*,
is e q u i v a l e n t
to the
of R-algebras;
in the sense of
we shall denote by
coefficients
the usual
If
R-algebra,
IRG: RG G
we shall w r i t e
ply the set of functions
and counit
> RG
is finite,
bra operations,
RG
GR = from
and c o a l g e b r a
in an e a s i l y - a s c e r t a i n e d
it implies [21, p. 265].
G
Hopf R - a l g e b r a with
case
augmentation). IRG(~)
RG
= ~-I
for
is a finite Hopf
(RG)*; note that
GR
is sim-
with the p o i n t w i s e
operations
fashion
condi-
the group algebra of
(i.e.,
G to R
S
If
is defined by
in w h i c h
of
of Galois A-objects.
R, a c o c o m m u t a t i v e
comultiplication
The antipode in G.
in
is the
structure
We turn now to some examples G
| A ----> R
A*
the algebra
of
S
(}:
(7b)
links
be a h o m o m o r p h i s m
measures
A*
=(~)U(l ) (x)u(2) (y)
(7b), which
to the c o a l g e b r a
(7a)
and antipode
alge-
arising
from the group o p e r a t i o n s
in
G. If be viewed,
S
via
is a GR-object,
(7a), as a left module
then g u a r a n t e e s morphisms
of
on which
G
that the elements
of
If, conversely,
S
S.
acts via R - a l g e b r a
the usual w a y corresponding,
with
a left RG-module, via
G
finite,
over G
RG =
S then , (GR) 9
(7b)
act as R - a l g e b r a
is a c o m m u t a t i v e
automorphisms, and the map
(6), to the structure
may
map
then
aS: S RG | S
S
auto-
R~igebra is i n
> | GR > S
15
renders 1.3e]
S
a GR-object.
and
[6, L e m m a
Finally,
2.5]
renders
scrutiny
apparent
is a G a l o i s
GR-object
if a n d
with
group
G
in the s e n s e
S
fields,
if
Galois R
only
and if
group
for
S G
G
is a n o r m a l ,
t,
Having
described
the
group.
integral for
the
RZ
If we w r i t e is
a free of
t.
If
any i n t e g e r
n
by
the
{x in S
It is t h e n e a s i l y i.e.,
S
is a ~ - g r a d e d
component.
algebra,
then
defined
b y the
If, S
formula
nally,
i t can be
S1
a projective
is
S
of
direct
condition
S
extension
about
G = Z,
basis
if a n d with
Galois
proof.
GR-objects,
at
the
least RG-objects.
the i n f i n i t e with
generator
consisting we
of a l l
define
Sn
-
= x | t n in S | BZ~.
that
S = 7+~ 9 with
Sn
Sn
S = Z+~_ |
an K - o b j e c t
with
and
S m S n c Sm+n;
as its n th h o m o g e n Sn
is ~ - g r a d e d
aS:
S t ~
R-
S | ~Z
-
= x | tn
(x in S n)
are p r e c i s e l y
the Z - g r a d e d
shown
R
is an R Z - o b j e c t ,
R-algebra
becomes
the K - o b j e c t s
an e a s y
conversely,
as(X) Thus
extension
with
that
GR-object
multiplicatively
I as(X)
verified
fact
In p a r t i c u l a r ,
a Galois
in w h i c h =
[4, T h e o r e m
is a G a l o i s
to i n q u i r e
R-module
powers
Sn =
eous
case
the
[4].
Galois
is t e m p t e d
special
S
of
separable admits
one
if
is
assertion
finite,
then
S
; this
We c o n s i d e r cyclic
are
only
of
that,
R-module
if
S
of r a n k
is
a Galois
one
R-algebras.
Fi-
RZ-object,
then
[3, C h a p i t r e
2,
54];
-16-
in fact,
the
the G a l o i s This 8,
correspondence
RX-objects
correspondence
our
S1
is o n e - t o - o n e
the p r o j e c t i v e
is d i s c u s s e d ,
last example,
of c h a r a c t e r i s t i c
A = k [ t ] / ( t P n) , of
t in A;
One
sees
the
structure
~A(Z)
p >
n
then
z
R-modules
for example,
in
let us r e p l a c e
R
xP n in k,
that
is
a purely
a Galois
formula
not
between
of rank
one.
[20, E x p o s e
~A(Z)
in
k.
following
fact:
extension
of
a Galois
A
[22, T h e o r e m
SI
who If
has K
then
A-object
Next
and
examples
k,
is
A
US:
z
be
given,
and
field
~A(Z)
> S | A
of
way,
such
= -z.
extension easily
zP n = 0.
a unique
Hopf k-algebra
and
S
in
the i m a g e
that
Now k,
let with
verified
that
is d e f i n e d
by
the
-
Sweedler,
for
= 0,
a field
the k - a l g e b r a Let
It is t h e n
if
by
as a k - a l g e b r a
can be
inseparable
A-object
complicated
M~ E~
A
commutative
a s(x) More
integer.
A
of a f i n i t e
~Dn-1
and c o n s i d e r
generates
= z | 1 + 1 | z, be
0,
a positive
immediately
S = k(x)
S
~
w For
k
and
S
= x | 1 + 1 | z of
this
type
have
proved
a theorem
is
finite
any
there
for s o m e
finite
which
purely
is a f i e l d
been
S
considered
implies
the
inseparable containing
commutative
Hopf
by
field K
which
k-algebra
6]. we
describe
a commutative, are A - o b j e c t s ,
the
group
cocommutative an R - a l g e b r a
X(A) Hopf
introduced R-algebra.
isomorphism
f: S
in If
(i), S
~
S'
-17-
w i l l be called The elements A-objects.
an A - i s o m o r p h i s m
of
~S If =
(f ~ l) a S: S ---~ S' 8 A.
are the A - i s o m o r p h i s m
X(A)
If
if
Sl, S 2
classes
are Galois A-objects,
of Galois
we set -
cl(S I) + cl(S 2) = cl(S)
(cl() tained
meaning
"A-isomorphism
as follows,
by the formulae
Define
class
maps
of
( )"), where
8,~: S 1 8 S 2
is ob-
S
> S 1 @ S2 @ A
8(x | y) =(~)x | Y(1)
8 Y(2)
~(x 8 y) =(Zx)X(1 ) @ y | x(2 ) for
x in SI,
y in S2,
The c o a s s o c i a t i v i t y common S | A,
of
A,
A-object;
of
Suppose
A-object,
of
8
S
a map
guarantees
US: S
) S | A.
is
X(A)
cl(A),
now that
cocommutative
where ~: B
Hopf
R-algebra
) S | A | B
that
the operation
uniquely Using
S =
(S,~ s)
group.
~A = AA: A
is a obtained
The zero
~ A | A.
is a h o m o m o r p h i s m
R-algebras.
If
homomorphisms
by the formulae
-
~(x | b) =(~)x | ~ ( b ( l )) | b(2 )
~(x | b) =(Zx) X(l ) 8 x(2 ) | b
through
the cocom-
"+" thereby
an abelian
) A
~(w)}.
that the
factors
and
it can then be shown
we define
~,~: S | B
~S2
and renders
X(A)
commutative,
~Sl,
furthermore,
is w e l l - d e f i n e d element
to
thus p r o d u c i n g
mutativity Galois
of
restriction
S = {w in S 1 @ S 2 / 8 ( w ) =
and let
S
of
is a Galois
-18N
and set
N
9(S)
algebra
of
= {z in S | Bl~(z)
S | B,
= ~(z)}.
and it is e a s i l y
9(S)
verified
is an R - s u b -
t h a t the r e s t r i c -
N
t i o n to
9(S)
factors
of the m a p
uniquely
:9(S)
S | AB:
~(S)
through
> 9(S)
S | B
>
(S | B)
| B, thus p r o d u c i n g
| B
a map
It c a n t h e n be s h o w n t h a t
| B.
9(s) 9(s)
= (gCs),
)
is a G a l o i s
B-object;
moreover,
the m a p
9(s) X(9) : X(A)
> X(B)
homomorphism
of a b e l i a n
tablish
that
X
defined
is an a d d i t i v e
H
antipode)
to the c a t e g o r y
of c o m m u t a t i v e ,
The g r o u p
J
a finite
previously Let
in
[ii],
X(A)
below
t h e n es-
functor
f r o m the
Hopf R-algebras
from a similar
for the s p e c i a l
group.
is a
(with
groups.
treated,
abelian
We d e s c r i b e c ases
cocommutative
was
= cl(~(S))
computations
contravariant
of a b e l i a n
X(A)
by D. K. H a r r i s o n
A = JR,
Routine
groups.
category
view,
X(9) (cl(S))
by
point
of
c a s e in w h i c h
for some of the s p e c i a l
considered.
k
be a field,
and
J
be a f i n i t e
abelian
group.
Then X(Jk) where
K
is the G a l o i s
group
of a s e p a r a b l e
and the r i g h t - h a n d
side d e n o t e s
the c o m p a c t
H
phism
group
is n a t u r a l
in
continuous
to the d i s c r e t e J,
(8a)
~ Homc(H,J) closure
k s of k,
homomorphisms
group
and m a y be d e s c r i b e d
J.
The
from
isomor-
explicitly
as
-19-
follows. SX
be
Let
•
ff
> J
the k - a l g e b r a
n(u(~))
= u(x(~)~ ) (u)(~)
defines
be
of all for
functions
all
of
in v i r t u e
of w h i c h
S
earlier.
A routine
exercise
that
S
is
•
; a,~
a Galois
such
The
that
formula
-
automorphisms,
a Jk-object,
in the G a l o i s
and
in J)
via k-algebra
becomes
X
> ks
a in J.
(u in S
J on S
homomorphism,
u: J
in if,
= u(~)
an a c t i o n
establishes
a continuous
as e x p l a i n e d
theory
Jk-object.
of
The
fields
isomorphism
x introduced
above If
discussed tive
then
R
sends
to
is a c o m m u t a t i v e
correspondence
R-modules
•
of r a n k
between
cl ( S ) X ring,
then
Galois
one e s t a b l i s h e s
in
X (Jk) .
the p r e v i o u s l y
RE-objects
and p r o j e c -
an i s o m o r p h i s m
-
X(PcE) ~ Pic(R) the
latter
denoting
projective
R-modules If
there
the P i c a r d
exists
R
of r a n k
of
U(R)
R,
and
one
is a c o m m u t a t i v e
an e x a c t
sequence
U(R) ( )n > U(R) where
group
ring
and
and
2,
classes
of
w
X n = ZInZ,
then
-
> X(RZ n)
left-most
of i s o m o r p h i s m
[3, C h a p i t r e
> Pic(R) ( )n> P i c (R)
is the m u l t i p l i c a t i v e the
(8b)
group
right-most
of i n v e r t i b l e maps
send
(8c) elements
an e l e m e n t
to its n th power. These pal homogeneous of
(8a),
facts
are w e l l - k n o w n
spaces.
in a m o r e
A somewhat
general
in the c o n t e x t more
situation,
detailed
can be
found
of p r i n c i -
treatment in
[7,
w
-
(8b) is d i s c u s s e d , (8c)
for e x a m p l e ,
can b e e s t a b l i s h e d
a l s o be o b t a i n e d
2 0
from
-
in
[20, E x p o s e
by an e a s y d i r e c t (Sb)
and s t a n d a r d
8,
w
argument;
it c a n
cohomological
techni-
ques. We s h a l l m o t i v a t e (i) by a b r i e f
consideration
the i s o m o r p h i s m
(3).
Let
prime
to a g i v e n n a t u r a l
tains
all n th r o o t s
of e x p o n e n t m i n e d by,
n,
of
hence
to d e f i n e
x
-i
a (x)
a subgroup we define
of
V(S)
is in
k
U(S)
a function
~x(a~)
in
is a f i n i t e
= x
that
k
(3) d e t e r m i n e s ,
let
S
and is d e t e r -
(9)
U(k)) be a G a l o i s
on w h i c h
for all
J
a in J.
contains on
-1
J
a(x)
Jk-object,
acts v i a k - a l g e b r a x in. U(S)
V(S)
U(k) .
If
by the f o r m u l a
=
such
is c l e a r l y x
is in
-
(a in J)
U(k) ; f u r t h e r m o r e ,
= x-i ~ { x ( x - i T ( x ) ) }
~Ox (a)~x (T)
con-
abelian group
w e see that,
for
-
= x-laT(x) =
J
and a s s u m e
b e the set of all
which sx
by
-
k-algebra
Let
takes v a l u e s
a,~ in J
If
n,
~ Ex~(Hom(J,Un),
9 x(a) 9x
number i.
case p r o v i d e d
b e a f i e l d of c h a r a c t e r i s t i c
this map,
a commutative
automorphisms. that
k
an i s o m o r p h i s m
to the i s o m o r p h i s m
of the s p e c i a l
the i s o m o r p h i s m
X(Jk) In o r d e r
our approach
(x-la(x))(x-iT(x)
V(S) ,
-21
whence
%ox
is
it f o l l o w s roots
of
object
a homomorphism.
that 1
in
%ox
takes
values
k.
Also,
the
guarantees
elements in
of
J
J
that
if and
only now
morphism.
It is t h e n
cohomology
of,
slight 2, p.
of
if
with
generalization 158] ; i.e.,
clear
that,
%O = %ox-
then
(where
the
0
>
arrows
respectively),
~
and
S
conclude
%ox(~)
is
Un
of n th
a Galois
JK-
fixed by
all
=
all
1
for
hence
J.
%O
is
90
[19, such
x
is t h e n
the
sequence
defines
map
VCS),
below
and
is e x a c t
x
> %ox'
of
E x t I { H o m ~ ( J , U n) , U(k) }. Of c o u r s e , termined (x,~) and
by,
the p a i r i n g
> %ox(~) k
the e x t e n s i o n
contains
that
V(S)
whose
n th p o w e r s
= x-l~(x). all n t h
is p r e c i s e l y
which
S
is
Galois
group
are
in
a normal, J,
V(S)
• J
Now,
roots
the p a i r i n g
'> U n
1
J in
defined has
for
field
introduced
and
exponent
elements
the s p e c i a l extension above
of
is de-
by
S, i t is e a s y
of n o n - z e r o
Thus,
separable
determines,
since
of
the s e t k~
~S
-
~ 0
the m a p
an e l e m e n t
a
that
in
and
by
Proposition
) H o m ~ ( J , U n)
the i n c l u s i o n
a one-
a coboundary
U(S)
homo-
to the G a l o i s
X],
Theorem
> V(S)
therefore
n,
an a r b i t r a r y
regard
in
that
is
left
Un
with
x
in
exponent
U(k) ~
U(S),
exists
denote
S
[19, C h a p i t r e in
U(k)
of
>
of H i l b e r t ' s
there
that
is in %o: J
for a l l
We m a y
fact
that
%OCt) = x-l~(x)
~S:
x
values
has
k, w h e n c e
for e x a m p l e ,
J
J
i n the s u b g r o u p
the s u b r i n g
is p r e c i s e l y
Suppose
cocycle
Since
n to see
of
case k
coincides
S in with with
22
that arising in the c l a s s i c a l
formulation
of K u m m e r
(see, e.g.,
The m a p p i n g
(9) assigns
cl(S)
[i, pp.
in
X(Jk)
Ex~(Ho~J,U
19-22]).
the e l e m e n t
n) , U(k)),
cl(~ s)
theory to
in
and the techniques
of
[i, pp.
19-22]
can then be used to show that this m a p p i n g is a w e l l - d e f i n e d isomorphism
of abelian
(3) becomes
more apparent
H o m ~ ( J , U n) ~ A(kJ,k)
groups.
The relation b e t w e e n
if one observes
= A((Jk)*,k),
A(
)
(9) and
that denoting k-algebra
h omomorphi s ms. In order to obtain in a quite tative
similar
fashion.
We define
such that
x-lu(x)
If
(7a) .
x
is in
formula
V(S)
V(S)
Routine
R
V(S),
be a commutative,
antipode,
for all
is a subgroup
and
we define
of
u in A*, U(S)
a mapping
S
be a Galois x
in
with
containing ~x: A*
cocommu-
> R
U(S) u(x) U(R). by the
-
computations,
just discussed,
algebras,
= x-lu(x) entirely show
that
(u in A*) similar ~x
to those of the special
is a h o m o m o r p h i s m
of R-
and the sequence b e l o w is e x a c t 0 ~>
where
A( )
arrows
denote
tively.
A
(i), we p r o c e e d
to be the set of all
is in
~x(U)
case
Let
finite H o p f R - a l g e b r a with
A-object.
as in
the i s o m o r p h i s m
denotes
U(R)
> V(S)
~ A(A*,R)
R-algebra homomorphisms,
the i n c l u s i o n
map and the map
(i0)
and the u n l a b e l e d x
> ~x'
respec-
In this case the latter map is not in g e n e r a l surjective.
- 2S-
Now,
the left A * - m o d u l e
in the u s u a l way,
structure
a right A*-module
on
structure
S
induces,
on
S* = H o m R ( S , R ) .
A
F o r the s p e c i a l
case
in w h i c h
the s u r j e c t i v i t y
of the m a p
f ollows.
generates
If
z
is an R - a l g e b r a
S* V(S)
S*
homomorphism,
as r i g h t A * - m o d u l e s ,
A*
> ~(A*,R)
m a y b e s e e n as
as an A * - m o d u l e we define
~: A* ---> R
and
f: S*
~ R
by the
equation f(zu) S
is a f i n i t e l y
exists
a unique
( ): S* @ S shown
that
generated x in S
> R x
is in
V(S)
T
and
in g e n e r a l
faithfully
can be c h o s e n
.
R-module,
f = (
and thus
.,x),
pairing.
there
where
It can t h e n be
9 = ~x" S*
n e e d n o t be i s o m o r p h i c
it is at l e a s t a p r o j e c t i v e
The techniques
be u s e d to s h o w t h a t for s o m e
projective
such that
as a r i g h t A * - m o d u l e ,
of r a n k one.
(u in A*)
is the d u a l i t y
Although A*
= ~(u)
of,
for e x a m p l e ,
S* | T ~ A* | T flat c o m m u t a t i v e
[5,
w
to
A*-module can then
as r i g h t A* | T - m o d u l e s R-algebra
T.
In fact,
to be of the f o r m r
T
=
H
(ll)
Rxi
i=l where
Xl,...,x r
the J a c o b s o n
are e l e m e n t s
radical,
is the l o c a l i z a t i o n subset g~nerated antee
easily
by
such that
of x i.
R
of
n o n e of w h i c h
R,
x I + ... + x r = i, a n d
at the m u l t i p l i c a t i v e l y The preceding
that a base-change
are in
from
R
paragraphs to
T
Ri
closed then guar-
renders
the
- 24
sequence
(i0)
a short
It is thus of s h e a v e s algebra
in t h e
S,
(ii)
easily those
and
that
form
x in S
this
collection
properties
thus
gives
rise
dual
to the
then
consider
category the
A
to a b e l i a n
groups
such
with
below
S | T
S
the
sheaves
for any
S
- F(d0),
(i = 0,i)
transformation
S
is a s h o r t
conditions
The
hold
exact
of
category,
0 . > F' - ~ in
[2]),
and
category We may in this F
d:
S
from --->
S
|
-
> S | T | T
is an a b e l i a n
to
a n d ele-
functor
covering
checks
dual
R-algebras.
(covariant)
F(S | T) F(d0)
a natural
to
on the
F(S
F
sequence
P>
| T 8 T)
defined
d O (s | t) = s | 1 | t, d 1 (s | t) = s | t | i. is s i m p l y
One
axioms
reader
of a b e l i a n
that,
is of the
S | T.
topology
is a
is e x a c t
> F(S) F ( d ) ) ~:
S
category of
0
refer
of c o m m u t a t i v e
object
the s e q u e n c e
we
T
satisfies
R-
an R - a l g e b r a
(for the d e f i n i t i o n
to a G r o t h e n d i e c k
An
topology.
of m a p s
sequences
a commutative
to be
in
groups. exact
where
x | 1
topology
of w h i c h
S
of
> S | T, to
short
Given
context.
S
goes
of a b e l i a n
to c o n s i d e r
a coverin~
of a G r o t h e n d i e c k
mentary
A
following
of the
sequence
natural
we define
homomorphism form
exact
-
by
A map
in
S
functors. and
F t'
a sequence
>
if a n d o n l y
-
0
if the
following
-
sequence
0
) F' (S)T(S);> F(S) p(S)-_ F'' (S)
(12a)
T,
-
is an e x a c t
Given there
sequence
of a b e l i a n
any o b j e c t
is a c o v e r i n g
s uch t h a t
S
of
d: S
p (S | T)(x)
II,
w
-
groups
A
and
for any o b j e c t
x"
> S | T
in F "
a n d an
S
of
(S) ,
A.
(12b)
x in F(S | T)
= F"(d)(x'~.
F or the p r o o f s Chapter
25
Note
of t h e s e
finally
facts we r e f e r
that,
since
S
to
[2,
is an a b e l i a n
*
category, Chapter then
we m a y d e f i n e
XII].
In p a r t i c u l a r ,
Ext~(F,G)
l ence c l a s s e s
E x t s(-,-)
m a y be v i e w e d
Now, A,
given
we n o t e
that
tative Hopf T-algebra A | T-object. cussion
It then
; G A
A | T with
is a s h o r t e x a c t the a b e l i a n
U(T)
> V( S | T)
V ( S | T) noting
T-algebra
and
of
G
in
> F
are a b e l i a n
as in
antipode,
sheaves,
g r o u p of e q u i v a of the f o r m -
(i0) a n d
and
easily
S
[13,
> 0
is a f i n i t e
> Vs
> U in
Vs(T)
T
an o b j e c t
commutative, S | T
cocommu-
is a G a l o i s
f r o m the p r e c e d i n g
dis-
S;
> A* where,
> 0 for
T
an o b j e c t
= V ( S | T)
and the m a p s
~ AT(HOmT(A
| T, T),T)
of
and
> A~*(T) = A(A*,T)
t hat the ro les
and
in
-
sequence group
S
follows
t h a t the s e q u e n c e
for e x a m p l e ,
as the a b e l i a n
; E
and
~S: 0
A,
F
of s h o r t e x a c t s e q u e n c e s 0
of
if
as,
homomorphisms) R, A,
S | T, r e s p e c t i v e l y .
S
are d e f i n e d
as in
(~ (i0),
are n o w b e i n g p l a y e d by The m a p p i n g
(i) s e n d s
deexcept
T, A | T,
cl(S)
in X(A)
-
to
cl(~s ) in Ext I(A*,U),
26
-
and c o m p u t a t i o n s
of an e s s e n t i a l l y
r o u t i n e n a t u r e e n s u r e that it is a h o m o m o r p h i s m of a b e l i a n groups. In order to o b t a i n the i n v e r s e map we t u r n to c o n s i d e r a t i o n s
Ext~(A*,U)
> X (A) ,
of a m o r e H o p f a l g e b r a i c nature.
G i v e n a short e x a c t s e q u e n c e ~: 0 ~ in
S,
U---~ E
> A*
we observe that the f u n c t o r
Hopf algebra
P~.
then guarantee R-algebra arises
H
The t e c h n i q u e s
that
E
U
is r e p r e s e n t e d by the
of f a i t h f u l l y
flat d e s c e n t
is l i k e w i s e r e p r e s e n t a b l e by some Hopf
(see, for example,
[14, p. III.
17-6]), w h e n c e
from a sequence A* ~
of a b e l i a n c o g r o u p objects
H in
> RZ A;
i.e.,
tative Hopf R - a l g e b r a s w i t h antipode. that
> 0
H
is an RZ-object,
with
commutative,
cocommu-
It is then easy to see
aH: H
> H | RZ
the c o m p o s i t e
map -
H AH-_ H @ H H @ ~> H @ I ~
Therefore,
with
Hn =
as r e m a r k e d earlier,
{z
in
H/all(z)
H
H =+~
|
= z |
tn}.
is a Z - g r a d e d R - a l g e b r a -
Hn
Since
it follows
of
is even a H o p f s u b a l g e b r a of
whence
the image of
H0 p
lies in
H0,
Hn
is a Hopf al-
gebra homomorphism, H,
that each
eH
thus p r o v i d i n g
is a s u b - c o a l g e b r a H.
Moreover,
a Hopf a l g e b r a
-
homomorphism morphism
p: A
2 7 -
We then obtain a c o a l g e b r a homo-
> H 0.
-
H I | A* H I |
P_~ H I | H 0
the u n l a b e l e d a r r o w d e n o t i n g the m u l t i p l i c a t i o n map of generated projective
the a p p r o p r i a t e r e s t r i c t i o n of
H.
Finally,
H1
is a f i n i t e l y
R-module, w h e n c e d u a l i z a t i o n of
yields an R-algebra homomorphism S = H I.
(13)
> HI
It turns out that
S =
aS: S (S,~ s)
> S | A,
(13) with
is a Galois A - o b j e c t , i
and the map cl(S)
(i) then sends
in X(A).
cl(~)
This c o m p l e t e s
of the i s o m o r p h i s m
in Ext~(~*,U)
to
our sketch of the c o n s t r u c t i o n
(I).
A final r e m a r k r e g a r d i n g
the G r o t h e n d i e c k
w h i c h w e use to d e f i n e our sheaves.
topology
N o t e that the c o v e r i n g s
in this t o p o l o g y are of a v e r y r e s t r i c t e d type;
in p a r t i c u l a r ,
the t o p o l o g y is m u c h c o a r s e r than,
the f a i t h f u l l y
flat t o p o l o g y on R
A.
is a local ring,
sheaf on groups)
A
Indeed,
for example,
for the s p e c i a l case in w h i c h
it is t r i v i a l l y v e r i f i e d that every pre-
(i.e., c o v a r i a n t
is a sheaf.
functor from
This is p r o b a b l y
in the d e d u c t i o n of the i s o m o r p h i s m
A
to a b e l i a n
the m o s t i m p o r t a n t step
(2) from the i s o m o r p h i s m
(i). The f o r e g o i n g m a t e r i a l
is a r e s u m e of a lecture pre-
s e n t e d at the B a t t e l l e c o n f e r e n c e on c a t e g o r i c a l cal algebra,
Seattle,
June-July
1968.
and h o m o l o g i -
D u r i n g the conference,
-
S. Shatz and D. Quillen isomorphism
28-
suggested
(i), the ingredients
Shatz formula, cohomological
a spectral
an alternate
of which include
sequence
classification
argument,
of principal
This method is similar to that of H. Epp using sheaves
in the faithfully
approach to the the Cartier-
and a well-known
homogeneous
spaces.
[8], who derived
flat topology,
for the special
case of group schemes whose duals are of multiplicative (i.e.,
A
has the property
fully flat commutative abelian group
J).
that
R-algebra
A | T ~ JT T
(i),
type
for some faith-
and finitely generated
29-
REF [1]
Artin, E., and Tate, J., Class Field Theory~ Princeton University mimeographed notes, Princeton,
New Jersey
(1960). [2]
Artin, M., Grothendieck Topologies~ Harvard University mimeographed notes, Cambridge,
[3]
Bourbaki,
(1962).
Chase, S. U., Harrison,
D. K., and Rosenberg,
"Galois Theory and Galois Cohomology
Chase, S. U., and Rosenberg,
Alex,
the Brauer Group, " Memoirs Amer 79, [6]
52; 15-33,
(1965).
"Amitsur Cohomology
and
Math. Soc., Vol. 52; 34-
(1965).
Chase, S. U., and Rosenberg, Kummer Theory,
Chase, S. U.,
Alex,
"A Theorem of Harrison,
and Galois Algebras,"
Vol. 2_~7; 663-685, [7]
Alex,
of Commutative
Rings," Memoirs Amer. Math. Soc. j Vol. [5]
(1962).
N., Alg~bre Commutativej Chapters I-II,
Hermann, Paris [4]
Massachusetts
Nagoya Math. J.,
(1966).
"Abelian Extensions
and a Cohomology
Theory
of Harrison, " 375-403 in Proceedings of the Conference on
Categorical Algebraj La Jolla (1965), Springer-Verlag, New York Inc., [8]
(1966).
Epp, H. P., Commutative Group Schemesj Harrison's Theorem,
and Galois Extensionsj Ph.D. Versity,
(1966).
thesis, Northwestern
Uni-
-
[9]
Harrison,
Columbia Uni-
(1965).
D. K., "Abelian Extensions of Arbitrary Fields,"
Trans. Amer.
Soc., Vol.
Math.
106; 230-235,
(1963).
, "Abelian Extensions of Commutative Rings,"
[11]
Memoirs Amer.
[12 ]
-
Giraud, J., Cohomologie Non-Abelienne, versity Notes
[lO]
30
Math.
Soc.
52; 1-14,
(1965).
Hasse, H., "Invariante Kennzeichnung Galoisscher K6rper mit Vorgegebener Galoisgruppe," Vol. 187; 14-43,
J. Reine Angew.
Math. ,
(1950).
[13]
MacClane, S., Homology, Academic Press, New York,
[14]
Oort, F., Commutative Mathematics,
[15]
Vol.
(1966).
Zeitsch., Vol.
105; 128-140,
(1968).
, "A Cohomological Description of Abelian Galois Extensions,"
to appear.
Shatz, S., "Cohomology of Artinian Group Schemes Over Local Fields," Ann.
of Math., Vol. 79; 411-449,
(1964).
Serre, J. P., Groupes Algebriques et Corps de Classes, Hermann, Paris,
(1959),
(Act. Sci. Ind. #1264).
, Corps Locaux, Hermann, Paris,
[19] Sci. Ind. [20]
Berlin,
Orzech, M., "A Cohomology Theory for Commutative Galois
[16 ]
[18]
(Lecture Notes in
15), Springer-Verlag,
Extensions," Math.
[17]
Group Schemes,
(1963).
(1962),
#1296).
Strasbourg University Department of Mathematics, Algebriques," 1965-66.
(Act.
"Groupes
Seminaire Heide lberg-Strasbourg Annee
-
[21]
Sweedler, Applied 276,
[22]
M. E.,
31
-
"The Hopf Algebra
to Field Theory, " J. of Algebra, Vol.
9 "Structure
Wolf,
as
8; 262-
(1968). of Inseparable
Annals of Math., Vol. 87; 401-410 9 [23]
of an Algebra
P. , "Algebraische
Theorie
Extensions,"
(1968).
der Galoisschen
Algebren,"
Deutscher Verlag der Wissenschaften, Math. Forschungsberichte III, Berlin,
(1956).
-
THREE
DIMENSIONAL
32
-
NON-ABELIAN
COHOMOLOGY
FOR
GROUPS
by Paul Dedecker
The
So far non-abelian cohomolog7 has been discussed in dimensions n < Z.
following is an effort to discuss dimension n = 3 and solves the m a i n difficulties sO H o w e v e r an n-dimensional theory for
that just polishing w o r k remains necessary. n > 3 remains
a remote
The present
report
I am grateful Wesleyan offered 1.
target. is a direct
continuation
for the hospitality
University,
of t h e C e n t e r
during the final phase
by the University
of [ 2 ] . for Advanced
of t h e r e d a c t i o n .
Studies,
Typing was
kindly
of K e n t u c k y .
Introduction. We shall assume
as developed few facts.
in e a r l i e r
to serve
dimensions
0 and
took so much
* Research belge
papers
as coefficient
1.
1958
group.
with the
[1],
category
time to take off.
[2], [ 3 ] .
theory,
Let us however
is
why the
1958 w e h a v e t o
22~ and
a
of G o n l y in
2 and that is the reason
g r a n t No.
n < 2,
recall
>7~1 o f G - g r o u p s
for a cohomolog 7 theory
As I have shown since
by NAT0 Research
et de T o p o l o g i e
n-dimensional
Then the category
I t i s n o t s o in d i m e n s i o n
supported
d'Alg~bre
is familiar
since
Let G be an arbitrary
satisfactory
theory
the reader
Centre
-
replace 3 1 7~2 -~ ~ i "
33
-
by the category 7~2 of crossed groups which has a forgetful functor This concept was introduced by 5. H. C. Whitehead under the n a m e of
crossed module.
V e r y roughly speaking a crossed group (or crossed module)is a
group endowed with an additional structure related to its group of automorphisms. In dimension 3 w e shall have to introduce a n e w category ~ 3
whose objects w e
shall call supercrossed groups and which has a forgetful functor ~ 3
-~ ~ 2 "
As is
well known, if the functor H0(G, -) has values in groups, this is not the case of the functor
HI(G,-)
whose values are often described
However
this can be made much more
category
of p o l y p i [1, I, II], [3].
a category functor
precise
Similarly
known as the category
as pointed sets
H2(G,-)
So f a r t h e s t r u c t u r e
to say enough to show that they are very curious
in t h e
t a k e s i t s v a l u e s in of v a l u e s
H 3 ( G , -) is n o t s u f f i c i e n t l y k n o w n t o g i v e t h e s e o b j e c t s a n a m e .
however
[6] S e r r e ) .
since this functor has values
the functor
of s p i d e r s .
(e.g.
and interesting
of t h e
I hope animals
with nice feathers. Of course and
the usefulness
(ii) it p r o d u c e s
sequence
q2 functorial
of t h i s c o h o r n o l o g y t h e o r y ~ i s t h a t
a cohornology exact sequence
in t h e c o e f f i c i e n t s
category,
associated
a notion respected
(i) it is f u n c t o r i a l
with a short exact
by the forgetful functors
Let us m e n t i o n h e r e that C i r a u d [5], [61 d o e s not p r o d u c e a t r u l y cohomology,
nor a true exact cohornology-sequence.
....
HI(A,,) _~ H2(A ,) -~o H2(A) ~ H2(A,,)
He has in fact a
s ection
associated
to a short exact sequence
of s h e a v e s
A' -~ A -~ A"
In t h e r e t h e
-
symbol
o
does not represent
theory
in 3 1 .
However
(showing that
our H 3 to the more
this defect in Giraud's
a nice paper
beautiful theory more
to generalize
This trouble
when one does not go into a good category
could be easily avoided and certainly
important
2.
occurs
but wants to remain
this and making Giraud's
-
a map but a relation.
t h i s k i n d of H 2 is n o t f u n c t o r i a l ) of c o e f f i c i e n t s
34
could be written
accessible.
general
clarifying
It w o u l d t h e n b e
situation considered
by Giraud.
Quick review of 3-coh.o_mology. We r e m i n d
a n d 13 a r e
that a crossed-group
(arbitrary)
groups,
of
13 o n t o H b y a u t o m o r p h i s m s ,
if
ae
13, h e H , (i) (ii)
H,
If a ~ 13,
morphisms commutative
A = (H, p,13, ~) i n w h i c h H
p is a homomorphism
p : H - ~ rl a n d 9 i s a n a c t i o n
subject to conditions
(i) a n d (ii) b e l o w .
t h e r e s u l t of a acting on (or t w i s t i n g )
If h , k (
A morphism
A is a system
then
h ( H,
h is d e n o t e d b y ah c H .
Pkh = k h k - 1 . then
p(ah) = a. p h . a
A -~ A' b e t w e e n t w o c r o s s e d
j : H -~ H' ,
In these,
y : 13-* 13'
compatible
-1
groups
is of c o u r s e
with the structures,
a p a i r of i.e.
making
the squares
ll
Y > If'
H
j >
13 X H
@>H
(2..1)
This defines the category
~2
H'
II'
X H' @'~l H'
and the forgetful functor ~2
-" ~ 1 "
(~7~I = the
- 35
c a t e g o r y of g r o u p s )
is d e f i n e d by
(H, p, rI,@) ~ T h e o r e m 2.1.
-
H .
T h i s f o r g e t f u l f u n c t o r h a s no s e c t i o n .
R o u g h l y speaking, this m e a n s g r o u p of a u t o m o r p h i s m s
that it is not possible to associate to a g r o u p a
in a functorial w a y .
If t h e t h e o r e m w e r e f a l s e , t h e r e w o u l d be no n e e d of c r o s s e d g r o u p s to d e f i n e non-abelian cohomology. exists.
To the b e s t of m y k n o w l e d g e no p r o o f of t h e t h e o r e m
H o w e v e r t h i s n o n - p r o v e d t h e o r e m is r e s p o n s i b l e f o r t h e s n a g g y
o in
Giraud's theory. For
G
an arbitrary group, w e denote C0(G,A)
= C0(G,H)
= H ,
0-cochains,
CI(G,A)
= CI(G,H)
= App(G,H)
C2(G,A)
= A p p ( G , II) X A p p ( G X G , H )
,
l-cochains, ,
2-cochains,
w h e r e App d e n o t e s t h e s e t of a l l m a p s in b e t w e e n t h e u n d e r l y i n g s e t s .
Moreover
we d e n o t e by Z 2 ( G , A ) t h e s e t of 2 - e o c h a i n s
c G):
(2.2.a)
~(S)h(t,u) 9 h ( s , t u )
(2.Z.b)
~(s).
~(t)
--
ph(s,t).
(~,h)
such that (s,t,u,...
= h(s,t) 9 h(st,u) , ~(st)
.
An a c t i o n (2.3) is d e f i n e d by
* : CI(G, A) • Z Z ( G , A ) --~ Z2(G, A)
- 86-
a*(~,h) ~'(s)
= (~',h')
,
a:G-~H,
= pa(s) 9 ~(s) ,
h'(s,t)
T h i s is a g r o u p a c t i o n , c o n s i d e r i n g
(~,h) e Z Z ( G , A ) ,
= a ( s ) . ~(S)a(t) 9 h ( s , t ) , a(st) -I .
CI(G, A) as a g r o u p in t h e o b v i o u s w a y .
The
orbits form a set (thick 2-cohomology) HZ(G,A)
= Z2(G,A)/CI(G,A) .
"
There are other actions
(z.3)
E] : 11 •
A) -.. CI(G, A)
(2.4)
: 11 •
-* Z 2 ( G , A )
w h i c h a r e d e f i n e d by t h e f o l l o w i n g f o r m u l a s in w h i c h ac
11,
aE
CI(G,A),
(~,h) c Z Z ( G , A ) : a
aWa
=
a',
~(s) = a . ~ ( s ) . a
a'(s)
-1 d e f = a~(s) ,
=
a(s);
h(s,t)
= ah(s,t) .
T h i s s a t i s f i e s t h e f o l l o w i n g " d i s t r i b u t i v i t 7 law!' av[a*~,h)]
= (~Oa) * ( a V ( ~ , h ) ]
w h i c h shows t h a t t h e a c t i o n V t r a n s f o r m s t h e c o r r e s p o n d i n g o r b i t of a V (~, h). WV2(G, A).
t h e w h o l e o r b i t of (4, h) u n d e r C I into
T h i s t h u s d e f i n e s a n a c t i o n of 11 onto
T h e s e t of o r b i t s in ]~2(G, A) u n d e r t h i s l a s t a c t i o n is c a l l e d t h e t h i n
2 - c o h o m o l o g y and d e n o t e d
-
3 7 -
H2(G, A)
= ]HIE(G,A)/n.
W e can also consider the crossed product (or semi-direct product) r
= CI(G,A)~I]
which has as underlying set the product CI(G, A) x 11 with multiplication law (b,~). (a,a) = (b. (~ •
a),13a) .
W e then define a group action
|
1- x ZZ(G, A) -~ ZZ(G, A)
by
(a, ~) | ($, h) = a * [ ~ V ( $ , h ) ] . T h i s o b v i o u s l y p r o d u c e s an i s o m o r p h i s m
H2(G,A)
~ ZZ(G,A)/F
.
A 2 - c o c y c l e (4, h) is said to be neutral if h : G X G w h i c h we s h a l l d e n o t e by 1, t a k i n g e v e r y p a i r ( s , t ) this case
-*H is the trivial ma p,
i n t o t h e u n i t e l e m e n t of H.
~ : G -~ TI is a good s t a n d i n g h o m o m o r p h i s m .
A 2-dimensional coho-
m o l o g y class in H 2 (G,A) or in H2(G, A) is said to be neutral if it contains a neutral Z-cocycle. L e t 8 : G -~ 11 be a h o m o m o r p h i s m .
~
(G,A)
= Hom0(G,A)
the set of e-crossed h o m o m o r p h i s m s
W e then denote by = ZlelG, A) C
CIIG, A)
f:G-~ H, i.e. the set of m a p s
f:G-~ A
In
-
38
-
such that f(st)
=
f(s)
9 els)f(t)
.
An a c t i o n is d e f i n e d : *:H•
Z1o(G,A) -~ Z10(G,A)
,
b y putting h*f
= f',
f'(s)
= h- f(s). O(S)h'l 9
The orbits for this action form the set
Hle(G,A) - zle(o, A)/H 9 We a l s o d e n o t e by 0 H i ( G , A)
-- H 0
t h e s u b g r o u p of H c o n s i s t i n g of t h e 0 - i n v a r i a n t e l e m e n t s
h E H, i, e.
such that
h = 0(S)h f o r a n y s E G. F i n a l l y w e s h a l l d e n o t e by
t h e c o r r e s p o n d i n g s e t s e n d o w e d w i t h t h e s t r u c t u r e of n o t ov_ly h a v i n g a " c l o u d " of n e u t r a l e l e m e n t s but a l s o h a v i n g a m o n g t h e m a p r i v i l e g e d o n e , n a m e l y t h e cohomology class containing the 2-cocycle
(0,1).
T h e s e t ]E2(G, A) is a l s o d e n o t e d by E x t l ( G , A) .
-
39
-
To be able to define cohomology exact sequences w e need to define the notion of a short exact sequence in the category ~F~z of crossed groups.
This will be a
diagram
A
(z.s) giving
rise
to the commutative
11'
I
diagram
)
.'T
) H'
I) A J>A,,
. )
t
II
~ > 11"
.T H
."T
. Y H"
J
) i
satisfying the following conditions: (i)
the lower
row is a short
exact
sequence
of groups
;
(ii) 11' -~ 11 is an isomorphism and ~:II -~ 11" is epimorphic. In view of condition (ii) w e shall put 11' = 11 and replace this diagram by
(Z.6)
(Z) 1
n
----
p'I
I
~ J
.)
H
) H'
n
:
)
n,,
:p" .) H "
' )
I
J
Let e : G -~11, e " : G - - 11,, be h o m o m o r p h i s m s then possible
to define
it is
maps
A 1 1 A = AE:H~
such that e" = ~/. e.
0
= AE:H
,,(G,A")-- H (G,A') I i I HZ(G, Extl(G, A') , A = AE:He,,(G,A")-" A')
such that the following canonical square c o m m u t e s
-
1 H o m e , ,(~, A"I ~k-~
Hle ' ' ( G ' A ' ' ) 1 - ~ A
40
-
E x t l ( ~ , A') = ~I 2 ( G , A ' )
H2(G'A')
Moreover Theorem 2.2. s e q u e n c e (E) i n ~ 2 1
All w h a t we h a v e d e f i n e d is f u n c t o r i a l a n d e v e r y s h o r t e x a c t g i v e s r i s e to c o h o m o l o g y e x a c t s e~uenc e s:
0 A") A0 _~ H~(G, A') _~ H0(G, A) _~ He,,(G, -~
(2.7)
2 -. HI(G,A ') -~ H Ie (G,A)-~ He,,(G,A,, i AO ) Al-- H ~ (G, A' ) -- H ~ (G,A) --* HO,,(G; A"), 1
, -~ Home(G,A' ) -~ H o m e ( G , A ) -~ Home,,(G,A,, ) &-~
( z . 8)
&l Extle(G, A, ) In t h i s s t a t e m e n t , endowed with structures
Extle(G,A )
Extle,,(G,A")
e x a c t n e s s m e a n s t h a t t h e s e t s in t h e s e s e q u e n c e s a r e w h i c h m a k e it p o s s i b l e to a n s w e r t h e f o l l o w i n g t w o
que s t i o n s : Problem
(a).
W h e n is an e l e m e n t in one s e t ( e x c e p t t h e l a s t one) t h e image
of t h e p r e c e d i n g a r r o w ? Problem
(b).
W h e n do two e l e m e n t s in one s e t h a v e t h e s a m e i m a g e t h r o u g h
the next arrow? F o r e x a m p l e e a c h s e t is p o i n t e d o r h a s a c l o u d of n e u t r a l " d r o p l e t s " w i t h a m o r e distinguished one.
T h i s a l l o w s to s o l v e q u e s t i o n (a).
But t h e s t r u c t u r e n e c e s s a r y
to s o l v e q u e s t i o n (b) is m o r e c o m p l i c a t e d a n d m o r e i n t e r e s t i n g .
For example each
-
41
-
H o m f ( G , A) is a polypus and each Ex===~tI(G,A) = I~Z(G, A) is sitting under an object which I would like to call a lobster, na me ly a bigger animal which has pinces.
3. Approach of three-dimensional cohomology. This aims at solving question (a) for the last object in the sequences (Z.6) and (2.7). 1 To be m o r e geometric let us remind that any ~ e Ext0(G,A) can be repres e n t e d by an i s o m o r p h i s m c l a s s of d i a g r a m s 11 A~
(3. I)
I'~ I
1
> H i'-~X
>1
>G
a homomorphism
with the r o w an exact sequence of groups and
s u b j e c t to
c onditions (i) (ii)
p = ~.i, f o r x ~ X,
h c H,
i(~Xh) = x . i h . x
-1
T h e s e d i a g r a m s a r e f u n c t o r i a l in t h e s e n s e the a m o r p h i s m s
(Z.l) produces a sort of push-out diagram 11 1
)
H
i
)
H ' ---->
)X
)G
>
(3.2) O
>i
1
(j, •j
in '~z
as
in
-
u n i q u e up to i s o m o r p h i s m .
42
-
In p a r t i c u l a r t a k e H' = H,
II'
= II, ~/ = id,
and let
a
j =j
: H - ~ H be t h e a u t o m o r p h i s m i n d u c e d by s o m e a e I I .
T h e n (3.Z) c a n be
c o n s i d e r e d as an i s o m o r p h i s m b e t w e e n two d i a g r a m s of t y p e (3.1).
These larger
i s o m o r p h i s m c l a s s e s c a n be i d e n t i f i e d w i t h the e l e m e n t s of t h e t h i n HZ(G, A). P r o b l e m (a) f o r the l a s t o b j e c t in the s e q u e n c e (2.7) is e q u i v a l e n t to: s u p p o s i n g t h a t in (Z. 1) j a n d ~ a r e e p i m o r p h i s m s
and we h a v e in (3.2) t h e
bottom part, namely I] !
1
>H'
is it p o s s i b l e to f i n d s o m e d i a g r a m
>X'
>G
> 1
(3.1) f i t t i n g into ( 3 . 2 ) ?
And s i m i l a r l y f o r t h e
l a s t o b j e c t of s e q u e n c e ( 2 . 6 ) . L e t us n o w go b a c k t o t h e e x a c t s e q u e n c e (Z.4) in ~7~2 and s u p p o s e t h a t A C I] is the k e r n e l of N : I I -~ I I " ,
l
so t h a t we h a v e t h e e x a c t s e q u e n c e
> A
> n
> n,,-->l
N e x t , l e t us c o n s i d e r a Z - c o c y c l e ( ~ " , h " ) e z Z ( G , A " ) . p o s s i b l e to f i n d a 2 - c o c h a i n (~,h) w i t h v a l u e s in A, w h i c h l i f t s ( ~ " , h " ) in t h e s e n s e t h a t
k:GXG
) A
n a m e l y (~,h) c C 2 ( G , A )
5 (4, h) = ( ~ " , h " ) .
#
,
k:GXGXG
It is t h e n c e r t a i n l y
~
We t h e n h a v e f u n c t i o n s
H'
such that (3.3)
~(s)~(t)
(3.4)
r
= Ms,t) ph(s,t)~(st)
h(s,tu) = k(s,t,u), h(s,t), h(st,u)
-43
-
It thus s e e m s that the pair (k, k) is a candidate to the title of 3-cochain and ultiH o w e v e r another lifting ( ~ ' , h ' )
mately of 3-cocycle.
of ( ( ~ " , h " )
is o b t a i n e d by
applying to (6, h) a deviation (a, a) such that
(~',h')
= ( a , a ) . (~,h)
= (a. r
a.
h)
where
a:G-~A are arbitrary
functions.
and
a:GX G-~H'
T h e n e w lifting ( ~ ' , h ' )
then produces a pair
(k',k')
and
one w o u l d l i k e to c o m p u t e i t a s a f u n c t i o n of t h e i n i t i a l (k, k) and t h e d e v i a t i o n (a, a). U n f o r t u n a t e l y s u c h a f o r m u l a d o e s n o t e x i s t and e a s y c a l c u l a t i o n s y i e l d : k'(s,t)
= a(s).~(S)a(t).k(s,t).ph(s't)a(st)-I
k'(s t,u) = a(s)~(S)a(t,u).~(S)L '
a(s)
(t,
T h e r e a l t r o u b l e is t h e p r e s e n c e ~(s) L a(s)
(t,u)
. p a ( s , t ) -I ,
u). [~(s) ph(t, u) ]a(s ' tu). k ( s , t, u). ph(s, t ) a ( s t ' u)-I .a(s, t) -1
of t h e e x p r e s s i o n
=
a(s)~(S)h(t, u) ~(S)h(t, u) -1
and w e r e a c h h e r e the f i r s t c r u c i a l p r o b l e m in t h e c o n s t r u c t i o n of 3 - c o h o m o l o g y . Moreover,
in b o t h f o r m u l a s a p p e a r the f u n c t i o n ~ w h i c h m e a n s t h a t we h a v e to add
it to t h e p a i r
(k, k).
But we a l s o h a v e to i n c o r p o r a t e
L:A •
a function
x G x G -~ H'
the v a l u e of w h i c h at (a, 6, s , t ) we w a n t to d e n o t e
L(a,~,s,t) = !L(s,t) = ~L a
s,t
-
44
-
and which is given by
~a L ( s , t ) This is actually ~h(s,t)
c H,
a nice function
when twisted
element
of the smaller
inverse
crossed
morphism
by
group
s i n c e it m e a s u r e s
H'
Also, L:A
for
the modification
-~ H '
(namely
undergone
being by necessity
~, s , t f i x e d ,
an
this function is
its inverse
by
is a crossed
an homo-
or one has
a~ L = a(L).
We can however
-I 9
a ~ A , this modification
homomorphism
A -~ H'
= a.~h(s,t).~h(s,t)
consider
the more
simple
function
K:A•
(3.5)
a L)
-~ H'
defined by (3.6) which measures
K(a,s,t)
= aK(s,t)
the non invariantness
of
= ah(s,t).h(s,t)-i
h:G x G-~ H by a c A.
!L(s,tl = This can be clarified
as follows.
Consider
C I = CI(A,H')
and define a group action c 1] and K ~ C I,
~*K:A
then have
_lo K] the set = App(A,H')
* of 1] onto C I (composition product) -~ H'
We
is defined by
such that for
-
45
-
or equivalently
Similarly,
if K is a s in ( 3 . 5 ) ,
then we get
it c a n be i d e n t i f i e d w i t h a m a p
K:Gx
G-~ C1 and
~*K:G XG-. C 1. With this formalism it is clear that ~aL(s,t) : a ( ~ * K ) ( s , t )
(3.7)
To simplify the notations w e shall a l s o w r i t e ~ * K
.
= ~K a n d
~L(s,t) = !K(s,t) .
(3.8)
W e then want to consider
a
3-cochain as a system (k, k, 4; K, ~) of five
functions k:GxGXG k:GXG ~:G
-~ H', -- A ,
-~ 11 ,
K : A x G XG -~ H ' , ~:GxG
-~ I I .
NOW, s t a r t i n g w i t h a l i f t i n g (4, h) c C 2 ( G , A ) of ( ~ " , h " ) c Z 2 ( G , A " ) , obtained such a system
Thus
[ w i t h 11 ( s , t ) = p h ( s , t ) ] w h i c h we s h a l l d e n o t e b y
(k, k, };K;T]) = [ ] (~,h} .
(3.9) D
represents
we have
s o r t of a c o b o u n d a r y o p e r a t o r
[7 :C2(G,A) -* C3(G, ?) .
- 46
-
At t h i s p o i n t t h e l e c t o r c a n e a s i l y c h e c k t h a t the p a i r s
(a, a) f o r m a g r o u p
C2(G, B) w h e r e B is t h e c r o s s e d g r o u p ( H ' , p' , A , ~ )
in w h i c h ~ is t h e a c t i o n of
A to H '
M o r e o v e r t h e r e is a g r o u p
r e s t r i c t e d f r o m t h e one ~' of II onto H '
action
cZ(G' B) x CZ(G, A) - CZlG, A), ((a,a),(~,h))~-~(a,a).
(~,h)
= (~',h')
We a l s o h a v e a g r o u p a c t i o n
(3.10)
* : C2(G, B) x C3(G, ?) -- C3(G, ?) ((a, a), (k, k, J~;K;rl))n,--~ ( k ' , k', +' ;K' ;1"1' )
such that (3.11)
0[(a,a).(~,h)]
= (a,a)*[O (~,h)] .
This action is explicitly defined by putting (3 12) 9
k'(s,t,u) = a(s)~(S)a(t,u).J~(S)K'- " [~(s)~t'U)]a(s , tu) a(s)
~,uj.
9 k(~ t , u).
rl(s, t ) a ( s t ' u) -1. a(s, t)-1 ,
(3.13)
k'(s,t) = a(s). ~(S)a(t). k(s, t).Tl(s' t ) a ( s t ) - l , pa(s, t) -1
(3.14)
(3.15)
(3.16)
~'(s)
i K'(s,t) ,l'(s,t)
= a(s).~(s),
= t~a(s,t).l K(s,t).a(s,t)-I = pa(s,t).n(s,t)
.
,
-47
-
T h e e x i s t e n c e of t h e a c t i o n (3.10) giving r i s e to f o r m u l a 3-cochain (k', k', ~';K' ;~ ' ) derived f r o m
(k, k, ~;K;~]) and t h e d e v i a t i o n (u, a).
(3.11) s h o w s t h a t t h e
(~', h') can be c o m p u t e d in t e r m s of
Thus the 5-uples
(k, k, ~;K;~) a r e r e a l l y
s e n s i b l e c a n d i d a t e s to t h e t i t l e s of 3 - c o c h a i n s and 3 - c o c y c l e s . N e x t we h a v e to f i n d out i d e n t i t i e s s a t i s f i e d by t h e 3 - c o c h a i n s d e f i n e d by
(3.9).
These identities will define a subset
Z3(G,?) C
C3(G, ?)
of 3 - c o c y c l e s and we s h a l l h a v e to k n o w w h a t
? means,
namely what the system
of c o e f f i c i e n t s is. We r e a c h h e r e the s e c o n d c r u c i a l p r o b l e m , t h e c o m p o n e n t k : G x G • G -~ H '
n a m e l y d e r i v i n g a n i d e n t i t y on
This identity should generalize the well known
c onditi on (3.17) (6k)(s,t,u,v) = ~(S)k(t,u,v)-k(st,u,v)+k(s,tu, v)-k(s,t, uv)-k(s,t,u)
of the c l a s s i c a l t h e o r y .
T h i s we do as f o l l o w s .
(3.18)
= ~(S)h(t,u).h(s,tu).h(st, u)-l.h(s,t)-I
k(s,t,u)
= 0
We h a v e
t o g e t h e r with
(3.19)
~(S)h(t,u)
= k(s,t,u).h(s,t).h(st,u).h(s, tu)-i .
We t h e n u s e (3.18) to w r i t e $(S)k(t, u, v) -- $(s)$(t)hlu, v). $(S)hlt, u v ) . $(S)hltu, v) -1. $(S)hlt, u ) - i This is then transformed using (3.3) and four different f o r m s of (3.19) to yield
- 48 -
~(S)k(t,u,v).k(s,t,u). 9H(s't'u)k(s,tu,v ) =
(3.2o.1)
[~ (s, t) ),(s, t). n (s, t)k(st, u, v). N( s, t, u, v).
pH( st, u, v) ]k(s, t, u v ) .
In t h e r e one has = h(s, t) 9h(st, u) 9h(s, tu)
-1
(3.21)
H(s, t,u)
(3.22)
N(s, t, u, v) = X(s, t). n (s, t)H(st, u, v). n (s, t)H(st, u, v) , I
, .
M o r e o v e r ~ ( s , t ) r e p r e s e n t s ph(s,t) and can be c o m p u t e d in t e r m s of the cornponents X, ~ by the f o r m u l a (3. ~0.II)
6(s) ~(t)
= x(s, t ) . ~ (s, t). ~(st) .
L a s t but not l e a s t the function N can be c o m p u t e d out of the function K, mo that (3.20. I) is actually an identity on the 5-uple (k, k, ~;K;~).
The a s s i d u o u s l e c t o r
will indeed c h e c k that N(s,t,u,v)
-1 -1 -1 = kX. nlkK1). ~nI(XK2 )n~lTI2n 3 ( k K ; 1 ) . n n l n 2 n 3 n (xK.I)
with
R e m a r k 3.1.
~K - ~(s, t) K(s' t) ,
XKI -- ~(s, t) K(st' u)
AK2 = X(s,t)K(stu, v)
xK 3 = X(s,t)K(st, uv)
It is useful to o b s e r v e that f o r m u l a (3.20.1) obviously has the
shape of (3.17) and it also has the shape of the d i a g r a m
49-
S @ (T@(U@V))
/\
S ~ k T , U, V
S~
((T~ U)(~ u /~R:S, T(~U, V
ks, T, U ( ~ V ~
(S(~)(T(~U)) (~ V
(S@T) (~ ( U @ V )
f kSxT'U'V~
S, T, U x v
((S~)T)(~)U) ~ ) V
the cornmutativit7 of which is a relation among the canonical isomorphisms kS, T,U:(~S~T) ~ U -~ S ~
(T~U)
expressing the defect of associativity of the tensor product of modules.
The only
snag is that the first m e m b e r of (3.20.I) should be written as ~(S)k(t, u, v). pk( s, t, u). pH( s, t, u)k( s, tu, v). k(s, t, u) .
But precisely the function k is an obstruction to the associativity of a product defined on the set H X G
by
(m. s). (n, t) = (m.~(S)n.h(s,t),st) where
m , n e H,
s,t e G .
R e m a r k 3.2.
Let h be an e l e m e n t of H and a an e l e m e n t of A .
We then
define the function 3h :A -~ H', putting (8h)(a)
=
8 h a
=
ah.h-I
.
The s a m e f o r m a l i s m applies when h is a function f r o m s o m e set E to H; then %h i s a f u n c t i o n f r o m
II x E t o H '
(e.g.
the function
K of ( 3 . 6 )
is just
%h f o r
-
h:G
XG
-* H).
50
We h a v e a l s o the f u n c t i o n
-
6h:A
-~ H '
(6h)(a) = 6 h = h. ah -I a
W e call the operators
6 and a the coboundary and antiboundar)r operators.
a crossed h o m o m o r p h i s m w h i l e Considering that the group
6h is
Oh = (6h) -I is an inverse crossed h o m o m o r p h i s m .
I] operates onto H by automorphisms
and onto CI(A,H ')
by the composition product ':', it is clear that O and 6 are equivariant, namely 8(~)h) = q~':' (8h), We c a n a l s o c o n s i d e r then observe
that,
6(~h) = ~ * (6h) .
8h a s a f u n c t i o n in b e t w e e n t h e l a r g e r
groups
I1 a n d H a n d
in t h i s f r a m e , = ( a ~ h . h - 1 ) . ( ~ ) h . h -1) -i = (a ~h). (0~h) "I .
8 a ( $ h ) = a~h.4)h-1
M o r e generally, if K : A
-~ H' is an inverse crossed h o m o m o r p h i s m ,
formula (justifying the writing
a
one has the
K instead of K(a))
(3.Z3)
a(~*K)
= a~K. ~K -I .
This can be seen as follows:
K verifies
a~K = a(K). aK so t h a t (3.24)
(aK)-I = a(_IK)
or
K(a) -I = a(K(a-l)) .
a
Then one proves (3.23) by w r i t i n g
a(fl*K) : ~( _lafl K) : fl[~-l(a~3K). _ i K] : aflK.(~K)
-I
-
A c c o r d i n g to p r e v i o u s n o t a t i o n s , A to
H'
A onto H '
51
-
the s e t of c r o s s e d h o m o m o r p h i s m s
is d e n o t e d Z I ( A , H ') = Z I ( A , H ' ) .
from
( R e m i n d t h a t @ is t h e a c t i o n of
in t h e c r o s s e d g r o u p B of ( 3 . 1 0 ) . )
We s h a l l d e n o t e b y
SI(A,H ,) = SI(A,H ,) t h e s e t of i n v e r s e c r o s s e d h o m o m o r p h i s m s .
Remark homomorphism,
3.3.
W e also want to show that if K : A - * H'
is an inverse crossed
so is also ~* K for any ~ ~ 11. Indeed
(~*K)(a~)
=
*[K(~-la~)]
=
~[~-la~K(~-l~).K(r
= a~K (~-I~). ~K(~'la~)
=
a[(~. K)(~)]. (~* K)(a) .
We n o w d e r i v e t h r e e m o r e i d e n t i t i e s on t h e 5-uple (k,k, ~;K;,]) = D (~'h)" One is j u s t t h e f a c t t h a t K is an i n v e r s e c r o s s e d h o m o m o r p h i s m , (3.ZO.III)
a~K
= a( K ) . a K
or
a~K(s,t)
To o b t a i n t h e n e x t i d e n t i t y we t w i s t f o r m u l a
a~(s)h(t,u)~ah(s,tu)
= a[~K(s,t)].aK(S,t)
namely .
( 3 . 4 ) b y a to g e t
= ak(s,t,u).ah(s,t).ah(st,u)
@
Then inserting in the spot 9 the trival expression ~( S)h(t, u) -i. k( s, t,u). h( s, t).h( st, u)" h( s, tu) -1 yields
~(s)a L(t, u)" k( s, t, u). h( s, t). h( st, u)- h(s, tu) -I. ah( s, tu) = a k ( s , t , u ) . ah(s, t). a h ( s t , u )
-
52
-
and finally, putting @(s,t,u) = 9H(s,t,u): (3.20.IV}
~(S)K(t'u)'k(s't'u)'e(s't'U)[aK(S'tU)]a
ak(s't'U)'aK(S't)'Vl(s't)aK(st'u)"
:
The analogy between this formula and the fundamental identity (2.2. a) on 2-cocycles is c l e a r . We f i n a l l y h a v e t h e f o l l o w i n g i d e n t i t y , w h i c h p l a y s a -role s i m i l a r to t h e s e c o n d identity (2.2.b) (B.20.V)
for 2-cocycles:
~(s)k(t,u).[~(S)rl(t'u)]k(s,tu).pk(s,t,u)
We d e r i v e it f r o m
= k(s,t).n(s't)k(st,
( 3 . 3 ) , c o m p u t i n g in t w o d i f f e r e n t w a y s t h e p r o d u c t r
u). ~(t). r
When a and ~ a r e e l e m e n t s of 1], w e h a v e w r i t t e n
a~ = a~a-1
4.
D e f i n i t i o n off t h r e e - d i m e n s i o n a l
cohomolo~y
We h a v e n o w o b t a i n e d g o o d c a n d i d a t e s f o r 3 - c o c h a i n s , of f u n c t i o n s
(k, k, ~;K;~) a s a b o v e .
(B.20. I - I I - I I I - I V - V )
are candidate
s y s t e m is n o t y e t c l e a r .
Among them, those satisfying the conditions 3-cocycles.
However so far the coefficient
A candidate would be the s y s t e m = (H',9',11,
in w h i c h A f = ( H ' , p '
n a m e l y the 5-uples
,I], @~) is a c r o s s e d
g r o u p and in w h i c h A is a n o r m a l
s u b g r o u p of 1~ s u c h t h a t
p'H'C
AC
~;A)
n
-53
-
H o w e v e r w e s h a l l f a c e m o r e d i f f i c u l t y w h e n t r y i n g to d e f i n e a c o n n e c t i n g m a p and a longer exact sequence,
T h i s w i l l o b l i g e u s to i n c o r p o r a t e
more
structure
in
the coefficient system.
Going back to a Z-cocycle (4",h") e Z 2 (G,A"),
we have lifted it into
(4 h) e C z (G, A) and formed ,
m (4, h) : (k, • 4;K;n) ~ Z3(G, A') However
(4,h) is defined up to a pair (a,a) , CZ(G,B)
(where B = (H',p',A,@)
is the crossed group appearing in (3.10)). Replacing (4,h) by (4' ,h') = (a.4, a.h) yields a system (k',k',4';K' ;T]') related to the preceding formulas
(3.12 ) to (3.16)
which precisely represent the action (3.10). They also represent what we shall call the first variation of the 3-cocycle (k, k, 4;K;T]). The second variation shall be obtained by moving the Z-cocycle (4", h") along its thick cohomology class, namely applying to it the action (2.3). To that effect w e consider b" :G -~ H" and produce b"* (4",h") = (~",h--"). Then b can be lifted into s o m e function b :G -~ H and the lifting (4,h) of (4",h") is then transformed into b* (~,h) D (~,~)
= (~,h'-) w h i c h is a l i f t i n g of ( ~ " , h " ) .
= (k, k , ~ ; K;n) in t e r m s
The problem
is t h e n to c o m p u t e
of [] (6, h) = (k, k, 4 ; K ; n ) a n d s o m e o p e r a t i o n to be
described inside of the coefficient system.
H o w e v e r it will turn out that such an
operation cannot be described in terms of our candidate A' : this will lead to the third crucial step in the development of 3-cohomology. Easy computations yield the following k(s, t, u) = b(s). 4(S)b(t) 94(s)4(t)b(u). k(s, t, u). ~ (s, t). 4(st)b(u)-l. 4(S)b(t) ~i. b(s)-i
- 54 or, putting ~(s) = pb(t) , (4.1)
~(s,t,u) = ~(s)'~(s)~(t)'~(s)~(t)~(U)k(s,t,u).~(s)'~(s)~(t)[8~(s)~(t,)b(u)]-l; =J.
),(s,t)I
(4. z)
~(s,t) = ~(s)"~(s) (t)k(s,t) ;
(4.3)
"~(s)
= ~(s).J~(s)
;
aK(S,--t) = [aab(S)]. ~(s)[aa~(S)b(t)]. ~(s). ~(s)~(t)[aK (s, t)].
(4.4)
9 ~(s). $(s) ~(t).~(s, t). ~(st)-I
(4. s)
[Oab(st)]-1
~s, t) = ~(s). ~(s)~(t).n(s, t). ~(st) -I
In t h e s e f o r m u l a s
we have set
(4.6)
8a~h = aa(~h)
= a[a(~h)] = a[~*(ah)].
There is finally a third variation if we allow the Z-cocycle (~", h") to m o v e into its thin cohomology class, specifically if we m o v e it under the action (Z.6) of If"
This however produces the very sweet formulas (4. 7) to (4.11) . Suppose
(Jp",h")
is brought by a"
If I'
into
',hfT):
~,,(s) = a"J~"(s),
~"'(s,t)
= ~"h(s,t) .
Lifting a" ~ I f " b y ~ e 11 brings the lifting (~, h) into a lifting (~, h~) of (~", ~") such that N
=
s) = ~.
jp(
s).
Then D (~,~) = (~,~,~;K;~) is given b y
-i
his,t)
= ~h(s,t) .
-
55
-
(4.7)
k(s, t,u) =
(4.8)
k(s, t)
(4.9)
~(s)
~k(s,t,u)
= ~•
,
,
= "~(s) , e~
(4. I0)
K(s, t) = a ( ~ * K)(s,t) o r K = ~ * K , -i
(4. n)
=
TI
m1~
=
Let us n o w g o b a c k to the s e c o n d variation.
We would have liked to eliminate candidate possible
coefficient) because
out of formulas
the group
(4.1) t o ( 4 . 5 ) .
we have to know the expression
of pb = ~ i s of c o u r s e
no trouble
H (which does not appear Unfortunately
8b w i t h b : G -~ H .
since it lies into
in our
this is not The presence
A,
T o go a r o u n d this difficulty w e a s k o u r s e l v e s u n d e r w h a t condition t w o elements
b a n d b'
of H
p r o d u c e at the s a m e
pb = pb'
Suppose
b' = b.x,
x e H.
px = I or
and
time
8b = 8b'
Then one should have xe
ZH
= Ker
p, the center of A
and a
for e v e r 7 a e A,
x = x.
T h e s e c o n d condition m e a n s
that x belongs to the II-invariant s u b g r o u p of
A - i n v a r i a n t e l e m e n t s of H.
T h u s the intersection of this g r o u p a n d
ZH
is a
-
normal
and even
quotient
group
rl-invariant
E = H/P
subgroup
and the quotient
P
56
-
of H. map
We thus
want to consider
• : H -~ E f i t t i n g
the
into the diagram
H
(4.1Z)
]3
which
defines
There
exists
well
as
uniquely also
8 and
a canonical
A in the sense
(
CI(A,H')
'% a n d i n w h i c h action
K'
o f rf o n t o
E
=
K i ,
8'
such that
= ai,
i:H'
-~ H .
K: i s e q u i v a r i a n t
as
that
Suppose n o w w e have a m a p
,%(r
= r
b:G--
H producing m
can then transliterate the delicate formulas
,
m ~ E , ~, =
11.
~.b:G--
E.
We
(4.1) and (4.4) as follows (we put
~(s) = era(s) and use a convention similar to (4.6)): (4.1)'
k(s, t, u) = ~(s)"~(s) ~(t)" ~(s)~(t) ~(U)k(s, t,u). ~(s)-~(S)~lt)[A~ls)~lt) mlu)]-i ;
Ms, t)-l
(4.4')
aE(s't) = [,%am(S)]'~(s)[,%~(S)m(s)]'~(s)'~(s)~(t)[aK(s't)]'a
~(s).~(s)~(t).n(s,t).~(st)-l[,%am( st)]-i Definition 4.1.
B y a super-crossed group w e m e a n A
as follows:
= (H,p,n,@;A;E,@,
<,8,~)
a system
-57-
(1 ~ )
A is a n o r m a l
subgroup
of I] s u c h t h a t
pHC A C
n;
(2 ~ )
A = (H, 9, I I , ~ )
(3 ~ )
E is a g r o u p e n d o w e d w i t h a n a c t i o n
(4 ~ )
~:H-~E,
is a c r o s s e d
e:E-*ll,
fitting into a commutative
group and thus B = (H,p,A,~)
as well;
of 11;
A:E-~CI(A,H)
are equivariant
homomorphisms
diagram H
e (5 ~
C = (E,8,1I,~)
is a crossed
group and
)4 :H -~ E i n d u c e s a m o r p h i s m
of A t o C i n ~ 2 . (6 ~ )
For
If A' morphism
m c E,
= (H'
will consist
A -~A'
subject to conditions
group
p a r t of d i a g r a m
~'
A
(4,12).
Then
= a.em.a
-1
.8m
(2~
a crossed
a
of morphisms ~A'
E ~ E'
We t h u s a r e a b l e t o s p e a k of t h e
groups.
Consider
condition
[a,eln]
is another s u p e r - c r o s s e d g r o u p ,
which we leave to the lector.
4 . Z.1.
A satisfying
of a s y s t e m ~ ~
--7~3 o f s u p e r - c r o s s e d
Example
is the commutator
a
, 9 ' , I J ' ;/~' ;E' , ~ ' , K ' , e ' A ' )
H ~H'
category
p(A m )
a ~ _~ ,
group
A : (H,
p,]'I,6)
This allows to construct
E, 9
together
with a
and the upper
-1
-58-
A is a n e x a m p l e A
= (H, p, n , ~ ; A ; E , ~ , < , e , a )
of a s u p e r c r o s s e d
group.
We could take
A = pH a n d w o u l d t h e n c a l l
t h e supercrossed group associated t o t h e crossed group A. Example
a s in ( 2 . 4 ) .
4.2.Z.
group
exact sequence
(E)
of c r o s s e d - m o d u l e s
Then with the above notations : (H' ,p',I],@;A;E,~,K
A' is another
a short
Consider
example.
We call this supercrossed
' ,0, A) group the first
supercrossed
associated t o (E). Definition 4.3.
B y a 3-cochain of G with values into At w e m e a n a system
(k, k, $;K;~) of functions k:G
•
•
k:GXG-~ ~:G-~
-~H ,
A
IT,
K:G• ~:G•
,
or
K:A•
If.
This 3-cochain is called a 3-cocycle if and only if it satisfies conditions (3.20.I to V).
W e denote by Z 3 ( G , A )
the set of these 3-cocycles.
(k,k, ~;K;~) is said to be neutral if its components
A
3-cocycle
k and k are the trivial m a p s
with a single value, the units in the groups H and A
respectively.
W e want to consider three actions onto Z3(G,A).
The first one ~ corres-
ponds to the first variation and is the one described in (3.10) and formulas (3.12)
59-
to (3.16), the acting group being C2(G,B).
T h e next one corresponds to the second
variation and has CI(G, E) as operating group; it shall be denoted
v:CI(G,E) • Z3(G,A)-* Z3(G,A) . It is described by formulas analogous to (4.1 '), (4.2;, (4.3),(4.4'), (4.5) . Finally w e allow for a third action by the group I], corresponding to the third variation, described by formulas (4.7) to (4.11). T h e orbits co of Z3(G, A) corresponding to the first action f o r m a set w h i c h could be called a super-thick
3-cohomology
I~3(G, ~k) but is not very interesting.
It turns out that the second action operates also on the set of orbits co. In other w o r d s an element m
~ CI(G, E) transforms an orbit co in an other one co' = m . co.
T h e union of these orbits m . co is then a bigger orbit ~ for a bigger action. set of orbits ~ then f o r m s what w e shall call the thick 3 - c o h o m o l o ~ y
The
set I~3(G, A )
e n d o w e d with a canonical projection lr: lm'3(G, A) -~ ] ~ 3 ( G ' A) .
Finany, words union
it t u r n s
out that the third
an element of these
orbits
The set of these 3-.cohomolo~)r
~ e 1! t r a n s f o r m s n. ~ f o r m s
big orbits
set
H3(G, A)
~
action operates on the set of orbits an orbit
~ into another
then a still bigger
constitutes
~
~'
with a canonical
-~ H 3 ( G , A )
.
= ~.
for a bigger
then a set which we shall
and is endowed
y:H3(G,A)
orbit
one
~.
In other The action.
call the thin
projection
-
60
-
In o r d e r to m a k e c l e a r the s t a t e m e n t t h a t an e l e m e n t m e CI(G, E) t r a n s f o r m s an o r b i t co i n t o a n o t h e r one of the s a m e t y p e , l e t u s m e n t i o n t h e f o l l o w i n g . co is t h e o r b i t of k = ( k , k , ~;K;T1).
T h e n m o v i n g k by (a, a) e C2(G, B) into
k'
by
= (a,a)*k
and t h e n m o v i n g k '
me
CI(G,E) into k '
= mvk'
Suppose
is e q u i v a -
l e n t to the f o l l o w i n g : m o v i n g k by m into k = m V k a n d t h e n m o v i n g k by s o m e ( a , a ) e C Z ( G , B ) i n t o Ca , a ) # k
where
a(s) = I3(s)a(s) I3(s)-I, a(s,t) : ~(S)rtAa('~(S)s) m~zJ''']"~(s). ~(s)I3(t)[a(s,t)] . T h e l a s t f o r m u l a c o r r e s p o n d s to an a c t i o n
V : CI(G, E) x CZ(G, B) -- CZ(G, B) such that the following distributivity law holds (4.12}
9 mV[r
:
r
r
.
S i m i l a r l y , so as to c l a r i f y the t r a n s f o r m a t i o n of the o r b i t ~ into a n o t h e r one ~,
= r ~ by t h e t h i r d a c t i o n , we o b s e r v e two t h i n g s .
One is t h a t m o v i n g k i n t o
(a, a)* k a n d t h e n t w i s t i n t o ~[(a, a)* k] is e q u i v a l e n t to f i r s t t w i s t i n g _k i n t o ~k and t h e n
(4.13)
m o v e t h r o u g h ( ~ a , ~a),
so t h a t we h a v e a n o t h e r d i s t r i b u t i v i t y l a w r e a d i n g
a) *k] :
a)*
A n o t h e r is t h a t m o v i n g k into m V k and t h e n t w i s t into ~ ( m V k) is e q u i v a l e n t to f i r s t t w i s t k i n t o ~___kand t h e n m o v e into (~rn) V (n___ k) w h e r e (~rn)(s) = wire(s)]. t h u s h a v e a t h i r d d i s t r i b u t i v i t y law:
(4.14)
~ ( m V k) = (~rn)V ~k)
We
-
61
-
T h e d e t a i l s i n v o l v e l ong c o m p u t a t i o n s w h i c h we l e a v e f o r a s u b s e q u e n t p u b l i c a t i o n . N o w , a n o r b i t ~, ~ o r ~ w i l l be c a l l e d n e u t r a l if i t c o n t a i n s a n e u t r a l 3-cocycle.
It s h o u l d be o b s e r v e d t h a t t h e f i r s t a c t i o n t r a n s o f r m s
in a n o n - n e u t r a l one in g e n e r a l .
a neutral cocycle
H o w e v e r n e u t r a l i t y is c o n s e r v a t i v e w i t h r e s p e c t
t o t h e s e c o n d and t h i r d a c t i o n s . If w e go b a c k t h e s h o r t e x a c t s e q u e n c e it a s u p e r - c r o s s e d
(E) of ( 2 . 6 ) ,
we have a s s o c i a t e d to
g r o u p @k' and a c o n n e c t i n g m a p
A 3 : Z 2 (G, A,, ) -- im 3 (G, A ' ) inducing finer connecting m a p s zZ(G,A)
(4.15)
l
A 3 and A3 fitting into the diagram )
Z Z ( G , A ,,) A 3 :, R - 3 ( G , A , )
l
HZ(G,A)
~ m~Z(G,A,,)
I.-I2(G,A)
~- H 2 ( G , A , , )
1
~ ~3(G,A,)
~ H 3 ( G , A ,) A3
M o r e o v e r t h e r o w s in t h i s d i a g r a m a r e " e x a c t "
if w e e n d o w
t h e s e t s in t h e l a s t
c o l u m n w i t h t h e s p e c i a l s u b s e t of n e u t r a l c l a s s e s . T h i s w a y , t h e p r o b l e m w h i c h m o t i v a t e d t he c o n s t r u c t i o n of a t h r e e d i m e n s i o n a l c o h o m o l o g y t h e o r y is s o l v e d .
But t h e l e c t o r w i l l e a s i l y d e f i n e s u p e r -
c r o s s e d g r o u p s A , @k" a s s o c i a t e d w i t h ( 2 . 6 ) s o t h a t , in s o m e s e n s e we g e t e x a c t p r o l o n g a t i o n s of t h e s e q u e n c e s
(Z. 7) and (Z.8)
by
- 62 -
(4.16)
....
(4. l?)
1~2(G, A) -~ HZ(o.A,,) -~ m~3(O.A, ) -~ H3(O.A) -~ H2(G, A)
HZ(G.A,,) ~ H3(G.A ,)-~ H3(G.A)~
~ H3(G.A,,). H3(G.A,,).
together with an exact sequence * -~ Z2(G,A') -~ Z2(G,A) -- ZZ(G,A") ~ H3(G, A') -* H3(G, A) -* B3(G, ~I 9 H e will also be able to define the general concept of a short exact sequence of supercrossed group A'
K
9 A
I
r A"
and construct the associated exact sequences generalizing the exact sequences (4.16) and (4.17). And then n e w problems (b) arise, the solution to which will require more
research. T o that effect it should be reminded that for a crossed group A = (H, p,ll , 4)
it is useful to consider the group ~ = ll/pH. Then it turns out that HZ(G, A) ha~ a natural m a p e :I~2(G,A)
-~ H o m ( G , ~ )
the fibers of which are acted upon freely by s o m e abelain groups HZ(G, Z H) of the classical theory where
Z H is the center of A and therefore an abelian group,
S o m e similar structure on I~3(G, •) will undoubtedly s h o w up. example,
For
if w e look at formula (3.20.II), it is clear that it will be useful to
consider the quotient group e a 2-cocycle.
= II/A in whichthe pair ($,11) induces obviously
Looking further at formulas
(4.3), (4.5) as well as (4.9), (4.11)
-63we o b v i o u s l y h a v e c a n o n i c a l m a p s
H3(G, A) ~ "~Z(G, O), H3(G, A) -- HZ(G, O) There,
of c o u r s e ,
O s h o u l d be r e p l a c e d by s o m e c r o s s e d g r o u p d e r i v e d f r o m a
f i n e r a n a l y s i s of t h e s i t u a t i o n .
W e have already seen that for s,tc G, morphism
K(s,t) is an inverse crossed h o m o -
and then formula (3.15) shows that there is a canonical m a p H3(G, ~) -~ A p p ( G • G, HI(A, B)) .
The latter certainly will induce m a p s going f r o m H3(G, A) and H 3 ( G , A )
to s o m e
quotient of App (G • G, HI(A, B)), the nature of this quotient being told by formulas (4.4) and (4.10) which should be interpreted.
A rough look at (4.4)
s e e m s to show that these quotients will look as an I~2(G, ?) and H2(G, ?) generalizing what w e already know.
M o r e precisely these sets should represent a sort
of t w i s t e d 2 - c o h o m o l o g y as a l r e a d y t o u c h e d b r i e f l y in [3]. We hope to h a v e s h o w n t h a t the s u b j e c t is w o r t h m o r e c a r e f u l c o n s i d e r a t i o n .
-
64
-
BIB L I O G R A P H Y
[i]
P. D e d e c k e r - 0, Comptes rendus, 247(1958), I160-62; I, ibid., 257(1963), 2384-87; II, ibid., 258(1964), II17-20; III, ibid., 258(1964)4891-93; IV, ibid., 259(1964), 2054-57; V, ibid., 260(1965), 4137-39.
[2]
P. D e d e c k e r - Cohomolog_ie non ab~lienne, mimeographi~, Fac. Sci. Lille, l~re ~d. 1963-64; 2 e m e ~d' augment~e, Sept. 1965.
[3]
P. Dedecker et A. F r e i - VI, Comptes rendus, 262(1966), 1298-1301; VII, ibid., 263(1966), 203-206.
[4]
J. Giraud- Comptes rendus, 260(1965), 2392-94 et 2666-68.
[5]
J. Giraud - Cohomologie non ab~lienne, Th@se, Facult@ des Sciences de Paris; mimeographed, Columbia University, 1966.
[6]
J. P. Serre - Cohomologie galoissienne, Lecture notes no. 5, Springer Verlag.
-65 -
H-SPACES
by R. R. Douglas,
P. J. Hilton,
1.
Let nected,
~
INTRODUCTION
be the h o m o t o p y c a t e g o r y of pointed,
finite CW-complexes.
is a pair
F. S i g r i s t
(X,m), w h e r e
m
Recall that an H - s p a c e
E ~(X
X • X
x X,X)
con-
(in
such that
-% x
X v x commutes.
( V
is the h o m o t o p y class of the "folding" map.)
In o t h e r words, m
is a m u l t i p l i c a t i o n on
sided h o m o t o p y unit.
If in addition,
X x X x X
l•
X, h a v i n g a two-
the f o l l o w i n g d i a g r a m
~- X x X
m
X x X
commutes,
then we say that
)
(X,m)
X
is h o m o t o p y - a s s o c i a t i v e .
In a t t e m p t i n g to c l a s s i f y the H - s p a c e s first asks w h i c h simple objects s u p p o r t an question,
in
(homotopy-associative) r e s t r i c t e d to spheres,
~
in
~,
(i.e. h a v i n g few cells)
H-space multiplication. is e q u i v a l e n t to the "Hopf
i n v a r i a n t one" problem.
Theorem:
(Adams
If
Sn
[i])
is an H-space,
one
then
n = i, 3, or 7.
This
86
Remark. known
S1
to s u p p o r t
Let
tively,
E
be
S3
S7
Theorem
E
q, n
is
a q-sphere
What
bundle
restrictions
is an H - s p a c e These
are (respec-
two q u e s t i o n s
two t h e o r e m s .
(ii)
(q,n)
=
(1,2),
These
or
SU(3),
2:
[5]
E
or
(3,4)
restrictions
o__rr (3,5) .
are the b e s t p o s s i b l e ,
bundle
in c a s e
(i), and
respectively,
in c a s e
(ii).
is a h o m o t o p [ - a s s o c i a t i v e
(i)
q, n
(ii)
(q,n)
Remark: best possible;
then either
6 {i, 3, 7},
m a y be the p r o d u c t
Sp(2),
of)
E
S7
multiplications.
H-space)?
is an H - s p a c e ,
(i)
If
or
in c a s e
and
[3,4]
Remark: E
(q,n)
by the f o l l o w i n g
If
as
space
positive).
homotopy-associative
l:
are L i e g r o u p s ,
(the t o t a l
(q, n
on the p a i r
are a n s w e r e d
Theorem
S3
no h o m o t o p y - a s s o c i a t i v e
o v e r the n - s p h e r e placed
and
E {i,3}, =
product
respectively,
are all Li e g r o u p s .
(3,5)
then either
o_~r (3,7).
the r e s t r i c t i o n s
bundles
H-space,
may be
or
(1,2),
Again,
E
in c a s e
in c a s e
on
(i) and
(ii), p r o v i d e
(q,n)
a r e the
SO(3), examples
SU(3) which
-
Until the c o m p a c t moreover,
recently
Lie g r o u p s ,
constructed
in
conjecture.
~
precisely,
bundle
over
It a r i s e s
examples
that t h e i r r e s p e c t i v e The n e w H - s p a c e The content
thus
2.
Whitehead
If
~ : Sn
E
If
as a d i r e c t
OF T H E O R E M S
products
on
search
E8
More
are
the p r o p e r t y
are d i f f e o m o r p h i c . factor
in a L i e group.
2
and
Sn
it f o l l o w s
are s e n t to W h i t e h e a d Thus, that
loss of g e n e r a l i t y ,
spectral
sequence,
of d i m e n s i o n
2n - i.
If
is
is an H - s p a c e .
Sq Sq
has t r i v i a l is an H - s p a c e .
we now assume
we compute
q = n - i, H*(E;Z2) , m a y be an e x t e r i o r
generator
and
having
1 AND
Sq
by a m o n o m o r p h i s m .
products,
the u s u a l
S3
> E, and t h e r e f o r e
Without Using
type,
with
to this
of m a n i f o l d s . E~
which
Sq > E > Sn is a f i b r a t i o n a n d E > q = n, it f o l l o w s t h a t t h e r e is a c r o s s -
Whitehead on
10-manifold
3 is j o i n t w o r k w i t h J. R o i t b e r g .
PROOFS
Suppose an H - s p a c e .
emerges
of S e c t i o n
of these;
of a s y s t e m a t i c
manifolds
products
were
A new H-space,
closed
in p r o d u c t s
homotopy
~
that these were the
type).
as a b y - p r o d u c t
of d i f f e r e n t
in
S 7, is a c o u n t e r e x a m p l e
two d i f f e r e n t i a b l e
constructed,
products
conjectured
(up to h o m o t o p y
for n o n - c a n c e l l a t i o n
Moreover,
S 7, RP 7, and p r o d u c t s
in this p a p e r as a s m o o t h
is a 3 - s p h e r e
section
the o n l y k n o w n H - s p a c e s
it h a d b e e n m i l d l y
only H-spaces
67-
q < n.
H*(E;Z2).
algebra
on one
H * ( E ; Z z) ~ H * ( s 2 n - I ; z 2 ) ,
-
then
it f o l l o w s
thus,
(q,n)
exterior
=
from
[i]
(1,2)
or
algebra
or
15,
then
P2E
three. with
8 and
~2(e)
lead
by
or
be
polynomial
~2~3(e)
=
_ ~2
[2],
algebra
(mod 2)
~2~3(8)
and
i;
E P-~
from
show
Lemma
(7,1),
If
S3
then
X
or
that
=
- B2 ~ 0 see
filtered, B
(of
at h e i g h t together
~3~2(S),
of p r o o f Sn
e,
(mod 2), [4]).
is a f i b r a t i o n
Theorem (q,n)
E;
2 will
and
E
follow
c a n n o t be
(3,4),
(7,7).
> X has
> S4
is a f i b r a t i o n
the h o m o t o p y
type
of
and
X
is an
S 7.
2
If or
can
n = ii
1
H-space,
Lemma
(7,3),
(q,n)
of the H - s p a c e
truncated
~2(B)
(for d e t a i l s
Sq
with
on t w o g e n e r a t o r s
and
~3~2(e)
if w e
l) ;
an
that
to s h o w
(torsion-free)
n + I, r e s p e c t i v e l y ) ,
suppose
i,
n >
H*(E;Z2)
plane
of as a
H-space.
(1,7),
then
is a f i b r a t i o n ,
is a h o m o t o p y - a s s o c i a t i v e Theorem
is n o t
H*(E;Z2)
it s u f f i c e s
> Sn
thought
~ 0
(recall
7
(7,15).
to a c o n t r a d i c t i o n
Now
If
be t h e p r o j e c t i v e
may
KU(P2E)
filtrations
(3,4).
S 7 ---> E
let
truncated
and,
(7,11)
If
2n - 1 = 3 or
on o n e g e n e r a t o r ,
H*(S q • sn;z2), c a n n o t be
that
68-
(7,1),
S q ---~ X
then
> Sn
the u n i v e r s a l
is a f i b r a t i o n , coverin@
X
of
with X
has
(~,n) the
=
(1,7),
-
homotopy
type of
S 7 .
Remark: associative
Lemma
The u n i v e r s a l
H-space
H*(X;Z)
X
of a h o m o t o p y -
a homotopy-associative
is a h o m o t o p y - a s s o c i a t i v e
~ H*(S 3 • sT;z)r
: H3(X;Z3 ) ~
Now exact
is a g a i n
covering
H-space.
3
If
pl
69
H7(X;Z3)
let
sequence
then
(q,n)
=
the S t e e n r o d
(7,3)
: HT(E;Z3)
and
implies ~
which
contradicts
Finally,
(q,n)
~
that
S7
(i.e.,
P~ = P~
9 P~
The S e r r e
that
the c o n c l u s i o n
for e s s e n t i a l l y
space
3 Steenrod
S3-bundles
by e l e m e n t s and
~n-I (S3) ~
of L e m m a
the same H-space
algebra).
S3-BUNDLES
principal
are c l a s s i f i e d
isomorphism
(7,7)
in the m o d
We consider
the c l a s s i f y i n g
X = E.
can not be a h o m o t o p y - a s s o c i a t i v e
3.
bundles
operation
H7(S7;Z3)
is an i s o m o r p h i s m ,
reason
and
is n o n - t r i v i a l .
for c o h o m o l o g y i*
H-space
let
~n(B) .
e <
over
6 ~n-i (S3). s0
We a d o p t
under
Sn . Let
Such B
be
the c a n o n i c a l
the n o t a t i o n
3.
-
70
-
S 3 ~ Eu
Sn ~ deliberately
Proposition
confusing
1
(S 3 U e e n)
follows
easily
Now
Proposition
$ = •
from Proposition
consider
i.
the fibre-product
Eu8
diagram
---> E 8
Eu
P~ --=>
and
Ee8
nSO S --~ B
2
Eu8 Theorem
U e n+3
3
E~ ~ E 8 < ~ > This
classes.
a CW-structure Eu =
Theorem
and homotopy
(James,Whitehead)
has
E
maps
B,
= Esu;
= E~
x S3
if
8 0 9 P~ = 0.
4
Let
8 = Zu.
Then
8 0 9 P~ = 0
if
ptp%~,~_1,.
--
where To p r o v e
~ 6 ~6(S 3) this we
measures consider
the non-commutativity
the Puppe
sequence
.
~3~
=
2
of
$3.
of t h e i n c l u s i o n
of
0,
-
the
fibre
in
E
readily
i> E
identify
q:
Y E ~ 7 ( S 4)
S n + S n+3
where
is
> Sn
s0
=
(Z~,
le
Whitehead s
if
/(Z-l) T
9 u,
=
Corollar[ I prime
to
k~
or to
9 u =
(leo,s
, and
Let
I ~ 1 mod
ko,
k = p,
where
s
is p r i m e
to
p
2.
E~ ~ E T ~
= Sp(2),
Then
9 74 ~>
is a d j o i n t
be
s O
of
Thusr
to
9 Pe
= 0,
,
ET~
s ~ •
is
. ~ 2 ,3 ,
and
mod
then
an
p,
if
E 8.
8 = 7~.
x S 3 = ET~
3 and
k 0 = gcd(k,24),
Then
a prime
~ = ~,
E~
k,
order
8 = s
E~ ~
Let
Theorems so
(s0,0),
[e,e]
hence
and
• S 3 = E 8 • S 3, a l t h o u g h
E~
= qj ,
" ~!4~)
• S3 = E8 • S3
if
Now
9 Y
e
example,
from
Pe
1 E ~ 3 ( S 3) 9
[e,e]
for
follows
Now
Z(s 2
Thus,
This
Z4~),
O.
i.
Corollary
as
'0>
(s
9 u
Ee
Ee
S4
components,
s0 9 q =
(s
product
73e~
9
9
its
map.
" q =
" q = s ~
of
u)
and
= the
y
Hopf
is a d j o i n t
e E ~4(B) s
Now
sn + sn~3
in terms
the
9 q = e
and
)
u, u
where
-
, S3
We
71
4 and H-space 9
Then
x S3 the
fact On
that
~r9 ( S 3 )
dimensionality
= Z39
-
grounds
it follows
it is the total
that
72
-
it cannot be any of the known H-manifolds;
space of a p r i n c i p a l
S3-bundle
over S 7.
73-
REFERENCES [1]
Adams,
J. F., "On the non-existence
of elements of Hopf
invariant one", Ann. of Math. 72; 20-104. [2]
Adams, J. F., "H-spaces with few cells", 67-729
[3]
(1960).
Topology ~;
(1962).
Dougla% R. R. and F. Sigrist, which are H-spaces"
"Sphere bundles over spheres
Rendic. Acc. Naz
Lincei 44; No.4
(1968). [4]
Douglas,
R. R. and F. Sigrist,
spheres and H-spaces",
E5]
Douglas,
"Sphere bundles over
(To appear) 9
R. R. and F. Sigrist,
"Homotopy-associative
H-spaces which are sphere bundles over spheres", appear).
(To
-
74-
CONSTRUCTION DE STRUCTURES LIBRES Charles Ehresmann
INTRODUCTION. Le but de cet article est de donner un crit~re d'existence de p-structures libres et des applications de ce critSre, qui fait intervenir un foncteur auxiliaire dont
p
est une restriction;
par exemple si
foncteur d'oubli relatif ~ l'univers M
p
P
est le
de la cat~gorie des O
homomorphismes
entre structures d'un certain type, P
ra @tre le foncteur analogue relatif ~ un univers ~
pourauquel
O
appartient M . Contrairement au th~or~me d'existence d'adO
joint de Freyd, ce crit~re impose des propri~t~s
(telles
que l'existence d'un "assez grand nombre" de produits) sur P, non sur le foncteur donn~ 8tre g~n~ralis~es
p. Les hypothSses peuvent en
(volt [i]), mais ici nous avons cherch~
indiquer des conditions simples, r~alis~es dans la plupart des exemples. Comme application,
nous obtenons des th~or~mes d'exis-
tence de limites inductives ou de structures quasi-quotients (voir aussi
[i]), et un th~or~me de compl~tion "maximale"
d'une cat~gorie en une cat~gorie ~ limites projectives et inductives d'une certaine espSce, avec conservation de limites donn~es. Nous avons montr~ ailleurs r~sultat a des consequences
[2] que ce dernier
int~ressantes pour l'~tude d'une
75
-
notion g~n~rale de "structure alg~brique" d~finie comme r~alisation,
dens une cat~gorie quelconque, d'une esquisse
(i.e. d'un graphe multiplicatif muni d'une famille de transformations naturelles).
O. QUELQUESP PPELS. Nous nous plaqons dens le cadre de la Th~orie des ensembles avec existence d'au moins deux univers M M
tels que
E M . N a i s en f a i r
0
seraient
0
aussi
valables
la
plupart
et
O
O
des raisonnements
dans une T h f i o r i e a v e c e n s e m b l e s
classes, en prenant pour M
la class, de t o u s l e s
et
ensembles
O
et pour ~l~ments de ~
des classes. Dens le dernier w
l'axi-
O
ome du choix est librement utilis~ dens M
O
La terminologie et les notations sont ceux de [~ , dens l'index
duquel
se trouvent
les
mots non e x p l i c i t e m e n t
d~fi-
nis ici. Les autres notions que nous allons rappeler figurent dens le cours
Univers. 1 ~ Si
[4].
Ensemble M
E
d'ensembles tel que: o appartient ~ M l'ensemble P(E) de ses parties O'
estun
~l~ment e t u n e
pattie de M O"
2 ~ Si (Ei)ie I e s t par un ~l~ment
I
une famille d'~l~ments de Mo index~e
de M , sa r~union appartient ~ M . O
O
3 ~ Ii exist, un ensemble infini appartenant ~ M . O
(Contrairement
~ la d~finition de Grothendieck,
geons pas que M
nous n'exi-
soit un ensemble transitif). O
Application9
Une application
d~sign~e par le triplet x § f(x)
f
de
(M',~,M), o~ de
M
M
dens
M'
est
est la surjection sur
f(M) C M'.
76
Si
f
-
est l'injection canonique de
~crit
M C M'
dans
M', on
f = (M' l,M) M d~signe toujours la cat~gorie pleine des applications
associ~e ~ un univers M . O
Cat~gor~e. La cat~gorie sition
(y,x) + y.x
de
M C C•
parfois, C ), et l'on pose de
C"
est
(C,<), o~ K est la loi de compodans
C, est notre
C" (ou,
M = C'*C'. La classe des unit~s
C" , les applications source et but e et 8 , le O
graphe
(C,8,e) sous-jacent
[C'], le groupo~de des ~l~ments
inversibles
C'. Si A et B sont des parties de C, l'ensemY ble des compos~s y.x tels que y e B et x e A est d~sign~ par
B.A~ on pose: {b}.A = b.A
et
B.{a} = B.a .
Foncteu2~. La surjection d~finissant le foncteur C"
vers
H"
est notre
F, de sorte que
F
de
F = (H',F,C'), et
l'on ~crit: ~(F) = C" , 8(F) = H" .
Transformation natumelle. des transformations H"
est notre p
est le foncteur h' e H.e, soit
T de
F) F =
C"
vers
un de ses ~l~ments H"
vers
(p.F',~t,p.F), est notre constant sur l'unit~
h'T
( B~,t'
Soit
est un foncteur
formation naturelle
De m@me, si
naturelles entre foncteurs de
~(H',C') ~.
(F',t,F). Si
La cat~gorie (longitudinale)
e
,
K', la transpT. Si
de
F'
H" et si
la transformation naturelle o~ et si
t'(i) = h'.t(i)
pour tout
h e e.H, on d~note
Th
i e C~.
la trans-
formation naturelle
(F' ,t",~), Produits. Si admettant
e
o~
t"(i) = t(i).h
pour tout
i e C~.
(e')ielI est une famille d'unit~s de
pour produit dans
H"
H', la projection canonique
- 77
de
e
vers
e.
~tant
1
duit naturalis~
darts
f
tel
((Pi)iei,e)
un pro-
H'; dans ce cas, pour route famille ^ de meme source et de but
fl9
(fi)iei de morphismes
nique
Pi' on appelle
e.,
Itu
-
l
que pour tout
f'Pi = f"1
i E I
est not~ [f~ieI" Soit
L
un ensemble d'ensembles; H"
si toute famille d'unit~s duit dans
H'. Un foncteur
duits si
H"
et
K"
(ei)iei, o~ p
de
H"
est ~ L-produits
~ ~ L, admet un provers
K"
sont ~ L-produits et si
((p(pi))iei , p(e)) est un produit naturalis~ dans K" lorsque est un dans
Une cat6gorie
(h',h), o~
h e H
H'. Un foncteur K"
dans
et p
H"
lorsque
de
j
est ~ noyaux si tout couple
h' e 8(h).H.e(h), admet un noyau dans H"
vers
sont ~ noyaux et si K"
((Pi)ieI,e) en
H'.
Noy~.
et
est ~ L-pro-
K"
p(j)
est ~ noyaux si
est un noyau de
est un noyau de
(h,h')
telle que et que
e
t(i)
soit une limite inductive de soit, pour tout
(p(h),p(h'))
dans
Limites [I]. Une transformation naturelle
H"
H'.
T = (8,t,F)
F (notre Lim F)
i e e(F) , l'injection canonio
que
de
F(i)
lisle (de relle
vers
e est appel~e limite inductive natura-
F ). Dans ce cas, pour toute transformation natu-
T' = (8',t',F), on note
l'unique
k
tel que
limTT ' (ou simplement
lim T')
kT = T'.
D~finitions duales: limite projective naturalis~e
T
et
limTT ' . § Soit ~ un ensemble de categories. Une application associant ~ certains (resp. ~ t o u s l e s ) vers
H"
foncteurs
F
de
C" e
une limite inductive naturalis~e de
F
est dite
78-
application
~-limite inductive naturalis~e partielle
naturalis~e)
sum
(rasp.
H'. MSme d~finition pour une application
1-1imite projective naturalis~e
p-structure libra. Soit
p
(partielle)
sur
un foncteur de
H'. H"
vers
K'. On appelle p-projecteur un couple
(s,g) e H'• tel qua o p(s) = 8(g) et qua, si (s',g') e H'• et si g' apparo tient ~ p(s').K.e(g), il existe un et un saul h e s'.H.s v~rifiant
p(h).g = g'; on dit alors qua
s
est une p-struc-
ture libra associ~e ~ a(g). Si, ~ touteunit~ ture libra
p
dite cat~gorie ~
p'
H"
j
ment
de
h
de
de
H
p
est
un
se prolonge en
p
admet un adjoint,
K"
K"
de
est
8(j).H
p(j)
un foncteur de
H"
vers
K'. Un
une p-injection si, pour tout
tel qua
p(h) = p(j).k, o~
h' e H
h = j.h'
j
e § p'(e)
H-projections.
existe un et un saul
Si de plus
est associ~e une p-struc-
p.
et si
p-injection. Soit P
K"
est le foncteur injection canonique vers
sa sous-cat~gorie
element
de
p'(e), l'application
un foncteur adjoint Si
e
~i~-
k eK,
il
v~rifiant : p(h' ) = k.
et
est un monomorphisme
de
K', on appelle
p-monomorphisme. Par dualitY, on obtient les p-surjections
etles
p-~pi-
morphismes.
Structures quasi-quotients. Soit vers la cat~gorie M d'applications. (ou p-structure d'~quivalence ne
H'"
de
sur l'ensemble
sur
de
Soit
p(s)), et
un foncteur de s
s
par
~ e H' r
(voir
S
une unit~ de r
.
H"
une relation
p(s). On se donne une sous-cat~gorie
H'. On dit qua
quasi-quotient
p
plei-
est une (H',p)-structure [5]) s'il existe un ~i~-
79
ment
j
de
~.H.s
(appel~
-
(H',p)-quasi-surjection)
v~ri-
fiant les conditions: 1 ~ p(j) 2 ~ Si
est compatible avec h e H'.H.s
et si
il existe un et un seul
r.
p(h)
h' e H'
est compatible avec
tel que
r,
h = h'.j.
Cette d~finition signifie que ~ de
est une H'-projection r dans une certaine cat~gorie p .(Voir [5]).
(s,r)
St, pour toute unit~ valence
r
sur
s
de
H"
p(s), il existe une
(i.e. une (H,p)-structure
et route relation d'~qui-
p-structure
quasi-quotient)
dit que
p
est un foncteur ~ structures
Si
j
est une p-quasi-surjection
p-structure quasi-quotient surjection canonique p(s)/r, on appelle Cas
~ ~
particulier:
de
de
s
de
K', et soit H"
et si
morphisme de
X
r, on
quasi-quotients.
r
et si
~
p(j)
comme est la
sur l'ensemble quotient s
par
r.
Categories quotients d'une cat~gorie. Soit
une pattie de
f e p(S).K S
par
une p-structure quotient de
Sous-mo~ohismes eT~end~8. vers
s
d~finissant
par
p(s)
de
quasi-quotient
p
H. Si
, on dit que
engendr~ par
f
un foncteur de
j
S
H"
est une unit~
est un (X,p)-sous-
s'il v~rifie les conditions
suivantes: 1 ~ j E S.X
2 ~ Si
et il existe un
j' e S.X
f = p(j').k',
il existe un et un seul
Graphe multiplicatif.
et
k' e K h e H
f = p(j).k. tel que
v~rifiant:
p(h).k = k'.
C'est un syst~me multiplicatif
pour lequel il existe un graphe y.x
tel que
et s'il existe un
j'.h = j
1 ~ Si
k e K
(C,8,~) ayant les propri~t~s:
est d~fini, on a
e(y) = 8(x) ,
C"
e(y.x) = e(x)
et
8(y.x) = 8(y).
80
2 ~ Pour
tout
x e C, les compos~s
sont d~finis et ~gaux ~ La classe
a(C)
x.a(x)
et
8(x).x
x.
des unit~s de
C"
est notre
C , 0
N~ofonot~ur. et
K"
C' est un triplet
F = (K',F,C')
sont des graphes multiplicatifs
une application I~
f(C')C O
2 ~ Si
f
et oh
, o~
C"
(K,F,C)
est
telle que:
K"
.
O
y.x
est d~fini dans
est d~fini dans K"
C', le
compos~
f(y).f(x)
et l'on a:
f(y.x) = f(y).f(x). Nous d~signons par foncteurs
N ~ (resp. par F) la cat~gorie des n~o-
(resp. des foncteurs) associ~e ~ l'univers M . O
I. THEOREMED'EXISTENCE PE STRUCTURES LIBRES. Soient unit~ de
p
un foncteur de
H"
HYPOTHESES. 1 ~ de
~"
vers
et
K"
de
2"
p
et
e
et
libre associ~e ~
une
pleines
I
g e p(s).K.e
est une unit~ de
H"
i E I, il existe un X
l'ensemble des couples ; le couple
de
et Pie
~ e P(S).~.e s..H.S
(X.H" O
P(X)
(s,g)
i = (s,g)
~"
S
, tel que, pour tout
v~rifiant
P(Pi ) g = gi de
}q"
soient des monomorphismes
, P)-sous-morphisme
, oh
sera not~
(S,~) , oh
un ensemble de monomorphismes
que les 61~ments de Ii existe un
H"
P
K" .
(si,gi). On suppose qu'il existe un couple
3~ Soit
e.
est la restriction d'un foncteur
~ des sous-cat~gories
2 ~ D~signons par O
K"
K'. Nous allons donner des conditions suffisantes
pour qu'il existe une p-structure
s e H"
vers
j
de
S
tel
de
engen-
~'.
81
dr~ par
~;
4 ~ Si
on note h
et
~ :
h'
-
e(j).
sont deux ~l~ments de
H
et de mSme but~ il existe un monomorphisme h.n = h'.n, X.n C X p(h))
dans
et que
p(n)
n de H" tel que
soit un noyau de (p(h'),
K'.
PROPOSITION I. Si les hypotheses I, 2, 3 vdrifi~es,
de source
8
et
4
est une p-structure libre associ~e ~
sont e.
A. D'apr~s la d~finition d'un (X.H~,P)-sous-morphisme engendr~, que
il existe un
e(k) =
e(~) =
k e ~
e
(~,k)
~" , on a
est un p-projecteur.
est un ~l~ment de
~ = P(j).k; puis-
et 8(k) = P(~) e K
une sous-cat~gorie pleine de que
tel que
I,
et que
K"
est
k e K. Montrons
En effet, si
i = (si,g i)
on trouve, avec les notations de
la condition 2,
et P(gl )'k = P(Pi )'P(j)'k = P(Pi )'~ : gi " D'autre part, supposons qu'il existe aussi
g E' s..H.~ ! i
i
avec
p(g~).k = gi" En utilisant la condition 4 pour
= gl
et
que
h' = g~, on obtient un monomorphisme
g~.n = -m~'n' X.n C X
(p(gl),p(g~))
dans
Ii s'ensuit
at
k'
v~rifiant
j.n e X
Par d~finition de = (j.n).n'. Or, j
soit un noyau de
p(n).k'
p(gl).k = gi = p(g~).k
P(j.n).k'
cette ~galit~
p(n)
tel
K'. Cette derni~re condition entra~-
ne l'existence d'un k e K
et que
n
h =
= k, car
.
et
= P(j).P(n).k' j, il existe un
= P(j).k = ~. n' e H
tel que
j =
~tant un monomorphisme, il r~sulte de -i n.n' = ~, d'o~ n' = n vu que n est un
-
82
-
$
H
s; P
H" 5
n
r K" A
monomorphisme. prouve que
Doric n
(~,k)
est inversible, et
gl : gi-"" Ceci
est un p-projecteur. V
CAS PARTICULIERS. A ~ Supposons que la condition 1 soit v~rifi~e et qu'il existe un produit naturalis~ ((Pi)ieI,S) lis~ darts
de
P (i.e. un produit natura-
H" tel^que ((P(pi))iEI,P(S))
naturalis~ dans merit
(si)iE I dans
~ = [g~iEI
K');
en prenant pour
tel que P(pi).~ = ~i
soit un produit ~ l'unique ~l~pour tout
i e I,
la condition 2 est remplie. B ~ La condition
4 est satisfaite lorsque
foncteur ~ noyaux et que
X.n C X
p
pour tout noyau
est un n.
C ~ Supposons v~rifi~es les hypotheses 1 et 2, et soit Q un foncteur de ~" vers L'. Soit X un ensemble de monomorphismes de ~" tel que P(X) soit form~ de monomorphismes de ~*. S'il existe un (X.H~,Q.P)-sous-morphisme j de S engendr~ par Q(~) et si P(j) est une Q-injection, alors j e s t un (X.H~,P)-sous-morphisme de S engendr~ par ~, de sorte
-
que l'hypoth~se tel que
il existe un unique
morphisme,
si
k
En effet, il existe un
~tant un Q-monomorphisme,
et, P(j) v~rifiant
~' =
Q.P(j).Q(~")
-
3 est remplie.
Q.P(j).k = Q(~)
Par ailleurs,
83
et
k
P(j).~'
et si
j' e S.X
= ~.
P(j').~"
trouve
= ~, on
= Q(~) et, par d~finition d'un (X.Ho,Q.P)-sousil existe un
r~sulte P(h).~'
h ~ }{ v~rifiant
j'.h = j; il en
= ~" , ce qui prouve l'affirmation.
2. EXISTENCE DE STRUCTURES QUASI-QUOTIENTS. Nous nous donnons deux univems M
et
~o
tels
.O
que
M
e ~ O
et
~{ c
O
M . Soient ~I et ~{ les cat6gories
O
d'applications
O
correspondantes.
Nous supposons que
Q
est
A
un foncteur de H"
vers
H"
de
H"
vers ~ et que
~{ restriction ~'. Soit
H'"
de
Q
q
~ la sous-cat~gorie
une sous-cat~gorie
Nous allons indiquer des conditions fence d'une o~
s
(H',q)-structure
est une unit~ de
lence sur
q(i)
pleine de
suffisantes
quasi-quotient et
r
de
H'.
pour l'exiss
par
r,
une relation d'~quiva-
= [i]iei
I
l'ensemble des
((Pi)iei,S)
i E H''.H.s O
soit compatible avec
duit naturalis@
tout
H"
pleine
q(s).
HYPOTHESES. 1 ~ Soit tels que
est un foncteur de
de
l'unique 616ment de
r. Ii existe un pro-
(8(i))ie I dans tel que
Q. Soit
pi. ~ = i
pour
ie I. 2~ X
~tant un ensemble donn~ de Q-monomorphismes,
existe un (X.H'o,O)-sous-morphisme posons
~ = e(j).
de S
il
engendr~ par Q(~,);
84
3 ~ Si but,
h' e H'
h" e H'
et
n
il existe un noyau
noyau dans
H"
tel
v6rifiant
dans ~I)
de
(h',h")
q(n)
que
ont mSme source dans
~
et mSme
q (i.e. un
soit un noyau de (q(h'),q(h"))
~(n) E H'
et
X.n C X.
PROPOSITION $. Si le8 hypotheses pr~c~dente8 8ont v~rifi~es, ~
est une (H',q)-structure quasi-quotient de
A. Par d~finition de un
k
tel que
Q(j).k = Q(~). Comme
phisme, il existe un et un seul si
i e I, dire que
signifie que
Q(i) = i'.r, o~
et
r
:
nonique de
Q(s)
entra~ne que Q(j)
s(j) e H'
j, on a
est un Q-monomor-
j
v~rifiant
g e H Q(i)
Q(g) = k
est compatible avec
est la surjection ca-
est compatible avec
r; l'application
~tant une injection, Q(g) est aussi compatible avec
r. De plus, pour tout
i c I, on a
pi.j e H'
(pi.j).g = pi. ~ = i. Par ailleurs, supposons h'.g = h".g, partiennent ~ j.n e X g'
et il existe
Q(s)/r; l,~galit~
sur
Q(~)
8 parr.
h'
et
h"
ap-
H'. En utilisant l'hypoth~se 3, on trouve
et, n
tel que
o~
et
~tant un noyau de
(h',h"), il existe un
n.g' = g. Ii s'ensuit Q(j.n).Q(g') = Q(j).Q(g) = Q(~),
de sorte que,
j
6rant un (X.H'',Q)-sous-morphisme de
S
O
engendr~ par
~, il existe un n'
On en d~duit
n.n' = ~
des monomorphismes.
v~rifiant (j.n).n' = j. -i n' = n , car j et n sont
et
Donc
h' = h"
quasi-surjection d~finissant quasi-quotient de
s
par
~
r. V
et comme
g
est une (H',q)(H',q)-structure
-
85
-
REfMRQUE. La proposition 2 peut se d~duire de la proposition 1 appliqu~e au foncteur canonique p de H'" vers r la cat~gorie q servant ~ d~finir les q-structures quasiquotients.
En effet, on montre facilement que, en identi-
fiant
~ une sous-cat~gorie
H"
(8(i))ie I dans
un produit de (h',h") dans de
S
q
et que
engendr~ par
nique de
~"
vers
de Qr
Qr que
l'~l~ment n
S
est
est un noyau de
j
est un (X.H'',P)-sous-morphisme o (~,r), si P d~signe le foncteur cano-
Qr.
Consid~rons toujours les foncteurs plus souvent nous aurons
H = QI(~I)). Soit
de Q-monomorphismes 9 Nous notons
Q X
et
q (le
un ensemble
~o la saturante de
~Io
dans
~ , c'est-~-dire l'ensemble des ~l~ments de ~ qui o o sont ~quipotents ~ un ~l~ment de ~Io; on montre que ~o est
un univers.
D~finition. On dit que
Q
est (fI,X)-engendrant si, pour
tout
S e H" et tout f e Q(S).~.~{ , il existe un (X,Q)o o sous-morphisme de S engendr~ par f.
PROPOSITION Z. Q
est (AIjX)-engendrant si, et seulement
8i, pour tout S E ~'o et toute pattie ~! de O(S) ~quipotente ~ un ~l~ment de ~{o~ il existe un (X,O)-sous-morpT,zisme de
S
engen~
par 1 'injection canonique (O~(S),~,M). A
A. Supposons
S e H" et f e Q(S).f~ . -Si Q est (I{,X)o engendrant et si f = (Q(S),I,M), oi] M e I] , il existe une o bijection g d'un M' ~ ~ sur M et un (X,Q)-sous-moro phisme j de S engendr~ par f.g. Ii est clair que j est un
(X,Q)-sous-morphisme
de
S
engendr~ par
f. - In-
86
versement supposons v~rifi~e la condition de la proposition et soit
f : (Q(S),f,A) e ~.~{ . En posant --
M e ~ , car
M
M : f(A), on a
O
est ~quipotent
au quotient de
A
par la
O
relation d'~quivalence
associ~e ~
tient ~ l'univers ~ . De plus
f, lequel quotient appar-
f = i f'
O
i = (Q(S),I,M)
et
f' = (M,~,A).
Comme il existe un (X,Q)-sous-morphisme i, on obtient
par
i = Q(j').k.
j" e S X.s
et
dans
Q(j")(Q(s))
IIen
r~sulte
M
dr~ par
Q(j")
i = Q(j").k"
de la bijection
j' = j".h, et
f = Q(j").k' et
j'
l'ensemble
M
est contenu
o~
9
k"
est la restriction h
Q
v~rifiant de
S
est (M,X)-engendrant.
engenV
i, 2 on d~duit:
soit ~ noyaux et que
~
pour tout noyau
joint et q
engendr~
.
est un (X,Q)-sous-morphisme
f. Ceci prouve que
S
est une injection (M',~",Q(s)).
COROLLAIRE. Supposons que
X.n c X
de
~,,-i. Donc il existe un
Des propositions
q
j'
II s'ensuit
f = i.f' = Q(j').(k.f') Si
o~
9
Q
8oit ~
II-produits~ que
soit (~I,X.Ho)-engendrant. Si n
dans
q, alors
q
admet un ad-
est ~ structures quasi-quotients.
En effet, les hypotheses des propositions ~videmment v~rifi~es.
1
et 2 sont
V
Application. Dans la plupart des exemples, les foncteurs et l'ensemble
X
q
= Q(s) ~] "~o pour tout s e ~" 9
X.H" C X . Y
et
poss~dent les propri~t6s:
(i) ^Q est fiddle, Q(S.~y.Ho) et
Q
Y
o
-
Dans la fin de ce w et nous notons
Y
87
nous supposons cette condition remplie l'ensemble des
j e X
tels que
Q(j)
soit une injection canonique ayant pour source un ~l~ment de Soit
S E ~" . Si
O"
j e S.X H', il existe un
O
tel que
q(~)
"
"
soit la bijection
Q(j), et l'on a
~ e S ~'.H"
O
O
d~finissant l'injection ^-i j' = j.g e Y. Par suite
j = j'.g, o~
~
X.H" = Y.~'.H'. I i e n r~sulte que, si M est une partie de o y o Q(S), alors j est un (X.H~,Q)-sous-morphisme de S engenare par
(Q(S),I,M)
sous-morphisme de
si, et seulement si, j' S
engendr~ par
est un (Y,Q)-
(Q(S),I,M).
Ceci permet
de poser la
D~finition.
Soient
S e ~"
et
M
une partie de
existe un (Y,Q)-sous-morphisme
j
de
(Q(S),I,M), on appelle ~(j)
S
Q(S). S'il
O
engendr~e par
une
S
engendr~ par
(X.H~,Q)-so~s-structure de
#~.
Grace ~ la propri~t~ (i), on d~duit facilement de la proposition 3 et de la remarque pr~c~dente que engendrant si, et seulement si, pour tout
Q
est (~{,X.H')o S e ~" et toute O
partie
M
de
Q(S)
sous-structure de
EXEMPLES.
appartenant ~ < , il existe une (X.H~,Q)S
engendr~e par
Prenons pour
Q
M.
le foncteur
de la cat~gorie F des foncteurs, pour foncteurs associ~ ~ 6{0, pour
X
H
PF
d'oubli vers
l'ensemble
F des
l'ensemble des PF-mo~omor-
phismes. La propri~t~ (I) est v~rifi~e. Soient
C" e F
et O
M
une partie de
C
appartenant ~ n o. La sous-cat~gorie
de
C"
engendr~e par
M
de
C"
engendr~e par
M.
structure de
C"
est une (X.Fo,PF)-sous-structure En effet,
engendr~e par
B"
est la PF-sous-
M. D'apr~s la proposition
B"
- 88-
est un quotient de la cat~gorie libre
2- 3-II, B"
[c.]
[M] est le sous-graphe de
engendr~ par
L[M], o~
M. Comme
O
est un univers, on a L[~] C I,..)~n e neN o ' ~ L[M] appartient aussi ~ M .
0 ~
de sorte que le quotient
PF
Donc
B
de
O
est (~,X.Fo)-engendrant.
En utilisant ce r6sultat,
on montre que les foncteurs d'oubli
PH
relatifs aux "struc-
tures alg6briques usuelles" (Chapitre II [4]) sont ( M , X . HO) engendrants,
si
X
est l'ensemble des P~-monomorphismes.
3. EXISTENCE DE LI~IITES I~tDUCTIVES. Soient encore j[{ et ~,I
associ~es aux univers
0
0
F,4 les categories d'applications
et
v~rifiant 0
0
0
Supposons donn6 un foncteur tion
q
le que
Q
O
~"
de
vers ~{ d'une sous-cat~gorie p l e i n e H
(A!,X.H'o)-engendrant, oz~ X phismes. Si
n
F
A. Soit l'ensemble
I
i = (~i,ti,F)
dans
H"
q
vdrifiant
tel clue X.n c X,
O"
un foncteur de
C"
vers
H'. Consid6rons
de routes les transformations de
F
s.1 e H;. La surjection
de m@me sour-
C'-limites induetives, pour
est une cat~gorie C"
H" tel-
soit ~ J~o-produits et
tout couple de morphismes de
toute cat~gorie
de
est un ensemble de Q-monomor-
ce et me'me but admet un noyau H"
O
H" ~.{O .
soit ~quipotent ~ un ~l~ment de
PROPOSITION 4. Supposons que
alors
vers ~] et sa restric-
~.
vers un foncteur
natumelles constant sur
1
i +
(t.(c)) i
ceC"
O
d~finit une bijec-
89
-tion de
I
sur une partie de
H C~. Comme
C" sont o ce produit l'est aussi, et
~quipotents ~ des ~l~ments de I{
H
et
0 *
il existe un produit naturalis~ dans
((Pi)igi,S) de (si)ie I est un foncteur de C" vers H" (resp. vers
Q. S i r
M), notons ~ le foncteur de
vers
C"
H" (resp. vers
d6-
fini par la surjection r ; nous utilisons une notation analogue pour les transformations
naturelles.
- Le foncteur
admet une limite inductive naturalis~e canonique est une cat~gorie ~ C'-limites
q.F
o , car
~I
inductives; d~signons par o C
l'injection canonique de
q.F(c)
dans
construction de o, la transformation
N = L~
naturelle
mite inductive naturalis~e du foncteur - Pour tout
q*F
c e C" , il existe un crochet
q.F. Par o est une li-
de
C" vers ~.
t(c) = ~i(c)]iei
0
de source
F(c), de but
S . Si
m r c'.C.c
, on trouve
t(c' ).F(m)=[ti(c' )] iel.F(m):[ti(c' ).F(m)] i c l = ~ i ( c ) ] iel:t(c), de sorte que et
pi T = I
T
=
une transformation
est
natureile
pour tout i e I. Soit k l'application
M dans Q(S) telle que
I~mOQT de
k~ = QT. Le foncteur Q ~tant (II,X.H')0
engen~rant et
M
sous-morphisme
appartenant j
de
S
~
~40
Q(j).k'
il existe un (X.H~,Q)-
engendr~ par
est une limite inductive de que
m
F. Or ii exlste
tel
= Q(j).k'.o C
j
k'
S"
= k; il s'ensuit Q(t(c)) = k.o
et,
k. Montrons que
C
~tant un Q-monomorphisme,
il existe
un unique t'(c)
v~rifiant
Q(t'(c)) ce pour tout
:
k'.o
et C
c e C'. Si
t(c)
=
j.t'(e),
m e c'.C.c, on obtient
0
j.t'(c').F(m) d'o~
t'(c').F(m)
ci montre que
= t(c').F(m)
: t'(c), car
T' = (~,t',F)
j
= t(c) = j.t'(c), est un monomorphisme.
Ce-
est une transformation natu-
-
~. ~ ~
/
90
-
S
~
P~
H"
r
F(c')
P
~ relle,
/
qui satisfait les ~galit~s QT' = k'~
Montrons que cela, soit
T'
et
jT' = T.
est une limite inductive naturalis~e.
i = (~i,ti,F)
e I. En posant
h = pi.j, on a
hT' = pi(JT ') = pi T = i , d'o~ Supposons aussi n
de
(h',h)
h'T' dans
= i. PaP hypoth~se, q
avec
h.t'(c)
hT'= i.
il existe un noyau
X.n C X . Soit
= t.(c)
Pour
c e C O . Comme
= h'.t'(c),
1
il existe un unique
t"(c)
v~rifiant
n.t"(c)
= t'(c).
Si
m e c'.C.c, on obtient n.t"(c').F(m) Ii s'ensuit, Par suite
= t'(c').F(m)
= t'(c) = n~
n ~tant un monomorphisme,
T" = ( ~ , t " , F )
t"(c').F(m)
= t"(c).
est une transformation
naturel-
m
le, et
nT" = T'. Si
(Q(j.n).k")~
k" = limaQT '', on trouve
= Q(j.n)QT"
= Q(j(nT"))
= Q(jT')
et, a ~tant une limite inductive naturalis~e, Par d~finition de
j, il existe
Cette relation a pour consequence
n'
= QT = k~
Q(j.n).k"
tel que (j.n).n' n' = n
, car
j
= k.
= j. et
n
-91 sont des monomorphismes.
Par suite
limite inductive naturalis~e de
h = h',
et
T'
est une
F. V
RE~RQUES. 1 ~ On pourrait d~duire la proposition 4 de la proposition i, en prenant pour de
H"
vers la cat~gorie
p
~(H',C') ~
le foncteur canonique de transformations
na-
turelles; dans ce cas, il faut montrer que les conditions de la proposition 1 sont satisfaites,
ce qui est aussi diffici-
le que de prouver directement la proposition 4. 2 ~ Avec les hypotheses de la proposition 4, et en supposant que
C"
est une cat~gorie discrete, on volt que
est une cat~gorie ~ M -sommes. La proposition
H"
2 affirmant
0
que, sous ces hypotheses, quasi-quotients, H"
est ~
v~rifiant
q est un foncteur ~ structures
on d~duit de cette seule assertion que
C'-limites inductives pour route cat~gorie C e ~I
~ l'aide de la proposition
C"
6-5-I [4]
0 ~
Mais la d~monstration de la proposition 4 ne se simplifie gu~re en y supposant
C"
discrete.
4. CO~PLETION D'UNE CATEGORIE. Soient M e t
M les categories d'applications
con-
sid~r~es dans les deux w precedents.
Nous supposons donn~s
deux ensembles [ e t
I"
J de categories
telles que
D~signons par S ~J l'ensemble des triplets
I e ~{o"
u = (C',~,9),
0
o~
C"
est un graphe multiplicatif,
o~
C E M
et o~ 0
(resp. o~ 9) est une surjection associant ~ certains n~ofoncteurs
F
de
I'e I (resp. I'e ] ) vers
fommation naturelle de la forme (~,t,F)), en notant
8
C"
~(F) = (F,t,~)
le n~ofoncteur de
I"
une trans(resp. ~(F)= vers
C" cons-
-
SIJ
(u',#,u)
SIJ
et o~ ' U' = (C' , ~ ' , ~ ' ) E o C" vers C'" v~rifiant la condition: .
ou" u = (C',~,~) r o est un n~ofoncteur de
u(F')
-
e e C~. Soit S IJ l'ensemble des triplets
rant stm
Si ,(F)
92
est d~fini, W'(#.F) est d~fini, u'(r
est dgfini et ~gal ~ #u(F); si est d~fini et ~gal ~ r
S IJ devient une cat~gorie pour la loi de composition: u)
:
u I' = u' . Nous identifions S oIJ ~ la
si, et seulement si,
classe de ses unit~s. La suPjection (u',~,u) § p ,(~
~
C',~,C)
p ''I~N de )s-j ( vers- I{, Soit F 'IJ la sous-
d~finit un foncteum
cat~gorie pleine de S I] ayant pour unit~s les triplets (C',~,9)
tels que
C"
u =
soit une categoric et que ~ soit une
application l-limite projective naturalis~e partielle,
v
une application J-limite inductive naturalis~e partielle,
F IJ la sous-cat~gorie pleine de F 'IJ ayant pour unit~s les triplets u = (C',w,u) ~ F ~,IJ tels que sum
C" . Soit
et u soient des applications l-limite projective naturalis~e et J-limite inductive naturalis~e respectivement, sur Nous notons
p,IJ et pIJ
les foncteurs de F 'IJ
respectivement vers ~{ restrictions de
et de FIJ
p,,IJ .
De mSme nous d~finissons ~ partir de l'univers ~ categories et
pIJ
~IJ, ~,I~
at
C'.
~IJ , les foncteums
p,,IJ o, p'~J
(toujours relatives aux mSmes ensembles I e t
PROPOSITION 5. Les foncteurs P IJ9 p,IJ
les
J ).
et p. IJ
sont ~ If -produitsj ainsi que les foncteurs injections canon~ques d ~ FIJ vers F 'IJ et de F 'IJ vers S IJ. A. Supposons oG
D c ~{o" Soit
u d = (Cd'Ud'~d) e S oIJ C"
pour tout
d ~ D,
le graphe multiplicatif produit d~DCd
-
et
Pd
93
le n~ofoncteur projection canonique de
Soit e(M) l'ensemble des n~ofoncteurs C" deD
tels que ~d(Pd.F) = (Fd,td,~ d) 9 Si
triplet
i e I " o' notons (F,t,~), o~
t(i)
C~.
I" E I vers
soit d~fini pour tout
la famille
F'
(td(i))deD; le
est un foncteur de
et si ~d(Pd.F') = ( ~ ,td,F ' d) '
C"
de
sum
e = (ed)d~D, est une transformation
naturelle ~(F). De mSme, si vers
F
C"
J" e J
est d~fini pour tout
d r D, nous d~signons par ~(F') la transformation naturelle (8',t',F')
d~finie par
e' = (e~)dg D
et
t'(i) = (t~(i))dE D
Nous obtenons ainsi deux surjections ~ et v e t est un ~l~ment de S IJ . Montrons que
u
pour i E I'. 0
u
= (C',u,~)
est un produit de
0
(Ud)dc D dans
S ~J. En effet, par construction, on a =
Id
(ud,Pd,U) E
Supposons ~d = (Ud'r
') E
(C' ,~',9'); soit r =[r C'"
S IJ
SIJ
Dle
pour tout pour tout
formation naturelle (r pour tout
d E D, o~
si
=
est d~fini, ~d(r
d e D. Comme Cd.F' : pd. r U(r
U !
n~ofoncteum canonique de
vers C" . Si ~'(F') = (F',t',8')
est d~fini pour tout
d E D.
la trans-
est d~finie et ~gale t{i) = (r
D = r
i e I" , c'est-~-dire ~ Cw'(F'). On volt de mSme O
que, si v'(F") r
est d~fini, ~(r
Par suite r = (u,r
~dl} pd. # p"
pour tout
est d~fini et ~gal appartient ~ S IJ et l'on a
d ~ D. Ceci prouve que le fonctettw
est ~
~I -produits. o - Supposons de plus u d E F 'IJ
notations pr~c~dentes, si ~(F) = (F,t,~) naturalis~e.
C"
pour tout
est une cat~gorie. Montrons que,
est d~fini, c'est une limite projective
En effet, soit ~ = (F,t',2')
tion naturelle.
d c D. Avec les
Etant donn~ que Wd(Pd.F)
une transformaest une limite
- 94 -
projective naturalis~e et que naturelle vers
pd @ est une transformation
Pd.F, il existe
k d = li+m pd @
tel que
Bd(Pd -F) k d = pd @ ; en posant de
C
k = [kd]deD, on volt que
v@rifiant
est l'unique @l@ment
k
est une limite pro-
p(F) k = @. Donc B(F)
jective natumalis@e.
On montre d'une mani~re analogue que v
est une application J-limite inductive naturalis@e partiel-le sur
C" . I i e n
produit de
r@sulte
u e F 'IJ , et
dans F' I J
(Ud)de D
. Ainsi
u
est aussi le
p,U
IJ
(SIJ, I,F '
et
sont des foncteurs ~ ~{ -produits. 0
9
~IJ
- Enfin s~
u d e r~
pour tout
est d@finie pour tout foncteum et v(F') vers
B(F)
de
C"
I" E I
est d@finie pour tout foncteur
C" . Par suite
duit de
F
d e D, la limite
de
J" e J
appartient ~ FIJ et c'est le pro-
u
(Ud)de D dans
F'
vers
F I].
V
PROPOSITION 6. Le8 foncteurs
pIJ , p
FIJ ,p"
IJ
sont
no~,aux, de re@me que les foncteurs injection~ canoniques de F IJ vers F 'IJ et de F 'IJ vers S IJ . A. Supposons que ~= (u,$,u)
et #' = (u,$',u)
S,IJ
et
deux ~l~ments de Soit
G"
, o~
u = (C',p,v)
le graphe multiplicatif noyau de
form~ des
x E C
tels que
Supposons
I" e I vers
p(~.F)
G"
G" , et
p(F) = (F,t,8)
- _ ~ = (C',p,v).
(#,~'), qui est
#(x) = ~'(x); soit q le n~ofonc-
teur injection canonique de foncteur de
soient
vers
C'. Soit
F
F'
le n6ofo~cteum
un n~oq.F'.
d~fini; alors
= ~(F)
at
~(~'.F)
= ~'p(F).
Les @galit@s r entra~nent
= r r
=r = @'p(F), d'oO
= r t(I') r G. Ii s'ensuit que 0
- 95
(F',t,8)
-
est aussi une transformation naturelle,
notons ~'(F')~ on a d~fini et ~gal ~
n~'(F') = B(F). De mSme si
Supposons
r
= (G',~',v')
= (u, r
~
S I Jo
~'
-
' n = (u,n,u')
E
r
r
r162
et
r"
La surjection ~" d~finit un n~ofoncteur v~rifiant
vers
C""
alors
F
n.r
r
tel que
=
Soit
~"(F")
F"
et F' : r " F" ~.F' . Puisque r
(u',r de
S 17 pour lequel
noyau de
(r162
~
dans
SIT
~
si u'(F')
vers
C""
~
S IJ
,
la
est d~fini et est ~gal s i v " ( F I) "
,,
est d~fini,
( El). " Par consequent
. . et e'est l'unique element r ,,
" = r . Ceci signifie que ~ est un
"rp
- Supposons de plus que montrons que
de
est d~finie et ~gale
est defln! " " " et vaut r
appartient
:
;
appartient
"" r " ~ "iF" ~ ). D 'une manlere analogue,
alors v' (~i".F ~')
.
un foncteur de
F = r
On en d~duit que ~'(F')
a"
S IJ
soit d~fini, et posons
transformation naturelle ~(F) r
et v', et
Slj, o~
~
u" : (C"',~",v")
G"
est
naturelle, d~sign~e par v'(F'). De cette
faqon, on construit deux surjections u'
v (F) est
(8',t',F')
(8',t',F), on volt que
une transformation
que nous
IJ i r
et r
appartiennent
F'IJ
~
, et
u' E F'I] 9 En effet, il suffit de montrer que
est d~fini c'est une limite projective naturali-
s~e. Or soit r une transformation
naturelle
(F',T,81) , et
F = n.F'. Etant donn~ que
r
leet
est une limite projective natura-
que
~(F) = n~'(F')
lisle, il existe ~(r162
k
tel que = r
r
naturel-
u(F)k =n~ 9 Les ~galit~s = (r162
=
ont pour cons6quence
est une transformation
r
= lim (r162 g-
= (r162 =
=
~(r162
= r
d'o~
k e G
-
et
~'(F')k = r
96
Ainsi u'(F')
est une limite projective na-
turalis~e. On prouve de mSme que 9'(F")
est une limite in-
ductive naturalis~e lorsque cette transformation naturelle est d~finie.
u' e F' IJet -~
Donc
dans F 'IJ . - Enfin lorsque r et r la construction
FIJ
de
u'
Soient
appartiennent ~
C"
noyau de
(r162
une cat~gorie et
dans
M
de
C
est 8 a t o n e
F
de
I"
vers
t(I') C
C"
M
V
C'. On dit
est d~fini pour un
et que et
IJ .
pou2 ~ si elle v~rifie
la condition: Lorsque ~(F) = (F,t,8) foncteur
p
~ une application
l-limite projective naturalis~e partielle sur qu'une partie
F [J,
entra~ne que c'est un ~l~ment de
, de sorte que ~ est un
Ddfinition.
(r r
est un noyau de
F(I) C M ~ on a
lim~(F)r e M
0
pour toute transformation naturelle ~ = (F,t',8')
v6rifiant
t'(I') C M. On d~finit dtune faqon analogue la notion d'une 0
C
partie de
8aiTxr~e pour ~ , si ~ est une application J-
limite inductive naturalis~e sur
Soit
X'
1 'ensemble des r
tels
C" (partielle).
f,IJ
=
que r soit un foncteur injectif, que ~'(F')
nl si, et seulement si, ~(r
est d~fini et que ~'(F')
soit d~fini dans le seul cas o~ ~(r
PROPOSITION teur de
G"
vers
7. Soit
est d6fini.
u = (C',~,~)
C" . On a
PF(r
~ F tIJ et r un fonc-
E P'IJ~
si, et
8eulement 8i, r est un foncteur injectif et si 8aturd pour ~ et pour v . Tout dl~ment de monomorphisme.
soit d~fi-
X'
r
est
est un p nIJ.
97h. Supposons
= ((C',p,~),r alors r est injectif et M"
de
c X';
M = r
d6finit une sous-cat6gorie
C'. Soit ~ l'isomorphisme de
par la bijection un foncteur F = r
F
o~
9
-i. de
M"
sur
G"
Supposons p(F) = (F,t,8)
I'e I vers
C"
tel que
d6fini d6fini pour
F(I) C M. Comme
F' = ~.(M',F,I'), par d6finition de
limite projective naturalis6e p'(F') p(F) =r
X'
la
est d6finie. L'6galit6
prouve que ~'(F') = (F',~._t,~) ,
o~
e = r
il s' ensuit t(I') = r
C r
= M .
0
Soit ~ une transformation naturelle t'(I') C M; si ~'
(F,t',~')
v6rifiant
est la transformation naturelle
O
(F',~t',~'), il existe
k' = limP'(F')~ ' - " r
= ~
et
o~
r
= e',
e G . Les relations #p'(F')
= p(F)
entraTnent limP(F)@ = lim p(F) #Q' = #(lim p '(F')~, ) = #( k' ) e M. 4-
Donc
M
g-
g-
est satur6 pour p. On montre de mSme que
M
est
satur6 pour 9. - Avec les notations pr6c6dentes, prouvons que ~ est un
~,IJ _ monomorphisme. Pour cela, soit
~' = ((c',p,~),#' , (K',~,~)) E S U
et
PF(r ' ) = P F ( # ) . f ''.
D'apr~s la proposition 4-2-II [4], f" d6finit un n6ofoncteur $"
de
p(Q'.F I)
K"
vers
G'. Si
~(F I) = (Fl,tl,8 I)
est d6fini,
est d6finie et l'6galit6 #'.F 1 = #.(#".F I)
re que p'(#".F I)
assu-
est aussi d6fini. Puisque
#p'(#".F i) = p(#'.F l) : #'~(F I) : #(#"~(F~)) et que # est injectif, on trouve p'(r
I) = #"~(FI). D'une
98-
mani~re analogue 9 si ni et ~gal ~ r
~(F~)
est d~fini9 9'(r
est d~fi-
~(FI). Par suite ,~ ' ),r (K" ,
est l'unique ~l~ment ~"
de
tel que
~.~" = ~' et PF(r = f". ,,[J Autrement dit, ~ est un P -monomorpnlsme. - Inversement 9 supposons que r soit un foncteur injectif de
G"
vers
C"
et que r
pour ~. L'ensemble Si
F
et que teuP de
M
~(F) = (F9 vers
soit satur~ pour ~ et
d~finit une sous-cat~gorie de
est un foncteum de
I"
= M
I'E I vers
C" tel que
soit d~fini9 et si M"
d~fini pap
F1
est le fonc-
est une limite projective naturalis~e ~ 9 car
projective naturalis~e,
M"
tel que ~(r
ralis~e ~'(~.(r teur de
I" E J vers
M
sur
(Fl9 est satur6 G"
d~fini
est aussi une limite
que nous noterons ~'(r
la surjection associant 9 ~ tout foncteum G"
F(I) C M
~ 9 le triplet
pour ~. En notant ~ l'isomorphisme de par _r , la transformation naturelle ~
C'.
F'
Soit ~' de
I" E I vers
soit d~fini9 la limite projective natu~'(F'). De mSme9 si G"
F"
et si ~(r
est un fonc-
est d~fini9 notons
Q'(F") l'unique limite inductive naturalis~e v~rifiant l'6galit~
Cv(F") = ~(r
Ii est ~vident que
X' et c'est l'unique 616ment de PE(r
u.X'
appliqu~ par
ce qui ach~ve la d~monstration.
COROLLAIRE. X = X ' C 3 F ~ morphismes.
P'I] sur
9
est l'ensemble des PlJ-~ono-
A. Soit ~ un P~J-monomorphisme r est un foncteur injectif. Soit
F'
((C',~,~),r un foncteur de
~')); I" e
99
vers
G'; comme
~ et ~' sont des applications
jective naturalis~es,
@appartenant
N(r
et N'(F')
~ 9~IJ , on a ~(r
I-limite pro-
sont d~finis et,
=r
De meme v'(F")
est l'unique transformation naturelle v6rifiant @~'(F")
pour tout foncteur
F"
Donc $ e X . - Inversement, monomorphisme, 7) et
de
I" e
J
tout @l@ment de
v(r
vers X
car c'est un P'IJ-monomorphisme
= G" .
est un pIJ_ (proposition
~IJ est une sous-cat@gorie pleine de ~,IJ .
v
PROPOSITION 8. Si I e t S sont 6auipotents ~ des dl~ments de ~ le foncteur P rIJ est (~4,Xr F'IJ)-engendrant et pIJ 0 ~
"
0
est (~X. FoI] )-engendrant. A. Soit
u = (C',u,v)eF'
et soit
M
une partie de
o
C
appartenant ~ la saturante ~
de ~{ o
l'ensemble des parties gorie de tersection
C"
B
C
o
d@finissant une sous-cat@-
et satur@es pour U et ~. Ii est clair que l'inG
de B appartient ~ B . Le foncteur
fiant la propri@t@
(i) du w 3, pour prouver que
(M,X'.fwIJ)-engendrant, o
de
dans ~ . Soit
o
p,17
v@ri-
p,IJ
est
il suffit de montrer qu'il existe
IJ
une (X' 9F'IJ,P ' 0
)-sous-structure
Or, d'apr~s la proposition 7, si existe, elle est de la forme
de
u
engendr@e par
une telle sous-structure
(G',N',v'), et
~ . Nous sommes donc ramen@s ~ montrer que o
M.
G
appartient G
est @quipo-
tent ~ un @l@ment de A4 . Pour cela, nous allons construire o
G
par r@currence transfinie. - Nous aurons ~ utiliser quelques notions sur les (nombres)
ordinaux que nous allons rappeler.
Soit I un ordinal et
OA
l'ensemble bien ordonn@ des ordinaux inf~rieurs ~ X. Si l n'a pas de pr@d@cesseur dans
O~, on l'appelle un ordinal
limite. On dit que X est un ordinal r@gulier si, dans
O
,
-
on a
100
-
sup l~ < ~ pour toutesuite transfinie ~<~,
(A~)~
que A tout ensemble
A, nous associons l'ordinal
plus petit ordinal ~quipotent ~
A, qui
est le
A; un tel ordinal est appe-
I~ ordinal initial. Ii existe une bijection y du sous-ensemble blen ordonn$ de rieurs ~
A
O~
form~ des ordinaux initiaux infg-
sum une section commengante de
0~; l'ordinal
y-l(A) est dit ordinal initial d'indice I , et not~ ~A " Si A
admet un pr~d~cesseur,
~
est r~guller. Si ~l est r~gulier
tandis que A est un ordinal limite, on dit que ~A est inaccessible; dans ce cas, on montre que A = ~A " Solt A l'ordinal borne sup~rieure des ordinaux aussi A = sup ~. Comme
A e M ; on a o est un univers, on prouve que
M
A ,
o~
o
O
A est un ordinal inaccessible.
Si
I" E I u J
, on a
soit A' l'ordinal borne sup~rieure des ordinaux d~e~it
I uJ
. L'hypoth~se
~ E ~
et
J g ~
O
< A et
i < A;
I , o~
I"
entraXnant O
A ~tant r~gulier, A' est strictement inf~rieur
A; a fortiori
A'+I <
A
, d'o~ mA~+l < ~A
= A . Notons
l'ordinal initial r6gulier ~A'+I " - Soit
M[
la sous-catggorie
avons vu (fin w 3) que
M1
nal inf6rieur ou 6gal ~ ~ transfinie (s)
M~
C" ; on a
de
C"
engendr6e par M. Nous
appartient ~ ~o" Soit ~ un ordiet supposons d6finie une suite
(M~)~<~ v~rifiant la condition: appartient ~ M~, C ~
~o et d~finit une sous-cat~gorie de
lorsque
~' < ~ < k .
D~finissons Mk comme suit: i er cas: Si I e s t un ordinal limite, on pose Evidemment
M l d~finit une sous-cat~gorie de
M l = ~)
C" contenant
- 101
M~
et, I ~tant l'ordinal associ~ ~ un ~l~ment de M , on a O
2~me cas: i = ~ +i. Si C"
tel que
notons
F(I) C M~
AF
F
est un foncteur de
et que
u(F) : (F,t,8)
l'ensemble form~ des ~l~ments
et des ~l~ments
I"
vers
soit d~fini,
t(i), o~
i E I'o '
h = lim~(F)~ , pour route transformation
naturelle ~ : (F,tl,8 I)
telle que
tl(I ~) C H~. La surjec-
tion telle que i § t(i),
(tl(i))ie I. + h 0
applique une partie
UF
de I U(M~) I~
et
M~
appartiennent ~ ~o' on a
Si
F'
est un foncteur de
air
F'(J) C M~
notons
A~,
et que
sur
A F. Puisque
A F r Mo .
U F E ~o' d'o~
J" e ]
vers
C"
I
tel que l'on
u(F') = (8',t',F') soit d~fini,
l'ensemble form~ des ~l~ments
t'(i), oh
i E J', 0
et des ~l~ments
limU(F')~ ' , pour route transformation na-
turelle #' = (8",t~,F') encore bles
A~, e ~o" Soit AF, oh
M~
la
M~ 9 La bijection tie de ~ % M
l'ensemble r~union des ensem-
sous-cat~gorie de
r~union
A~,, o~ C"
M~
~
F' e a(9).
engendr~e par
F § (F(i))ie I appliquant
)I e ~o, on a
Afortiori, -Dans
M~'
t~(J')o C M~; on trouve
F e a(~), et des ensembles
Nous noterons
semble
v~rifiant
a(,) sur une par-
< A. II an r~sulte que l'en-
appartient ~
~o' car, de mSme,
a--~)
M 1 e ~{o"
les deux cas, nous avons ainsi construit une suite
transfinie
(M
v~rifiant (e)
Par r~currence transfi-
nie, on obtient de cette faqon une suite transfinie croissante Posons
(M~)~
de sous-cat~gories de
A = M~; par d~finition de
contenu dans tout ~l~ment de
C"
telle que
A, on volt que
~. Montrons que
r~ pour ~ et pour ~, d'o~ il r~sultera que
A
A
M~ E ~4o" A
est
est satu-
est l'inter-
102
section
G
de
B. En effat, supposons ~(F) = (F,t,8)
ni pour un foncteur Pour tout
-
F
de
I"
vers
C"
tel que
i e I, il existe un ~'l < ~ tel que
~tant donn~ que
I e ~{
entra~ne
d~fi-
F(1) C A.
F(i) e M~i ;
I < ~ et que
~
est un
O
ordinal r~gulier, on a et, par construction de
= sup ~i < ~. Par suite iEI M~+I, on obtient
t(I')o C M~ C M~+IC
A .
De plus, si ~ est une transformation naturelle telle que
F(I)C M~
(F
o t'
,
S')
t'(I') C A, il existe pour la mSme raison un orO
dinal ~'
v~rifiant ~ < ~' < ~ tel que t'(I')o C M~, ,
Ceci prouve que gue montre que
A A
d'o~
lim~(F)~§ E M~, C A.
est satur~ pour ~. Un raisonnement analoest satum~ pour ~. Donc
que
G = A, c'est-~-dire que
est
(~|,X'.F'IJ)-engendrant.
A e B , de sorte p,I] appartient ~ ~o" Ainsi
G
O
- Si l'on suppose dans ce qui pr~cSde que o
u
appartient
, le corollaire de la proposition ? entralne que
d~finit ~galement une (X.F~J,PI])-sous-structure ~
G
(G',u',~')
p I] est (~(,X.FIJ)-engen o drant, ce qui achSve la d~monstration. V de
u
engendr~e par
M. Par suite
PROPOSITION 9. Si I e t
J appartiennent d f]
, la catd-
o gorie S I J est ~ F ,I -T- p r o @.e c t ~ o. n s et ~ FI J -~rojections$ la cat~gorie F rI'I est d FIJ-projections.
A. Nous nous ram~nerons au th6or~me g6n6ral d'existence de structures libres. Pour cela, montrons d'abord que F 'IJ est 6quipotent ~ un 61~ment de ~ . En effet, soit
u =(C',~,~)
O
un 616ment de
S IJ . Notons O
U
l'ensemble des couples U
103
-
(F'(t(i))iE=(F)o tels que
~(F) = (F,t,8)
-
) E N'x
U
C Ig = U c
I'EI
soit d4fini, et
U
l'ensemble
des couples )
u
(F', (t' (i))iE=(F,)0 tels que
(8',t',F') = ~(F')
u § (C',U ,U~) U = N'• 0
soit d6fini. La sumjection 7:
est une bijection de S IJo sur une pattie de
(Uc•
G ~,,
c I~ :
I'EJ
Comme
M
0
est un univers, les relations
0
N' E H , 0
I" ~ M 0
si
I" E
0
luJ
IuJ~
,
,
M
c
0
0
entraTnent UC s
o Par suite l'ensemble o
et
. Le foneteur
at U r U C' E ~ 0 S IJ est 6quipotent ~
o p,,IJ6tant
FIJ ) est 6 q u i p o t e n t
fiddle,
SI J
(eta
. ~un ~l~ment de 0
F' I J
fortiori
~ un 6 1 6 m e n t d e ~ . 0
-
Posons
P =
(SIJ
la proposition 5, P
,I,F' IJ )
at
P = (S IJ I,F' 17) . D'apr~s
est ~ ~ -produits, de sorte que
P
est
O
aussi ~
SI]-produits. Par ailleurs
p
est ~ noyaux en ver-
tu de la proposition 8; la d~monstration de cette proposition signifie que tout noyau
n
appartient ~
sant une sous-cat~gorie de
~,IJ
X'.nC
X'.X'
X' et,
X'
d4finis-
il s'ensuit C
Enfin la proposition 8 affirme que
X'
.
p,IJ = p,,IJ.p est un
foncteur (Id,X'9 F'I7)-engendrant o , et la proposition 7 que P(X') C X'
est form~ de
~
"-monomorphismes. Des cas patti-
cullers (w I) de la proposition i, il r6sulte que un adjoint, i.e. que
p
admet
S ~J est ~ F'IJ-projections. - On d~mon-
ire les deux autres assertions d'une fagon analogue, en prenant pour de
FIJ
p
vers
respectivement le foncteum injection canonique S IJ et de
FIJ
vers F '
IJ .
V
-
Soit
104
-
-IJ
(Q,~,u)
un (~
,S
IJ
)-projecteur, o~
(C',~,9). En g~n~ral, r n'est pas injectif. Mais,
partient ~ F'
IJ
si
u = u
ap-
, on montre que ~ est injectif, de sorte qu'il
existe ~galement un (FIJ,F'IJ)-projecteur (u',n,u),
tel que
C"
o~
u' = ( c " , ~ ' , ~ ' )
soit une sous-cat~gorie de
le foncteur injection canonique de cas, on appelle
u'
C"
,
C'"
et que n soit
vers
C''. Dans ce
une (I,J)-compl~tion de u . Supposons
de plus que ~ et 9 soient les surjections vides; u' est aussi appel~
(I,J)-compl~tion libre de c'. Cette d~finition
est justifi~e car, si nonique de F ~J
vers
q
d~signe le foncteur projection ca-
F, on volt facilement que
u'
est une
q-structure libre associ~e ~ C'.
REFERENCES. i. Sur l'existence de structures libres et de foncteurs adjoints, Cahiers de Topologie et g~om~trie diff~rentielle, IX, 1 et 2, Dunod, Paris (1967), 33-186. 2. Esquisses et types de structures alg~briques , Bul. Inst.
Polit. Iagi , XIV (XVIII), 1-2 (1968), 1-14. 3. Categories et 8t~ctures, Dunod, Paris (1965). 4. Ma~trise de Mathdmatiquesj C3, Alg@bre, Centre de Documentation Universitaire C.D.U.-S.E.D.E.S., 5. Structures quasi-quotients,
363.
Paris (1968).
tlath. Ann. 171 (1967), 293-
105
CATEGORIES
-
OF GROUP E X T E N S I O N S
by K. W. G r u e n b e r g
w 1
Our aim is to s k e t c h the b e g i n n i n g s that studies a fixed g r o u p
G
that can be c o n s t r u c t e d over
in terms of all the e x t e n s i o n s G.
We
limit o u r s e l v e s h e r e to
the e x t e n s i o n s w i t h a b e l i a n kernels. connexions
of a t h e o r y
E v e n so, w e u n c o v e r
b e t w e e n k n o w n b~t a p p a r e n t l y u n r e l a t e d a s p e c t s of
g r o u p c o h o m o l o g y and g r o u p t h e o r y proper.
The t h e o r y seems to
p r o v i d e the right kind of f r a m e w o r k w i t h i n w h i c h to seek extensions
of t h e s e results and to p o i n t the way to new areas
that m a y be w o r t h exploring.
D e t a i l s of e v e r y t h i n g that follows w i l l be f o u n d in a set of l e c t u r e notes topics
(probably to be c a l l e d
in g r o u p theory")
"cohomological
that I am p r e p a r i n g for this same
series of S p r i n g e r L e c t u r e Notes.
w 2
If
G
is a given group,
let
C-G1
d e n o t e the c a t e g o r y
w h o s e o b j e c t s are all e x t e n s i o n s
(A]E): with
A
abelian,
1--9
A
) E
) G
and in w h i c h a m o r p h i s m
) i
(AIE)
,
- ~ (A~IE~)
-
106
-
is a p a i r of g r o u p h o m o m o r p h i s m s c: E --~ E l course,
and the a p p r o p r i a t e
e
is then n e c e s s a r i l y Let
determined Take
co(AIE)
by
denote
(AIE).
so that
diagram
) AI,
is c o m m u t a t i v e .
a G-module
this
Of
homomorphism.
the c o h o m o l o g y
We can o b t a i n
~: A
class
class
in
H2(G,A)
as follows:
any free p r e s e n t a t i o n
(i) .
(i.e., 8R
(~,~),
F
1
) F
is a f r e e group)
(restriction
gives
) R
rise
to
to
R)
(1)
"i s
lift
to
8: F
is a h o m o m o r p h i s m :
co(AIE ) .
Der(F,A)
and
) G
(Use M a c L a n e ' s
> Horn (R,A) F
--~ E.
R
) A
theorem
) H2(G,A)
Then
and this
that
. ) 0
is exact.)
Remark. spondence exactly
with
co(AIE)
(~,o) :
so that objects
(AIE)
introduce
A morphism
x~* = x I" of
elements
in the sense
(A,x), w h e r e
x E H2(G,A).
is free on a set in o n e - o n e
--~
of
G, t h e n
8R
corresis
of S c h r e i e r .
(A 11EI ) , t h e n
clearly
i ) c o ( A 1 IEI).
We now are p a i r s
F
the n o n - u n i t
a factor-set
If e*:
If
Mod G
(~G
a second
A 6 Mod G
category,
(= r i g h t
is a m o d u l e
F : (AIE)
G-modules)
homomorphism
is the c a t e g o r y
in M a c L a n e ' s
~G' w h o s e
of
terminology:
---~ (A, co(AIE))
H2(G,
objects
and
~: A
) A1
)-pointed
[4] , p.53.)
Then
-
is a functor from (2).
,r
[~I
107
-
%.
to
is surjective
(= full and representative).
This is a completely elementary result. Nevertheless,
[i], p.179.)
it leads to very rapid proofs of some important
results in group theory: the centre-commutator tation theorem for
e.g., the splitting theorem of Schur,
theorem of Schur, the Magnus represen-
F/[R,R].
The surjectivity of what follows~/
(Cf.,e.g.,
In particular,
F
is also constantly needed in
for the characterization
of
epimorphisms and monomorphisms: (3). if r a
(~,~)
is an epimorphism in
is an epimorphism in
epimorphism in
Mod G.
OG
~!
if~ and only
if, and only if, a
is an
Ditto for monomorphi_sms.
We may now define projectives
and injectives in the
usual manner.
(4). r
tive in
(A] E)
is injective in
is in~ective in
[G!
if, and only if,
ifLand A
only if,
is injec-
Mod G. Thus both our categories have enough injectives.
But the injectives do not seem nearly as interesting as the projectives
and we shall say nothing more about injectives here.
It is easy to manufacture examples of projectives.
108 -
Take any free presentation, = R/[R,R], and
F = F/[R,R].
(R,x), where
these objects sense.
as in Then
(i) above, (RIF)
X = co(R[F),
is projective
is projective
in
in
~.
6GI
In fact,
can be regarded as "free" in a certain natural
In any case, both categories have enough projectives. It is difficult
the relation between
to say anything n o n - t r i v i a l
different
is a simple consequence
of Schanuel's
lemma: are free objects,
R--I~___ V2 ~'------R2 | Vi
then
'
IS G-free of rank = rank F i. A more profitable
comparison
about
The following
free objects.
[5).
whe re
and write
project turns out to be the
of mi'nimal pro~ectives:
called minimal if every e p i m o r p h i s m
of
is n e c e s s a r i l y
A ~ 0 ).
an i s o m o r p h i s m
(and
The following statements
(i)
(AIE)
(ii)
r CA[E) =
(ii!)
there exists (R,x) =
Moreover,
s arily G-projective.
in
~;
is ~ro~ective
a free pair
(A,x) 9
the module
of projectives-
are equivalent:
is projective (A,X)
(P,O). P
is
to a projective
(ALE)
we first state a characterization
C6) 9
(AIE)
The projective
in
in~;
(R,x) (%
so that
has productsl)
(iii) is neces~
-
It follows
109
-
that a projective
(A,x)
is minimal
if,
and only if, there does not exist a splitting of the form (A,x) = (B,y) @ If
G
is finite,
H2(G,
(P,0)
.
projective)
= 0
and so we
have: (7). minimal
For finite
G, the projective
if, and only if, A
scalars.
Let
of all pairs
K
give
x
QG
to restrict the discussion
corresponding
be a commutative
A 6 MOdKG
let
Then
(A,x)
QKG
the category
and morphisms for
A(K ) = A | K Z
to
to a change of
ring and
A similar definition
A 6 MOdG,
' / ~ X(K ). ,
is a functor:
%
(A,x), with
KG-homomorphisms.
If
of
is
has no projective direct summand.
It is often necessary certain subcategories
(A,x)
to be
IKGI.--
and
A
> A,K , ~J
) (A(K) ,X(K )) = (A,x) (K)
) ~KG"
Everything we have done so far works, with suitable modifications, free objects"
in in
~KG ~KG
and
IK-~GI. In particular,
are of the form
(R'•
(K)"
w 3
H e n c e f o r t h we shall assume
G
is finite.
the "K-
-
110
-
Our concern will be with rings:
fields,
Z, ~(p)
following
= local ring at
coefficient
p, Zp = p-adic
integers.
IGI
and
tives. phisms
Suppose
K
(A,x),
(B,y)
If
(R'•
to
is a field of c h a r a c t e r i s t i c are finitely
(K) ' (S,T) (K)
(A,x) , (B,y),
generated
dividing
minimal
projec-
are K - f r e e pairs with epimor-
respectively,
then by
(5) and
(6)
(iii), A 9 P |
where
P, P'
projective
(KG) m
B 9 P' | (KG) n
are KG-projective.
summands
(8).
A
~
(by
(7)),
Since
is minimal
have no KG-
the K r u l l - S c h m i d t
t h e o r e m implies
B.
(If the c h a r a c t e r i s t i c (A,x)
A,B
projective
if,
of
K
is prime
and only if,
to
x = 0
IGI, then and
A
is inducible. )
(9). IGI,
If
K
is a field of characteristic
then ~ny two finitely
generated
dividing
minimal p r o j e c t i v e s
are
isomorphic. To complete gical
information
about
In view of same as that in
the proof of
R(K)"
A
and
(6) (iii),
(9), we need more
cohomolo-
B.
the cohomology
Now we have
in
the following
A
is the
result.
-
(i0).
Let
K
111
-
be any commutative dZ ~-~ Hq(G,K),
(i)
H q+2 (G,R(K))
(ii)
H 2 (G,RcK)) - ~
K/n~j,
ring.
for all
where
Then
q > 0;
n = [G[,
and
d (iii) H!(G,R(K)) ideal of
where
~
is I the augmentation
KGo If we had used Tate cohomology,
Remark, and
gG r
~
(iii) would be special cases of (i).
due to Tate
(cf. Kawada
0 ---~ r ---) KF ----) KG --~ 0 mentation ideals of
MOdKG. where
{/~r
We know (xi)
~: F ----) G
K = ~
is
and so, if
{,g
yields are the aug-
KF, KG, respectively,
0 ---> rl~r ----) {l{r ---) g --~ 0
(11).
(ii)
[3]).
The homomorphism
Proof,
The case
then
is KG-free on all
freely generate
F
and that
(i-
is exact in
x i) + fr,
~/{r ~ ~(K).
we
also have the exact sequence (12). and (12) yield
0 --~ g --~ K G - - ~ (i);
(ii) since
0.
Obviously
(Ii)
Finally,
(ii)
also
CKG)G ---~ K - - ~ gives
K--~
Hl(g) - ~
(KG) G = KT, where
H 2(RCK )- ) T = Z x. xs G
112
yields
(iii)
(~/{r)G __) gG
since
We may now complete characteristic basis
of
dividing
H 2 (G,A)
the i s o m o r p h i s m
-
and
(8).
(9) .
JG J . y
in the
zero map.
So
is a f i e l d of
nK = 0
Thus
a basis
K
a n d so
H2(G,B).
of
x
Let
is a #
be
Then x~* = ky
a nd h e n c e
~
gives
Remark 9
Zp, f o r any
is
(A,x)
an i s o m o r p h i s m :
The proof
of
N\
J
(9) o b v i o u s l y
(B,y)
9
also works
if
K
P.
w 4
Result characterization
(I0) c a n be u s e d of p r o j e c t i v e s
We first determines
e
in
E,
way
(AIE) , let
of r e p r e s e n t a t i v e s
in
QKG' w h e r e
show that every
in a n a t u r a l
Given
to g i v e a p u r e l y
element
an e l e m e n t
T =
is a field 9
(A,x)
in
in
QKG
HI(G,A).
be a t r a n s v e r s a l
(t i)
of the cosets)
K
cohomological
A
of
in
E
a n d for e a c h
let tie = ai, e ti(e)
Put ed T = n t -I i i(e) Then
dT
is a d e r i v a t i o n
of
ai
E
,e
ti
in
(e) A.
If
S
( = set
is s e c o n d
113 -
transversal,
then
an element, that when transfer char K
ds
call A
is cohomologous
it the transfer
is central
in
homomorphism.) dividing
class,
E, then
dT
of as an element
Ker d T
conjugate:
another (13).
proof
If
HI(E,A).
have this
of Schur's
~ = Z x 6 G,
(Note
is the ordinary K
a field with
in
class
HI(G,A).
When
is a subgroup
and since all such subgroups
So we obtain
and so the transfer
Remark due to B. Wehrfritz: an automorphism,
d T.
in
In our case of
JGJ, Ad T = 1
can really be thought
to
A ~
A
is
complementary form,
to
A
they are all
theorem!
then the isomorphism
x6G
(i0) (iii) maps
~
to the transfer
class
of
(R,x)
-
(14). IGJ, then
If the field
(A,x)
is projective
(i)
HI(G,A)
K
has c h a r a c t e r i s t i c in
QKG
H2(G,A)
dividing
if, and only
has d i m e n s i o n
if,
one over
K
and
has d i m e n s i o n
one over
K
and
is a trivial
consequence
the transfer (ii)
9 (K)
class ~ 0;
x~O. Proof. (10) and
(6).
epimorphism Then
#
One d i r e c t i o n
Conversely, r
induces
(R'•
assume
(K)
isomorphisms
(i) and
of
(ii) and pick an
) (A,x). Hi(~,(K))
N)
Hi(A),
i = 1,2,
114
and hence
H2(p)
projective
(cf.
P
(K
= 0
where
[5], chaper
-
P = Ker~.
Thus
9) and hence
P
is KGsplits over
R(K)
is a field!). Somewhat (15).
if, for all
similar arguments yield
(A,x)
is projective
p_ and any Sylow p-subgroup (i)
H I (Gp,A) = 0,
(ii)
H2(Gp,A) order
(iii) A
is
(16). is projective
If
in
~pG'
(A,x)
~
for all
if, and only
Gp ,
is cyclic O n
xRes,
of
IGpl ; and T-free.
There is also a localization
tive in
~_
in
principle:
is finitely generated,
if, and only if,
(A,x) (Tp)
then
(A,x)
is projec-
p.
w 5
/-%
ZG (B,y)
~I
P/IGI
We return now to minimal projectives. Z(p)
and write
are in the same genus (17).
M G = M | Z G. if
(A,x) G ~
Let
We shall say
(A,x) ,
(B,y) G.
Any two finitely generated m i n i m a l projectives
are in the same genus.
-
We omit the proof having
had crucial
115
here,
-
but I wish
help from Irving
to a c k n o w l e d g e
Reiner with one part of
this proof.
Finally, of this
theory with m u c h
Let
K
We shall call Frattini
(AIE)
extensions
This was
[2].
It follows
the following
is m i n i m a l
of some
extension
The Frattini Frattini
to
if
(AIE)
IGI.
A & Fr(E),
is m a x i m a l
(AIE)
It is not at all obvious
dividing
the
if any
is n e c e s s a r i l y
that m a x i m a l
Frattini
In fact they do and any two are isomor-
established
by Gasch~tz
from our theory
(for
K = ~p)
as a consequence
of
in 1954 (9) and
theorem:
(18). then
E.
even exist.
phic.
connexion
earlier work of Gasch~tz.
a Frattini
from another
an isomorphism.
a surprising
be a field of c h a r a c t e r i s t i c
group of
epimorphism
IGI,
we m e n t i o n
(AIE)
If
K
is a field of c h a r a c t e r i s t i c
is m a x i m a l
projective.
Frattini
if, and only
dividing if,
(AIE)
-
116
-
REFERENCES
[1]
Artin,
E. and J. Tare,
Class Field Theory, Benjamin,
New York, 19679 [2]
Gasch~tz, Gruppen,
W.,
"Uber modulare Darstellungen
die von freien Gruppen induziert werden",
Math. Z. 60; 274-286. [3]
Kawada,
9; 417-431.
J. Fac
Bull
Amer
Math9 S o c ,
(1965).
Serre, J.-P.,
SQ
(1963).
S , "Categorical Algebra"
71; 40-106. [5]
(1954)9
Y., "Cohomology of group extensions"
Univ. Tokyo, [4] MacLane,
endlicher
Corps locauz, Hermann,
Paris,
1962.
-
ALGEBRAS
117
-
GRADED BY A GROUP
by Max A. Knus*
INTRODUCTION
A Brauer
theory
for ~ / 2 X - g r a d e d
d e v e l o p e d by C. T. W a l l in on a l g e b r a i c we
K-theory,
try to define
an arbitrary simple
algebras Before
Bombay,
such a theory
abelian
algebras
group G.
(See also H. Bass, 1967,
[3]).
a structure
algebras.
a class of graded algebras w h i c h
Clifford
algebras.
many s u g g e s t i o n s
class of
graded tensor product.
we study
thanks
This
group, we prove
t h e o r e m for graded central simple
My sincere
graded by
We first define graded central
a suitable
the Brauer
Lectures
In this paper,
for algebras
and give some examples.
is closed under
defining
[7],
algebras was
Finally,
generalize
are due to Michel Andre
and discussions.
*This w o r k was c a r r i e d out under a grant from the N a t i o n a l Science F o u n d a t i o n and a fellowship from the S c h w e i z e r i s c h e N a t i o n a l f o n d s .
for
- 118
io
-
GRADED CENTRAL SIMPLE ALGEBRAS
By algebra, w e s h a l l m e a n a finite d i m e n s i o n a l associative field
algebra
A
w i t h u n i t over a c o m m u t a t i v e
K.
Let
G
not necessarily
be a group, w r i t t e n a d d i t i v e l y , b u t abelian.
A G-@raded al@ebra
A
is an
a l g e b r a w h i c h is given t o g e t h e r w i t h a d i r e c t s u m d e c o m p o s i t i o n as a m o d u l e A =@ A g6G g w h e r e the A ' s g
are s u b s p a c e s
of A, in s u c h a w a y
that
AA cA g h g+h The e l e m e n t s
of
K
are h o m o g e n e o u s
of d e g r e e
zero.
A
h o m o m o r p h i s m of g r a d e d a l g e b r a s is a h o m o m o r p h i s m of algebras A
subspace
~: A-4~ B I
of
such t h a t A
of the i n t e r s e c t i o n s
~(Ag) c Bg,
is g r a d e d if it is the d i r e c t su/n INA . g
We call the g r a d e d a l g e b r a are n o p r o p e r g r a d e d
g 6 G.
(two-sided)
a l g e b r a is n o t n e c e s s a r i l y
A
ideals.
s i m p l e if there A graded simple
a s i m p l e algebra, b u t the
f o l l o w i n g e a s y g e n e r a l i z a t i o n of the T h e o r e m of M a s c h k e is true.
-
Theorem
characteristic over
K,
A
be
of
then
x
0
A.
Since
= i.
Hence
does is
Let
A
graded
not
algebra.
divide
a semisimple
x = r. x , gEG g
is s i m p l e Trace(x)
the
dimension
= Dim A
of
algebra.
x ~ 0, b e
graded,
If the
we
in
may
is n o t
the
radical
assume
zero
that
and
x
is
K
not nilpotent.
Let
Therefore
~:
~(gl
group
In p a r t i c u l a r
Let
to s i m p l i f y
~(0,h)
K.
= ~(g,0)
A = 9 A gEG g
the n o t a t i o n ,
a, d e g r e e
graded
algebra
x
A
or
of
be
such
A
This
must be
of
G
zero.
in
means and for
g i ' h.1
in
G.
= I.
a G-graded
we
A
algebra.
shall write
~(a,b)
In o r d e r
for
b).
~
be
is
central
that,
ax = ~(a,x)xa
be
of
a pairing
= ~ ( g , h I) ~ ( g , h 2)
Let n o w
of
K*
+ g2 'h) = ~ ( g l 'h) ~ ( g ~ , h )
~ ( g , h I + h 2)
~(degree
the r a d i c a l
G x G ~
the m u l t i p l i c a t i v e
A
a simple
K A
Proof.
of
-
1.1
Let
A
119
a fixed pairing. if
the
xa = ~(x,a) ax
We
say
that
only homogeneous
the elements
for all h o m o g e n e o u s
for a l l h o m o g e n e o u s
a
in
A
a
in
are in
K.
-
Examples of ~raded 1.2
central
Let
G
trivially
on
class of
H z (G,K*).
Take
al~ebras
be the second
coefficients
K*.
-
simple
H 2 (G,K*)
group of
on the space
with
1 2 0
in
K*, where
a normalized
We define
of all formal
cohomology
cocycle
an associative
linear
G
operates
f
in any
multiplication
combinations
g6G g g with
coefficients
~
6 K
g
by setting
x We shall denote KfG.
If
f
group algebra cohomologous resulting K.
See
g~hf (g,h) Xg+h
=
g6G g g
h6G
g,h6
the G-graded
is the trivial KG
of
G
cocycles
algebra
thus
cocycle,
KfG
over
define
class is called
by
is simply
the
We remark
isomorphic
that two
graded
an algebra e x t e n s i o n
algebras. of
G
over
[8] for more details. An algebra e x t e n s i o n
Suppose
K.
constructed
now that
cocycle,
G
is abelian
and that
graded
f
simple.
is an abelian
i.e. f(g,h)
The algebra e x t e n s i o n satisfies
is obviously
= f(h,g) is central
the following Non-de@eneracy
g ~ 0, there exists
g,h 6 K
.
if the given p a i r i n g
condition: condition:
h 6 G
such that
for every ~(g,h)
g 6 G, ~ i.
The
121
-
In p a r t i c u l a r ,
the
group
algebra
then
is g r a d e d
central
simple.
It is e a s y if the means
group
G
that,
is
if
a primitive
G
n-th
to c o n s t r u c t finite
has root
direct
Pi p r i m e s Choose the
of
i).
1
a primitive
p~i-th
where
gi
s ~'
component 9:
of
g
)
K*
G • G
satisfying version ing.
of this
ug =
~u gi i i
space
root
if
~i
and
by
we
V K,
(this
K contains
be G
P l ~ P2 ~
all
i
=
of the
appearing
(*). = u gh
condition.
in
of
of the i - t h
The m a p is
a pairing
In the
only
the u s e
this
an a b s t r a c t
group.
therefore to see
first
special
dimensional
abelian
difficult
class
The
pair-
pairing.
G-graded algebra
graded
simple.
that
EndK(V)
"'''
"'" = Pq"
u gh = ~ugi hi i 9
a finite
any
that
de fine
considered
graded,
it is n o t
then
P m = Pm+l
for
~(g,h)
suggested
Let
G-cyclic
n,
such
the d e c o m p o s i t i o n
paper,
over
G
g,h 6 G
defined
is s i m p l e ,
Furthermore
of
a representative
Andr~
1.3.
EndK(V)
For
in
is
of o r d e r
" > rq
the n o n - d e g e n e r a c y
Michel
vector
(*).
is
K
G = 9 z/prlz
rm
sum
and
pairing
Let
sum decompositioa
and
direct
abelian
an e l e m e n t
(*) be t h e
a nondegenerate
is
-
graded
central
(for any p a i r i n g
Remark. algebraically a graded
A
is
which
9).
a finite
T h e map
V.
n-th
to
9 A
graded
J
is t r i v i a l ) .
automorphism
The
and
K
show that
(non-graded)
EndK(V)
is
is
for s o m e
as f o l l o w s 9 n
Let
is the o r d e r
defined by
---> ~ g a g
The r e s u l t
group
can e a s i l y
simple
One p r o c e e d s
,
g is a n o n - t r i v i a l
t h e n one
r o o t of I, w h e r e
~: A : a
cyclic
is c e n t r a l
algebra
space
be a primitive G.
G
as a g r a d e d
G-graded vector
of
If
-
c l o s e d and G - c y c l i c ,
algebra
isomorphic
1 2 2
of
ag 6 A g
A
is c l a s s i c a l
(if
A
is n o n - t r i v i a l l y
if the g r a d i n g
a l g e b r a is c e n t r a l
simple,
of
A
therefore
is i n n e r , Sag = u a ~ u -I for some
u 6 Ao
The e l e m e n t
scalar
multiple.
of
is c o m p l e t e l y
A
By d e f i n i t i o n
Ag = In p a r t i c u l a r , ~n
is t r i v i a l ,
to an e l e m e n t
w e see
determined
with
K
by
u,
A
of
u belongs
K.
to
is c e n t r a l Since
u
w e may a s s u m e
is a l g e b r a i c a l l y
the e n d o m o r p h i s m
4, the g r a d e d s t r u c t u r e
that
un
closed,
algebra
up to a
of
9
therefore ~ ~ 0
is d e t e r m i n e d
{a s A I ~ g a u = ua}
up to a s c a l a r m u l t i p l e , field
u
End
(V)
The a u t o m o r p h i s m
a n d thus is e q u a l
is o n l y d e t e r m i n e d that
hence we K
o
u n = i.
The
can i d e n t i f y
of a v e c t o r
space
A
-
V
over
K.
endomorphism Qu(t)
are
1 2 3
-
The m i n i m a l p o l y m o n i a l u ~g,
is
clearly equal
g 6 G.
Q
(t)
of the
t n - i.
to
The r o o t s
of
Set
Vg = {x 6 V I u(x) Since
Q u (t)
is a p r o d u c t
= ~gx}
of distin~
9
linear
factors
over
U
K, V=|
. g6G g
Finally
one v e r i f i e s
Ag = {
that
End K
l ~: V h ~
V g + h , V h 6 G}
as it s h o u l d be. 2% G R A D E D T E N S O R
A graded a igeb ras
A
and
tensor product B
is the u s u a l
PRODUCT
A | B
of two G - g r a d e d
graded vector
space
A @ B = ~ (A @ B) g6G g whe re (A @ B) g = h+h'| = g A h with
|
KBh S
the m u l t i p l i c a t i o n (a | b)
(a' | b')
a,a I 6 A The m a p defined
~
is a p a i r i n g
= ~(b,
and of
G
b,b' in
a j
)aa'
6 B K*.
| bb' .
The p r o d u c t
thus
is a s s o c i a t i v e .
Example.
Let
G = ~/2Z.
The u s u a l
graded
tensor
-
p r o d u c t corresponds
Let
~
to
respect
and
over
9.
@ C --~ A @ (B | C)
b u t in general not commutative,
V
to
All t e n s o r Let
A, B and
The tensor p r o d u c t is associative,
(A @ B)
v e c t o r spaces
(-1) gh.
now be a fixed pairing.
be graded algebras.
Let
-
~ (g,h) =
p r o d u c t s w i l l be taken w i t h C
124
Va
K.
be
,
A @ B ~ B | A.
finite d i m e n s i o n a l
The algebra
EndK(V)
G-graded
is g r a d e d and
~ n ~ (v~ ~ ~ n ~ (v') -~ ~ n ~ (v ~ v' ) if we de fine (f | f')
Proposition
A
A | B
and
x) fx | f'x'
B
are graded central simple
algebras,
is also graded central simple.
Proof. for example
~(f'
2.1
If then
(x @ x')
in
The proof
for the n o n - g r a d e d
cases
given
[4], can be used w i t h some trivial
modifications.
Let A*
of
A
A
be a graded
is i d e n t i c
to
A
algebra.
The opposite
as a vector space
algebra
and has the
multiplication a'b* = ~ ( a , b ) b a The algebra
A*
is obviously
.
central simple if
A
is so.
-
1 2 5
-
There is a n a t u r a l h o m o m o r p h i s m ~: A | A* ~ d e f i n e d by
of graded
algebras
EndK(A)
(a | b*)x = ~(b*,x) axb
if
G
is abelian.
P r o p o s i t i o n 2.2 is an i s o m o r p h i s m
The n a t u r a l map central
A
is
simple.
Proof. example
The same
as in the n o n - g r a d e d
See for
case.
[4].
3,
A STRUCTURE
A pairing ~(g,h)
if
= ~(h,g)
T H E O R E M FOR G - G R A D E D ALGEBRAS
is called s y m m e t r i c
q0: G • G --~ K*
for all
g,h
in
Suppose
G.
that
G
if
is abelian.
T h e o r e m 3.1
Let
~
be a s y m m e t r i c p a i r i n g
the n o n - d e g e n e r a c y of
G.
Then a graded
char K ~ d i m K A
H
and
H'
f
to a graded
Kf(H)
)
| of
graded by
G
is an abelian
G
A
such that
algebra of the type
.
such that
class
in
extension
H 2 (H,K*) of
H
is a m a t r i x a l g e b r a over a d i v i s i o n H'.
satisfying
subgroup
algebra
is i s o m o r p h i c
is the c o r r e s p o n d i n g
~n(D,H')
for any finite
central simple
are subgroups
G ~ H x H'. Kf(H)
condition
of
over ring
and K. D,
-
Proof. Z = The s p a c e
H =
Z
of
Therefore
Z
0
be
Z is
contains
is the
contained
have
Zh =
h
G,
defines
an a b e l i a n
Another
choice
of
cohomology
9
is
graded
subgroup
the
graded
Zh, h
is d i f f e r e n t
hence
simple. H
dimensional.
in
~,
homogeneous
in e a c h
zero
n a n d is
We
~ ~ 0.
Z h.
of
If
Zh, h 6 H,
all
space
center
6 H. from
~ 6 K,
The
are
We know
= f ( h , h ' ) X h + h,
x's h
class.
A,
the e q u a l i t y
2-cocycle the
be
xh ~ 0
therefore
a 6 A}
The
in
~n =
Z0x h = Kxh,
A
finite
xn h
of
Z = @ Zh, w h e r e h6H
G.
x h ~ 0~
in
all
center
of two n o n - z e r o
contained
X h X h,
same
of is
Choose
# 0,
for
since
Z 0' t h e r e f o r e
ZhX h c
Xh~,
A
Let
dimensional.
that
zero,
a n d is
of
in
the u n g r a d e d
The p r o d u c t
because K
-
graded,
a subgroup
Z 0 = K.
order
be
certainly
is n o t
finite
A, h e n c e
one
is
H
Z
{x 6 A I xa = ax
{h 6 G I Zh ~ 0}.
elements
must
Let
126
f
of
would
We have
H
with
define
thus
value
in
a cocycle
constructed
K*.
in the
the p a r t
Kf (H). Let n o w
B
B = zx
As
zx = xz
for
=
be
the
graded
centralizer
{x 6 A I xz = ~ ( x , z ) z x ~(z,x)xz,
all
z
in
Vz
Z,
E
z}
9
Vz 6 Z
of
Z or
in
A,
127
-
B = (9
{x 6 A I ~(z,x)
is s y m m e t r i c
element
! ).
f 6 HOmZ|
= 1
We s h o w
(Z ,A)
f((z I @ z*)x) 2
-
for all
that
is a m a p
z 6 Z}
B--~ H O m Z | f: Z
An
> A
such
= ~ ( f , z I @ z*)z | z*f(x) 2 I 2 = ~0(f,z I | z ~ ) q 0 ( z 2 , f x ) z l f ( x ) z 2
L et
P: H O m Z |
It f o l l o w s zf(1) B.
(Z ,A)
from
) A
= ~ ( z , f ( 1 ) ) f(1) z
for all the map
= zb, b 6 B, is a
non-degeneracy
condition
graded
central
simple
Hence
B ~--
n ~ (z)
z fb:
in
~
f(1).
that
Z, h e n c e
Z ~
f
A
f(1)
is i n
d e f i n e d by
Z | Z*-homomorphism.
By h y p o t h e s i s ,
Hom E
the m a p d e f i n e d b y
f((z | i) i) = f ( ( 1 | z) l)
On the o t h e r side,
f(z)
be
the p a i r i n g on
H.
~
satisfies
Therefore
and, by P r o p o s i t i o n (Z,A).
the
Z = Kf (H) 2.2,
By P r o ~ o s i t •
is
Z | Z* = E n d K ( Z ) .
A6 of
[2],
the map (Z,A)
: Z | K g i v e n by
0 (x | f) = f(x)
over
The
K.
corresponding
x | f(1)
=
which
an i s o m o r p h i s m
Z | B,
is ZB
(x | l) f(1)
and
A.
~ A
E n d K (Z) is an i s o m o r p h i s m isomorphism
= xf(1)
(see
of a l g e b r a s .
of v e c t o r
spaces
Z | B ---~ A is d e f i n e d b y
[2'] T h e o r e m Hence we
3.1.)
can i d e n t i f y
-
Let now or
x
be
in the g r a d e d
bx = ~(b,x)xb
xz = zx or
for
for
of
central, be in
because and
not difficult
therefore
is g r a d e d
and
Z
to see t h a t
B, xb = ~ ( x , b ) b x M(x,z)
= 1
and
xzb = ~ ( x , z b ) z b x x
is in
central.
any e l e m e n t B
of
Since
z 6 Z, w e can w r i t e
A, B
Z,
-
center
b 6 B.
zbx = ~(zb,x) xzb,
center
128
B
K,
the g r a d e d
is a l s o u n g r a d e d
of the c e n t e r
of
B
would
K.
It is
are d i s j o i n t
over
B
simple.
is g r a d e d
Since
char B
K ~ dimKA , char K ~ dim B a n d b y P r o p o s i t i o n 1. i., K is s e m i s i m p l e . A s e m i s i m p l e c e n t r a l a l g e b r a is s i m p l e ,
hence
B
is
(ungraded)
Let H' = of
H'
be
{g s G I ~(g,h) G.
Suppose
there exists
of
B,
Zh
know
Z
and
B
H N H' = 0. G=
Remark
would
B,
is a s u b g r o u p
h 6 H N H', h ~ 0. be
contained
are d i s j o i n t
that,
H'
of
if
over
Ag ~ 0
By
in
~,
K.
Hence
for a l l
but we
g 6 G, t h e n
H • Kso
Suppose The c o n s t r u c t i o n the n o n d e g e n e r a c y if
the s e t of d e g r e e s
= i, V h 6 H}.
definition that
simple.
K
showed
that
of E x a m p l e condition
is a l g e b r a i c a l l y in E x a m p l e
K
is a l g e b r a i c a l l y 1.1 g i v e s
closed
a pairing
for
Ho
Furthermore
closed.
If
G
1.2 t h a t
M
n
(K,H')
and G-cyclic.
satisfying H z (H,K*)
is c y c l i c , w e is the e n d o m o r p h i s m
= 1
129
-
algebra
of a vector
Corollary
order.
simple
to an algebra
be a l g e b r a i c a l l y
Suppose A
that
K
such that
End K(V)
2)
KG | End K(v)
where
V
Remark.
The hypothesis
Let
3.2.
~: G • G---~
G-graded
central
if there exist
simple
central
G
is ungraded.
that
above.
char K ~ dimKA
A direct proof
K*
4 in
is
can be
[7].
GROUP
be a fixed p a i r i n g
field
algebras
two G-graded
of
is isomorphic
immediately
of Lemma
in a commutative
A G-graded
End K(V)
from the remarks
the proof
G cyclic
is graded by
where
in C orollar~
and
types:
follows
following
G
closed
char K ~ dimKA
4. THE B R A U E R
group
H'.
is G-cyclic.
of the following
i)
not necessary given,
K
algebra
The proof
graded by
3.2 Let
prim~
space
-
vector
K. A
of abelian
We say that two and
B
spaces
are similar V
1
and
V
2
such that
A
En
Cv I)
as graded
algebras.
Let
classes.
By P r o p o s i t i o n
induces
a group structure
B
B(K,G)
En Cv2> be the set of e q u i v a l e n c e
2.1 and 2.2, on
B(K,G).
the tensor p r o d u c t The classical
-
(nongraded)
Brauer
group
1 3 0
-
is certainly
B (K)
c o n t a i n e d in
B (K,G). Exan~le and
G
that
4.1,
Let
cyclic of prime B(K,G)
of
of primes
order.
n
copies
of
9
Let
G
field.
Let
K G
contains
ALGEBRAS
abelian
group
and ~
K
a
constructed
be the n - f o l d graded
tensor
n
If
G
to
~)
of the group algebra
is cyclic
el, i = i, 2, m
of order
...n
is a p r i m i t i v e
subject
KG is d e s c r i b e d
to the relations
n
i < j,
m-th root of i.
cyclic, was
m, KnG
i = i, 2,...,
eie 4 = ge.e. ]l J
K G, G
B(K,G)
G.
OF C L I F F O R D
be a finite
ei = 1
of
is any finite
m
(with r e s p e c t
by generators
The
(non-graded)
d e s c r i b e d by Morris
structure
[6], w h e n
m
n
odd or
m
is even
and
shall give a complete abelian
groups.
3.2
is the n u m b e r
We shall use the p a i r i n g
in E x a m p l e 1.2.
w i t h itself.
G
n
closed
from C o r o l l a r y
If
Z/2Z, w h e r e
a p p e a r i n g in the order of
i
product
It follows
char K = 0, one can see ~hat
5. A G E N E R A L I Z A T I O N
G-cyclic
be a l g e b r a i c a l l y
is cyclic of order two.
cyclic group and product
K
~
has
a square r o o t in
description
for a r b i t r a r y
K.
We
finite
is
a
-
Let that
KG*
KG*
131
-
be the opposite
is t_h~ algebra
algebra
extension
of
of
G
to the cocycle
9)
n-fold
tensor p r o d u c t
KG* | . .. | KG*.
K G n
and
K G* q
are connected KnG* nG means
The proof
[i].
Clifford
algebras
K n G*
by the following
l| K2G*
ungraded
The algebras relations:
Kn+2G*
to the proof
given
and
and
!) tensor products.
(without
The algebras C ni
K
[| I K2G ~- Kn+2 G
[5] is similar
algebras
.
(remark
for the
corresponding graded
Write
over
KG
if
Cq
for Clifford
~G*
G
are the
is the cyclic
group
of order two.
KqG*, for
TO describe
completely
the algebras
KnG
and
it is therefore
sufficient
to know
and
KiG*
i = 1,2.
Using
the results
KIG = K • ...
KiG
of Morris
x K
[6], one obtains
n copies
K2G = M n (I{) n
G.
is the order of
that cyclic
KiG = KiG*.
If
n
is odd,
A more i n t e r e s t i n g
of order a power
contains
a primitive
not have
a square
of 2,
2r-th
root in
KIG* = K(~)
x...
K.
case is w h e n
n = 2 r.
root of
it is easy to see
We k n o w
1, ~.
that
Suppose
Then
• K(~)
2 r-I
copies
is
G K
does
-
where ~2
K (~ )
132
is the q u a d r a t i c
-
extension
of
K s u c h that
~.
=
K2Ge = ~42r-- I ( ~ ) where
is the q u a t e r n i o n i c
generators
and
n
algebra
subject
If
,,, h a s
results
a square
and p r o o f s
root in are g i v e n
given by
and
9
K, t h e n in
K
to the r e l a t i o n s
~2 = n2 = ~ ~n = -n~
over
[5].
K i G = KiG*.
Complete
-
1 3 3
-
REF~
[i]
Atiyah, M. , Bott, R. , and Shapiro, A. , "Clifford modules",
Topology, 3, [2]
Auslander,
(Supplement i) ; 3-38.
Mo, and Goldman,
(1965).
O., "Maximal orders",
Trans~ Amer. Math. Soc., 97; 1-24.
(1960).
[2~' ] Auslander, M. , and Goldman, 0. , "The Brauer group of a com/~utative ring", Trans. Amer. Math. Soc., 97; 367-409.
(1960).
[3]
Bass, H., Lectures on algebraic K-theory, Bombay,
[4]
Herstein,
Z., Noncommutative rings, Carus Publ. #15, MAA,
Providence,
[5]
(1968) .
Knus, M. A., A generalization of Clifford algebras, i0 p.
[6]
(unpublished) .
Morris, A. M., "On a generalized Clifford algebra",
Quart. ~. Math., [7]
(1967).
(Oxford Ser.), 18; 7-12. (1967).
Wall, C. T. C., "Graded Brauer groups",
J. Reine Angew.
Math., 213; 187-199. (1964).
[8]
Yamazaki, K,, "On projective representations extensions of finite groups", J. Fac. Sci.
Sect. I, i0; 147-195. (1964).
and ring
Univ. Tokoyo,
134
-
Diagonal Arguments
-
and Cartesian Closed Categories
by F. William Lawvere
The similarity between the famous arguments of Cantor, is well-known,
and suggests that these arguments
Russell,
G~del and Tarski.
should all be special cases of a sin-
gle theorem about a suitable kind of abstract structure.
We offer here a fixed-point
theorem in cartesian closed categories which seems to play this role. Cartesian closed categories
seem also to serve as a common abstraction of type theory and propositional
logic, but the authorts discussion at the Seattle conference observation will be in part described pear in Dialectica, Adjoint Functor",
elsewhere
["Adjointness
and "Equality in Hyperdoctrines
of the development in Foundations",
and the Comprehension
of that to ap-
Schema as an
to appear in the Proceedings of the AMS Symposium on Applications
of
Category theory]. 1.
By a cartesian closed category is meant a category C equipped with the follow-
ing three kinds of right adjointsz
a right adjoint 1 to the unique C
a right adjoint • to the diagonal
+ I
,
functor C ~ CxC,
and for each object A in C, a right adjoint C Ax(~ The adjunction transformations be denoted by
6,x
( )A to the functor C.
for these adjoint situations,
in the case of products and by
by A. Thus for each X one has XAA X
A >(AxX)
and for each Y one has Ax~ Given
f-AxX ~ Y, the composite morphism
YEA~Y.
AA,~ A
also assumed given, will
in the case of exponentiation
-
XAA~
(A•
x
will be c a l l e d transform
the
"A-transform"
135
-
A
of the m o r p h i s m
f. A m o r p h i s m
h:X ~ yA
is the A-
of f iff the d i a g r a m AxX
YeA is c o m m u t a t i v e , transform.
showing
Taking
cf~:1 ~ ~
in p a r t i c u l a r
the case
(d~pping
1 -- ~
is of t h a t
the indices
A , Y on
< a,~ one calls sume
~ the
in g e n e r a l Although
categories
"evaluation" that
natural
f is d e t e r m i n e d
we do n o t m a k e
as a l g e b r a i c
f can be u n i q u e l y
X = 1, one has t h a t e v e r y
and that e v e r y
a:l ~ A one has
that
form
~ when
f:A ~ Y
gives
for a u n i q u e
f.
from its ~rise
to a u n i q u e
Since
for e v e r y
t h e y are clear)
> ~ = a.f,
transformation!
note h o w e v e r
b y the k n o w l e d g e
use of it in this paper,
versions
recovered
of type t h e o r y
c a n be
of all
its
t h a t we do not as"values"
the u s e f u l n e s s further
a.f.
of c a r t e s i a n
illustrated
closed
by assuming
t h a t the c o p r o d u c t 2 = I+1 also exists
in C. It then
follows
(using
the c l o s e d
structure),
that
for e v e r y o b j e c t
A A• and so in p a r t i c u l a r
that
= A+A
2 is B o o l e a n - a l g e b r a - o b j e c t
in C, i.e.
that among
the m o r -
phisms 2x2x...x2 in C there truth
are w e l l
tables,
of B o o l e a n
determined
and that these
algebra.
morphisms
satisfy
Equivalently,
"C-attributes
X along
for e a c h X the
of type X" b e c o m e s
any morphism
of C induces
corresponding
to all the
all the c o m m u t a t i v e
Pc(X) of
~ 2
diagrams
finitary
(two-valued)
expressing
the axioms
set
= C(X,2)
canonically
an actual
contravariantly
Boolean
a Boolean
algebra,
homomorphism
and v a r y i n g
of a t t r i b u t e
136
-
algebras. among
The m o r p h i s m s
these
For a n y
I ~ 2
form
are the two c o p r o d u c t
"constant
truth-value.
Now noting
PC(1)
the B o o l e a n
injections
x:l ~ X
of type X"
-
algebra
of " t r u t h - v a l u e s " ;
w h i c h p l a y the r o l e s o f
and a n y a t t r i b u t e
"true"
and
"false",
~ of type X, x . ~ is t h e n a
that X• x
> 2 (2) CX
is a " b i n a r y o p e r a t i o n "
we c o u l d w r i t e
it b e t w e e n
X E ~'I
an e q u a l i t y of t r u t h v a l u e s ; the s u b s e t presses
of X c o r r e s p o n d i n g
the u s u a l
Returning surjective
iff for e v e r y
domain
an e v e n w e a k e r
for our
fixed p o i n t
there
exists
of the
x:l - X
of Z",
since
section)
inverse
n o t i o n of s u r j e c t i v i t y theorem.
a morphism
g is p o i n t - s u r j e c t i v e
from an e l e m e n t
as the c o n s t a n t
naming
@, o n e sees that the a b o v e e q u a t i o n
we d e f i n e
(as in the n e x t
transformation
limit of Z comes
concern,
"onto the w h o l e if
c~1 :i ~ 2 X
ex-
axiom.
z:i ~ Z
I; for e x a m p l e
then a natural
~,
to the a t t r i b u t e
to our i m m e d i a t e
so t h a t w e h a v e
= X.q),
if w e t h i n k o f
"comprehension"
i m p l y that g is n e c e s s a r i l y with
thus
its a r g u m e n t s ,
g:X ~ Z
with
xg = z. T h i s
t h e r e m a y be
X and Z are
does n o t
few m o r p h i s m s
set-valued
if e v e r y e l e m e n t
limit o f X.
to be p o i n t -
functors,
o f the i n v e r s e
In c a s e Z is o f the
c a n be c o n s i d e r e d ,
which
form
in fact s u f f i c e s
Namely g x
w i l l be c a l l e d w e a k l y p o i n t - s u r j e c t i v e for e v e r y
iff for e v e r y
there
is x s u c h that
a:l ~ A ( a,xg
>e = a . f
F i n a l l y we say t h a t an o b j e c t Y has the morphism Theorem
f:A -- Y
t:Y ~ Y
there
In a n y c a r t e s i a n
point-surjective
is
y:l ~ Y
closed
with
category,
fixed-p.oint p r o p e r t y
iff for e v e r y e n d o -
y.t = y. if t h e r e
exists
an o b j e c t A and a w e a k l y
morphism g
t h e n Y has Proof:
the f i x e d p o i n t p r o p e r t y . Let ~ be
the m o r p h i s m
whose
A-transform
is g. T h e n
for a n y
f:A ~ Y
there
-
is
xtl ~ A
such that
for all
any endomorphism
t of Y a n d
>g = a.f. let f b e the c o m p o s i t i o n
A6 A thus t h e r e
is x s u c h t h a t
a(Ab) The
=
< a,a
famed
Cantor's
theorem
Corollary
then
If there
A does
>. But t h e n
"diagonal
there
type
theory!
argument.
argument"
< a,a
exists
< x,x
>gt is c l e a r l y
a fixed p o i n t
just the c o n t r a p o s i t i v e
t:Y ~ Y
such that
yt 9 y
for all
could have been
relation,
( no e x p o n e n t i a t i o n )
v i t y as a p r o p e r t y
y:l ~ Y
then
for no
morphism).
that
set t h e o r y be f o r m u l a t e d
universe,
w e do n o t n e e d
In fact w e n e e d o n l y a p p l y the p r o o f o f our theorem,
ly, our t h e o r e m
o f our theorem.
morphism
for A the s e t - t h e o r e t i c a l
membership
for t.
Y = 2.
P a r a d o x d o e s not p r e s u p p o s e
t h a t is,
>Y!
>g
is o f c o u r s e
follows with
set-theoretical
products
>g =
exist a point-surjective
Russells
t >Y
y =
( or e v e n a w e a k l y p o i n t - s u r j e c t i v e
2.
g >AxA
for all a < a,x
since
-
a:l ~ A < a,x
Now consider
1 3 7
dispensing
with
with g entirely.
stated and proved
by simply phrasing
That
2 A for the
g:AxA ~ 2
as the
is, m o r e
general-
in a n y c a t e g o r y w i t h o n l y the n o t i o n
of
(weak)
as a h i g h e r
finite
point-surjecti-
of a m o r p h i s m A x X ~ Y!
however
discovering
require
thinking
elements
of X,
the l a t t e r
form
(or at l e a s t c a l l i n g
of such a m o r p h i s m
suggesting
it s u r j e c t i v i t y ! )
as a f a m i l y of m o r p h i s m s
that a closed category
is the
A ~ Y
"natural"
seems
to
i n d e x e d b y the
setting
for the the-
orem. In fact the m o r e products)
follows
the f o l l o w i n g full c l o s u r e
form of the t h e o r e m
from the c a r t e s i a n
remark. under
general
Notice
that
finite p r o d u c t s
just a l l u d e d
to
(for c a t e g o r i e s
closed version which we have proved,
it w o u l d
suffice
to a s s u m e
of the two o b j e c t s
A,Y)
C small
with
by virtue
(just take the
of
-
Remark
138
-
Any small category C can be fully and faithfully embedded in a cartesian closed
category in a manner which preserves any products or exponentials which m a y exist in C. Proof:
We consider the usual embedding C~__ ~ Cop
which identifies an object Y with the contravariant set-valued functor x ~
(x,Y).
By "Yoneda's Lemma" one has for any functor Y and any object A that the value at A of Y AY ~ ~C~ where the right hand side denotes the set of all natural transformations
from
(the
functor corresponding to) A into Y, so that in particular the embedding is full and faithful.
It is then also clear that the embedding preserves products
(in particular
if 1 exists in C it corresponds to the functor which is constantly the one-element set, which is the 1 of ~C~
For any two functors A,Y the functor C~C~
plays the role of ~ .
In particular if B A exists in C for a pair of objects A,B in C
then (C)BA - C(C,B A) J C(AxC,B)
-~COP(AxC,B)
showing that the embedding preserves exponentiation. Theorem
Let A,Y be any objects in any category with finite products
(including the
empty product I)~ then the following two statements cannot both be true a) there exists such that for all
~:AxA ~ Y
such that for all
f:A ~ Y
there exists
x:l ~ A
a=l ~ A < a,x >~ = a.f
b)
there exists
t:Y ~ Y
such that for all
y:l ~ Y
y.t&y. Proof:
Apply above remark and the proof in the previous section.
Of course the "transcendental" proof just given is somewhat ridiculous,
since the in-
compatibility of a) and b) can be proved directly just as simply as it was proved in the previous section under the more restrictive hypothesis on C. However we wish to
139
-
take the o p p o r t u n i t y o f an a r b i t r a r y denote
to m a k e
(small)
the s m a l l e s t
"definable"
braic ries
with definable alternative are
it u s u a l l y
to be at l e a s t p a r t l y definition.
a natural
rator have
to the)
call a natural
transformation given objects
operator
for all of $ C o p u n l e s s a partial
for C itself, Recall
result
is d e t e r m i n e d
c a l l the e l e m e n t s
(let the
latter
which
contains
C). One
structure
is d i f f i c u l t
theories
all p o s s i b l e
semantics
of alge-
if the e l e m e n t a r y
transformations for d e f i n a b l e
are o b j e c t s
is to con-
to o v e r s e e
enumerating
functorial
in the p r e s e n t
case.
theo-
are i d e n t i c a l ones.
Thus
The
latter
for e x a m p l e w e
in a c a t e g o r y
"values"
of the set
between
the e x p o n e n t i a l
in $ C~
(hence in C).
functional.
C = I~ h o w e v e r
in m o r e
"1 is a g e n e r a t o r
b y its
C
C with
fi-
of the
(func-
>D C
a natural
in t h a t d i r e c t i o n .
we c a n d e s c r i b e
that
(e.g.
embedding
operator
shall be s i m p l y a n a t u r a l
we would
this
substitute
If A , B , C , D
BA
tors c o r r e s p o n d i n g
of ~ C o p
that n a t u r a l
true
category
requires
of elementary
ones or at l e a s t a r e a s o n a b l e
led to the f o l l o w i n g
closed
canonical
into a h i g h e r - o r d e r
situations
semantics
the a b o v e
etc.! h o w e v e r
one h a s c o m e to e x p e c t
seems
nite products,
in m a n y
about
subcategory
a structure
p o i n t of v i e w since
or f u n c t o r i a l
are complete)
closed
operators,
On the o t h e r h a n d
theories
remarks
into a c a r t e s i a n
of e m b e d d i n g
functionals,
from a s i m p l e - m i n d e d definitions.
category
further
full c a r t e s i a n
of the s t a n d a r d w a y s sider
some
-
it m i g h t
In fact,
familiar
Note
C(I,X)
for
of p o i n t s
in p a r t i c u l a r
that
1 will
conceivably
be
if
C = I
not be a g e n e -
so for C, and we
in the c a s e t h a t 1 is a g e n e r a t o r
terms w h a t
for C" s i m p l y m e a n s
x.f:1 ~ Y
functors
a natural
operator
that a morphism
x : l ~ X.
In that c a s e
is.
f:X ~ Y
in C
it is s e n s i b l e
of X a l s o the e l e m e n t s
of X. T h e n a
function C(1,X) is i n d u c e d b y at m o s t o n e C - m o r p h i s m language
t h a t the
Proposition rator,
operator
are o b j e c t s
it is, w e
say by abuse of
o f C.
t h a t C is a c a t e g o r y w i t h
and t h a t A , B , C , D
I) a n a t u r a l
X ~ Y, a n d in c a s e
f u n c t i o n i_~s a m o r p h i s m
Suppose
~ C(1,Y)
finite p r o d u c t s
of C. T h e n
in w h i c h
I is a g e n e -
to
-
140
-
r
BA
>D C
is entirely determined by a single function Ir C (A,B)
> C (C,D)
9and 2) such a function determines a natural operator iff for every object X of C and for every C-morphism
fsAxX ~ B, the function (f) (Xr
C(1,CxX) is a C-morphism, where
(f) (Xr
~ C(1,D)
is defined by
,o,x
for any
=
co,
c:l ~ C, xzl = X, f denoting the composition x A
Proofz
f
~AxX ~AxI ~ Axx
> B.
We are abusing notations to the extent of identifying a morphism with its
A-transform via the bijections of the form
C(AxX,B) ~ C(AxX,B) ~ C(x,BA). Actually the given operator
r is a family of functions Xr C(X,B A)
> C(X,D c)
one for each object of C! the "naturalness" condition which this family must satisfy, is, via the abuse, that for every morphism
xsX ~ ~ X
of C,
the d i a g r a m
Xr
C(A~X,B)--- ~ C x C(A•
should commute. Now let tion Xr at a given
(CxX,D)
, B ) - - - ~ C ( C x X ' ,D) X'r
X' = I. Since I is a generator for C, the value of the func-
fzAxX ~ B
is determined by the knowledge,
for each element x of X
and element c of C, the result reached in the lower right hand corner by going across then down in the commutative diagram
-
141
-
Xr
c
C (AxX,B)
> C (CxX,D)
C(A,B)
> C (X,D)
>C(C,D)
But since
the
same r e s u l t s
X ~ are d e t e r m i n e d tion
is t h e n clear,
proposition
since
is just
To m a k e exponential
object
perfectly
c a n be d i s c u s s e d
thing
to do.
sional m a n i f o l d s categories
3.
Experts may wish
in the above
fixed-point
theorem
In order concerning
itself,
we
ducts. ples
of)
Thus
iff their
the m o r p h i s m s
are
(classes
of)
of)
formulas
with
ses of)
sentences
responding falsez
1 ~ A
(classes
of the theory.
rise
two
of)
constant
provable
to the class
There
is an
is h o w e v e r
and m o s t
to c o n s i d e r
then
and c o n s i d e r -
between
"natu-
finite-dimenparticular
whether
to o b t a i n
the
Tarski's
for a t h e o r y w i t h i n to a c a t e g o r y
C with
be e q u i v a l e n c e
formulas
terms,
in the t h e o r y whose
pro-
of
(tu-
in the theory.
An ~ 2
is a m o r p h i s m
finite
are c o n s i d e r e d
the m o r p h i s m s
morphisms
the t h e o r y
classes
(or terms)
is p r o v a b l e
of s e n t e n c e s
exist.
does not exist,
C to be these
while morphisms
there
of the
codomain
smoothest
section
in p a r t i c u l a r
In p a r t i c u l a r
asser-
in those cases.
truth
(or equality)
so that
of s e n t e n c e s
corresponding
where
two free v a r i a b l e s ,
n free v a r i a b l e s
to the class
I ~ 2
equivalence
are
terms w i t h
of taking
and let the C - m o r p h i s m s
of the theory,
whose
object
functions
of the p r e v i o u s
h o w a t h e o r y gives
A,2
logical
or C ~
the
second
statement
is an e x p o n e n t i a l ,
They may also wish
of d e f i n i n g
The
its v a l u e s
is a product.
domain
the r e s u l t
the t h e o r e m
in the
that m o r p h i s m s
one has a n y a p p l i c a t i o n s
the i m p o s s i b i l i t y
or terms
whose
given
the e x p o n e n t i a l
domain
all the f u n c t i o n s
first assertion.
of X r p r o v i d e d
notice
functions
considerations.
two o b j e c t s
formulas
equivalent
to c o n s i d e r
of s e c t i o n
the
is in m a n y c o n t e x t s
on r e c u r s i v e
first note b r i e f l y
Consider
whose
then across,
(f) (Xr
even t h o u g h
operators
to a p p l y
theorem
clear,
the m o r p h i s m s
ing t h e m to be the n a t u r a l ral"
of
naturality
instead morphisms
of d e t e r m i n i n g
down
1~, p r o v i n g
such as to assure
just b y c o n s i d e r i n g the p r o b l e m
function
b y going
the d e f i n i t i o n
the s i t u a t i o n
).
c
are o b t a i n e d
b y the one
!o
>C(1
1r
1 ~ 2 true:
and s i m i l a r l y negation
AxA ~ A are
(classes
are
(clas-
1 ~ 2
cor-
a morphism
is p r o v a b l e
in
-
the t h e o r ~ M o r p h i s m s make
2n ~ 2
no use of that e x c e p t If the t h e o r y
not n e e d
the n a t u r e
of those
correspond
to s u b s t i t u t i o n
~:A ~ 2
category
C with
theory.
Models
hom-sets
finite
introduction.
t h e o r y was
to those
with
a higher-order
arguments morphisms
We t h e n formula stant
arise
czl ~ A
~ 2
spelled
b u t we w i l l
such that
Anx2
~ not %
one, with
trivial
more
from the v a r i a b l e s
satisfaction
C ~.
We m a k e
cited
that the
of C are i s o m o r -
we could have
started
of a n y s i g n i f i c a n c e
note
no
b u t the cate-
assumed
all o b j e c t s
modifications
explicit
o f the
to the
that the p r o j e c t i o n
of the theory.
is d e f i n a b l e
in C such that
case
we get a
o f the two p a p e r s
tacitly
to
a unary
category
in the theory,
t h e o r y w i t h no c h a n g e
somewhat
with
functors
of C w e h a v e
composition
1 ~ 2, etc.)
the L i n d e n b a u m
to r e a d e r s
in w h i c h
that d e t e r m i n e s
Defining
composed
a~ not:
as c e r t a i n
w i l l be c l e a r
TM, but
one p o i n t
such that
a:l ~ A
by quantification
above
although
out above.
the s e n t e n c e
can t h e n be v i e w e d
or s e v e r a l - s o r t e d
say that
sat:AxA
2 ~ 2
2 = I+I,
w h i c h m i g h t be c a l l e d
in C i n d u c e d
form
fact that
w i t h not gives
single-sorted
To m a k e
An ~ A
not:
a constant
In our c o n s t r u c t i o n
of the
below.
is a m o r p h i s m
the
of this o p e r a t i o ~
a first-order
phic
to use
products
use h e r e of the o p e r a t i o n
in the
operations,
case:
not e x p l i c i t l y
of the t h e o r y
description
there
(for e x a m p l e
composed
gorical
all p r o p o s i t i o n a l
~:1 ~ 2
we w i l l
formula
include
for the f o l l o w i n g
is c o n s i s t e n t
for all m o r p h i s m s In p a r t i c u l a r
would
142
in the t h e o r y
for e v e r y u n a r y
for e v e r y c o n s t a n t
a the
formula
following
iff there ~:A ~ 2 diagram
is a b i n a r y there
is a con-
commutes
in C
a
i. < a,c
>A
>~
~
AxA
>2 sat
Here we i m a g i n e
taking
condition
traditionally
would
for c a G6del
number
be e x p r e s s e d
for
(one of the r e p r e s e n t a t i v e s
by requiring
that the
of)
~. The
sentence
a sat c w-~ a~ be p r o v a b l e
in the theory,
but
tegory
amounts
same thing.
this
to the
if C a r i s e s
from our c o n s t r u c t i o n
o f the L i n d e n b a u m
ca-
-
Combining the t h e o r e m Corollary
mean,
If s a t i s f a c t i o n
which
-
the a b o v e n o t i o n w i t h our r e m a r k a b o u t
of the p r e v i o u s
In o r d e r
143
section we have is d e f i n a b l e
immediately
often realizable.
to r e q u i r e Namely we
some
further
suppose
of c o n s i s t e n c y
assumptions
that there
and
the
in the t h e o r y t h e n the t h e o r y
to s h o w t h a t T r u t h c a n n o t be d e f i n e d w e
seems
the m e a n i n g
is n o t c o n s i s t e n t .
first n e e d to say w h a t T r u t h w o u l d on the theory,
is a b i n a r y
which
are h o w e v e r
term
subst AxA in C a n d a
("metamathematical")
>
A
binary relation
r ~ c(1,A)xc(1,2) between
constants
I)
For all
and s e n t e n c e s ~:A ~ 2
for w h i c h
there
is
the
c:l ~ A
following
holds.
such t h a t
for all
a:l ~ A
(a subst ic) F(a~) For e x a m p l e w e c o u l d o f the
sentences
applied unary
imagine
which
to a c o n s t a n t
formula
that
represent
dFu
means
G, a n d t h a t
a a n d to a c o n s t a n t
~, y i e l d s
the G 6 d e l
t h a t d is the G S d e l
subst
is a b i n a r y
c which happens
n u m b e r of the s e n t e n c e
number
operation
to be the G ~ d e l
of some one which,
when
number of a
obtained by substituting
a
into ~. Given a binary relation definable
in the t h e o r y
F ~- C(1,A) xC(1,2)
(relative
to
we
F ) provided
say that Truth
there
is a u n a r y
(of sentences)
is
formula Truth:A ~ 2
such t h a t 2) Again
F o r all
~:I ~ 2
the t r a d i t i o n a l
and
formulation would ~ a~
be p r o v a b l e ,
but
= ~.
Theorem
If the t h e o r y
binary relation
F
to the same b i n a r y
Proof-
If b o t h
I) a n d
dF~
require
category
is c o n s i s t e n t between
if
T r u t h ~-* ~; for
in the L i n d e n b a u m
~Truth
relative
d:l ~ A,
this
then t h a t the ~
and
just a m o u n t s
t h e n the d i a g r a m
to the e q u a t i o n
is d e f i n a b l e
sentences,
relation 2) h o l d
sentence
F~
and substitution
constants
dTruth =
then Truth
relative
to a g i v e n
is not d e f i n a b l e
144-
-
a
i,
>A
Ax
A
> 2
subst shows
Truth
that subst AxA
> A Truth
is a d e f i n i t i o n
We w i l l predicate. vability
of s a t i s f a c t i o n ,
also prove
Given
contradicting
an " i n c o m p l e t e n e s s
a binary
relation
is r e p r e s e n t a b l e
theorem",
F between
in the t h e o r y
the p r e v i o u s
using
constants
iff there
result.
the n o t i o n
and sentences,
is a u n a r y
formula
of a P r o v a b i l i t y we
s a y that Pro-
Pr:A ~ 2
such
that 3)
Whenever
dFa
then dPr = true
Theorem
Suppose
that
C, s u b s t i t u t i o n not c o m p l e t e Proof: ency implies
for a g i v e n b i n a r y is d e f i n a b l e
iff
u = true
relation
and P r o v a b i l i t y
F between
constants
is r e p r e s e n t a b l e .
and s e n t e n c e s
Then
the t h e o r y
of is
if it is c o n s i s t e n t . Suppose that
By completeness
But a) a n d b')
b)
on the c o n t r a r y
false
9 true.
that
C(i,2)
Condition
= {false,true}.
3) states
that
for
a)
u = true
implies
dPr = true
b)
~ 9 true
implies
dPr 9 true
b')
~ = false
implies
dPr = false
dFu
implies
together
with
completeness
mean
that w h e n e v e r
1
A p----~-~r 2
dF~
Our n o t i o n
of c o n s i s t -
-
is commutative,
145
-
i.e. that Pr satisfies condition
2) for a Truth-definition,
which by
our previous theorem yields a contradiction.
Notem
Our proposition
in section two can be interpreted as a fragment of a general
theory developed by Eilenberg and Kelly from an idea of Spanier.
-
146
-
F O U N D A T I O N S F O R C A T E G O R I E S A N D SETS
by S a u n d e r s Mac Lane*
I
INTRODUCTION
A p r e s s i n g p r o b l e m c o n f r o n t i n g c a t e g o r y t h e o r y is t h a t of p r o v i d i n g an adequate, approaches
precise,
and f l e x i b l e foundation.
Two
are c u r r e n t l y in use; n e i t h e r is really s a t i s f a c t o r y . One c u r r e n t a p p r o a c h uses the c l a s s - s e t d i s t i n c t i o n
p r o v i d e d by G ~ d e l - B e r n a y s
a x i o m a t i c s e t theory.
Here a small
c a t e g o r y is d e s c r i b e d as a s e t of m o r p h i s m s e q u i p p e d w i t h operation
of c o m p o s i t i o n
satisfying
an
the u s u a l p r o p e r t i e s , w h i l e
a large c a t e g o r y is a class of m o r p h i s m s w i t h c o m p o s i t i o n , h a v i n g the same p r o p e r t i e s . familiar
large c a t e g o r i e s
gory of all
(small)
This a p p r o a c h does p r o v i d e (the c a t e g o r y of all sets,
groups,
and for the f u n c t o r c a t e g o r y does n o t p r o v i d e
and
for the
the cate-
the c a t e g o r y of all s m a l l categories) AB
for
B
for the f u n c t o r c a t e g o r y
small. AB
However,
for
B
it
a large
category. A n o t h e r a p p r o a c h uses G r o t h e n d i e c k ' s n o t i o n verse
U - - a set s u c h t h a t the e l e m e n t s
membership
relation
of Z e r m e l o - F r a e n k e l t h e n a set
x
x 6 y
x 6 U
between such elements
a x i o m a t i c set theory.
A
w i t h the g i v e n form a model
U - c a t e g o r y is
of m o r p h i s m s w i t h c o m p o s i t i o n s u c h t h a t
* U n i v e r s i t y of Chicago. The i n v e s t i g a t i o n s s u p p o r t e d by an ONR grant.
of a uni-
x 6 U.
reported here were
-
147
-
This n o t i o n is e s s e n t i a l l y t h a t of a s m a l l category;
(better,
a U-small)
the c a t e g o r y of all U - c a t e g o r i e s is then a U ' - c a t e -
gory for some
larger u n i v e r s e
f u n c t o r categories,
U'.
In this a p p r o a c h w e may
p r o v i d e d one adds to set t h e o r y
t h a t e v e r y set is a m e m b e r of a universe. a x i o m of infinity.
Also,
form
the a x i o m
This is a s t r o n g
on this approach, n o one to my k n o w -
ledge has a d e q u a t e l y e x a m i n e d the r e l a t i o n c a t e g o r y of all rings in one u n i v e r s e
U
(say) b e t w e e n
the
and that of all rings
in some l a r g e r universe. C o m m o n to b o t h a p p r o a c h e s is the i d e a of u s i n g categories of d i f f e r e n t sizes In effect,
(small and large,
this is a use of c a t e g o r i e s w h i c h
models of set theory
is o n l y one set theory,
concern repeatedly all sets,
set theory).
it is a c o m m o n d i c t u m that there
t h o u g h p e r h a p s one n o t y e t c o m p l e t e l y
d e s c r i b e d by the u s u a l axioms Kowever,
U, in U').
lie in d i f f e r e n t
(that is, in a "variable"
A m o n g a x i o m a t i c set t h e o r i s t s ,
extent).
or in
(Zermelo-Fraenkel
the s t a n d a r d d e v e l o p m e n t s
plus
axioms of
of c a t e g o r y
theory
c o n s t r u c t s s u c h as the c a t e g o r y of all groups,
or all c a t e g o r i e s .
The only v i s i b l e a p p r o a c h to such
c o n s t r u c t s is some use of "variable"
m o d e l s of set theory.
Once the i d e a of a v a r i a b l e set theory is accepted, it b e c o m e s
clear that the set theory n e e d n o t be as s t r o n g as
Zermelo-Fraenkel axiomatics purposes
(ordinal n u m b e r s
To d e f i n e a category
require.
aside)
For m o s t m a t h e m a t i c a l
Z e r m e l o s e t theory is adequate.
a very m u c h w e a k e r s e t theory s u f f i c e s
a theory w i t h e l e m e n t s
and sets, b u t n o sets of sets.
-
This weak
axiomatic
called will
paper
be
which
because
an a b s t r a c t
the
general
is d e v o t e d
set theory.
a "school"
usual
form
A model
B,
C.
items
Everything
and p o s s i b l y to b e
read
there
is
and
x, y,
y
both.
"the
items
x
and
y.
lowing
axioms :
Extensionality
If
Ordered
will
is
a collection
z,
.--, w h i l e
x pair
item
adjoint
axioms
chief
functor be
will
development theorem,
replaced
is is
These
S
given
others
data
are
are
of the
which
called
a more
of
which
classes
class
to b e s u b j e c t e d
A",
x 6 A, and
to i t e m s
pair
B
if a n d o n l y
if
are
classes,
x 6 B,
then
and if
x
of the to the
given fol-
f o r all i t e m s
A = B.
Pair
If
<x,y)
= <x',y'>,
then
x = x'
and
A,
or a class,
relation
assigns
the o r d e r e d
some
an i t e m
a membership
an e l e m e n t
called
of t h i n g s ,
is e i t h e r
operation
<x,y)
and
in
by
for C l a s s e s
A
be
SCHOOLS
the s c h o o l
item
The
of one s u c h
distinction.
There
an o r d e r e d a new
x, x s A
in
classes.
distinction
II
are c a l l e d
of t h e s e w e a k
of F r e y d ' s
subschool-school
S
to the i n v e s t i g a t i o n
it has
set-class
A school
148
y = y'
.
-
Empty
149
-
Class
There
is
a class
g
with
no items
as m e m b e r s .
Unit Classes
To e a c h
item
x
there
is
{x}
class
with
x
as its
only element.
Cartesian
Product
To c l a s s e s whose
elements
are
A
all
and
B
there
the o r d e r e d
exists
pairs
A
a class
(x,y)
x B
xs
with
A
and
y s B.
Comprehension
If every
bound
low),
then
with
x 6 A
A
a class
variable there
The class
is
usually
is
and
is
that
property use
C
is
a formula
item with
consisting
in which
a "limit"
(see b e -
of all t h o s e
items
x
~(x).
class
C
constructed
i n this
axiom
schema
is the
written
{x]x ~ A and
it is a s u b c l a s s ~
~ (x)
a variable
a class
c = Note
and
used
to d e s c r i b e
only quantifiers
each quantifier
of
~(x)}.
the g i v e n
class
the e l e m e n t s
which
are
"limited".
is t o h a v e
one
of the
of
A
and that
the
the
subclass
is
By this we
forms.
mean
to
that
-
(Sy) y 6 B --for some given class
or
(u
ables
~ (Xl,... ,xn)
for items
of items may be defined
and c a r t e s i a n p r o d u c t n
with
t_here exists
C = {<Xl,.--,x n} Indeed,
limit of the quantifier.
recursion.
for each n a t u r a l n u m b e r
a formula
implies ----
( X l , . - . , x n}
using c o m p r e h e n s i o n show
z & B
B, called the
Ordered n-tuples by the usual
150
that to classes
one can
A1,..- , A n
all bouxld variables
and
limited vari-
a class
Ix! 6 A I , . . . , x n 6 A n and ~(Xl,--.,Xn)}
the c a r t e s i a n
product
r e p l a c e d by this s t r o n g e r
and c o m p r e h e n s i o n
comprehension
Here
are some e x a m p l e s
(i)
Any r e a l i z a t i o n
axioms
may be
a x i o m for n-tuples.
of schools.
("model")
of the Zermelo axioms
for set d%eory w i t h i t e m = class = set
(of the Zermelo model),
and w i t h the usual m e m b e r s h i p
and the usual d e f i n i t i o n
relation
of ordered p a i r in terms of membership. (2) the i t e r a t e d (U • U)
If
U
is any set
(in some set theory),
cartesian products
x U,-.-
of
U
w i t h itself,
s u b s e t of one of these p r o d u c t s one of these products.
U,U
• U,
and take
and "item"
The axioms
U
(U • U),
x
"class"
for a s c h o o l in the "atoms"
u' 6 U)
There
pairs
(u,u')
of atoms,
which simultaneously triples.
sense:
to be any
to be any e l e m e n t
that this school is "homogeneous" in the f o l l o w i n g
form all
then hold. (elements
are classes
of Note
u,
of o r d e r e d
and so on, b u t there n e e d be no classes
contain b o t h
Such h o m o g e n e o u s
o r d e r e d pairs
set theories
and o r d e r e d
are i n d e e d
appropriate
to
-
many
mathematical (3)
U, V,
and
theory)
form all iterated
factors
is o n e
of one
of t h e s e
of
This
"itsm" N;
Let
f: N
k, m E N
F: A
thus
if
is
By the
of two
Two
other S,
B
TWO
functions phism.
One
function phisms.
M
with
called
hand,
the
MXoM
Take
pair
ordered to b e
much
axiom,
of
any
of
pairs,
take
any s u b c l a s s
of
of the we
S
c a n be d e f i n e d a function
familiar may
all
the
define
the u s u a l
and
what
of
the
com-
collection as m o r p h i s m s ,
the
school.
is m e a n t
procedures.
by
a
Such
a
data O
domain
giving
The
functions
the c a t e g o r y
properties
construct
it a f u n c t i o n .
following
giving
M
function
with
one m a y
(of m o r p h i s m s ) > O
A
• B
and prove
following
M
these
are c l a s s e s
F c A
of
of one
numbers.
ordered
"class"
school.
comprehension
consists
classes
and
any s u b s e t
hold.
a category,
within C
N
of the s c h o o l ,
On the
category
and
functions
constitute
category
of
to b e
of the
school.
of n a t u r a l the
set
each
any e l e m e n t
a homogeneous
With
(in s o m e
in w h i c h
"class"
to b e
and define
a fixed
A
take
"item" is
g i v e n sets, products
and
= f(k,m).
a subset
of all c l a s s e s then
> N
be
are
the s e t
for a s c h o o l S
of a graph. posite
and
be
Let
) B
or W
any e l e m e n t
the a x i o m s
as usual;
N
W
cartesian
again
• N
to be
to be
U, V,
products
these products.
bijection
-
purposes.
If
(4)
151
the
(of objects) and cod.main
composite
in
So
of a m o r -
of t w o m o r -
-
Here
Mx0M
usual
in terms
the
is the
class
of
of d o m a i n
comprehension These
axiom data
1 5 2
-
composable
morphisms,
and codomain; for
are
described
it e x i s t s
as
in virtue
of
a school.
required
to s a t i s f y
the u s u a l
axioms
for
a category. This gory
may be
description
described
as t h e
theory
between
such
evident
ways.
categories
formulation
of n o n - e m p t y classes when H c A
• B
an i t e m
is
another
with
(a,b>
known
a function
G: A - - - >
axiom
then be
of
choice
of a c h o i c e
say
that
S
that
(a,b> that
If
F: B
B
with
the previous
(a,b>
with
Fb = a
then
for e a c h
This
apply
classes
to e a c h
E H,
form > A
or w i t h
c 6 A of
the
has
and
and
If
B: there
a function
exists (i.e.,
is e x a c t l y
of c h o i c e B ~ g,
To
this
G, w e
of a l l
those
find H a
not in
one
implies
domain
to the s e t
b s B
of school
there axiom
The
a choice
a 6 A is
in the
for a s e t
classes
A
item
there
F G F = F.
axiom
is
such
a school.
function
Hence we
any t w o
theory,
transformations
for
are n o
for
a cate-
described
there
simply
F.
may
how
set
and natural
for
6 G.
form:
weak
do,
such
G c H, s u c h
with
the
not
holds
a class
b 6 B
a graph) b 6 B
sets w i l l
following
S
the e x i s t e n c e
in a s c h o o l .
the
within
exactly
in a very
Functors
consider of
indicates
adequately
of a s c h o o l .
Next we usual
thus
there
the
is
pairs
image
of
-
Ill N O R M A L
A subschool
T
153
-
SUBSCHOOLS
of a s c h o o l
of a c o l l e c t i o n ,
closed
under
of the i t e m s
S
a collection
such
that
these
(under
the
if
is
T
though of
of
and
a subschool
this
class
elements unit
in
S
classes
school
T
pecularities normal
when
differ
restrict
do n o t
arise.
for e v e r y
B c A
in
that
the empty
unit
class
N any
is
their
class
{x}
in
that
the
of
of
of N
constructed
A N
of and
S
of
S).
Thus
class
empty
gT
'
class
of
T
none
T.
Similarly, in
to s u b s c h o o l s N all
of
N.
of w h o s e the
the sub-
A
an a p p l i c a t i o n
items
N • B
of
of the
x
B
N,
is its
of
of
remark
S
S
imply
that unit
such
to b e
the class
of two c l a s s e s
A similar
S.
is s a i d
conditions
class
of
in w h i c h
classes
These
x
S
those
those
product in
S
a school
of c l a s s e s
is t h e e m p t y
product
of
S.
all
of
of s o m e
classes
an e m p t y
and
N
of an i t e m
by
T
A subschool
cartesian
cartesian
S
in
form
original
products
are c l a s s e s
class
the
attention
class
with
S
~T
is i n
consists
operation,
relation
of the s u b s c h o o l
will
are i t e m s
and
n o t be
from those
x ~ A
S,
need
cartesian
with
in
there
are i t e m s
may
themselves
S
a class
and the
We
of
pair
of s o m e of the
of t h e m e m b e r s h i p
gT
S; it is s i m p l y
by d e f i n i t i o n
the o r d e r e d
two s u b c o l l e c t i o n s
restriction
S
holds
of for
comprehension
axiom. A normal
subschool
N
of
S
is
completely
determined
-
1 5 4
-
by the s p e c i f i c a t i o n of w h i c h c l a s s e s of
S
belongs
to
N,
in the f o l l o w i n g sense.
Proposition 1
If
S
is a school,
than a s u b c o l l e c t i o n
c o l l e c t i o n of all c l a s s e s of
S
classes of a n o r m a l s u b s c h o o l c o n t a i n s the e m p t y s u b c l a s s e s in
S
class of
subschool detail, x s A if
N
of
S
if a n d only if
any class of
and w i t h any two classes
A
and
S
x
of
for some class is in
S A
B
of
S
W h e n these c o n d i t i o n s h o l d the
is an i t e m of of
L
all its
is u n i q u e l y d e t e r m i n e d b y the c o l l e c t i o n
an i t e m
{x}
N
S.
of the
is the c o l l e c t i o n of all the
S, w i t h
t h e i r c a r t e s i a n p r o d u c t in
L
L
N
L.
In
if and only if
or, e q u i v a l e n t l y ,
if and only
L.
The p r o o f is left to the reader. This n o t i o n of a n o r m a l s u b s c h o o l i n c l u d e s many b a s i c examples,
m o s t n o t a b l y the n o r m a l s u b s c h o o l
sets w i t h i n let
G
a
G6del-Bernays
be any G ~ d e l - B e r n a y s
classes,
s o m e of w h i c h
axioms.
Then
the sets of
G
u n i v e r s e of sets and classes. universe;
a school
SG
in w h i c h
the s t a n d a r d c o n s t r u c t i o n
the
G-B
the items
are the c l a s s e s of
w i t h the g i v e n m e m b e r -
ship relation.
S i n c e w e h a v e c h o s e n to r e g a r d e a c h s e t as a and since the e m p t y
are
of o r d e r e d pairs,
and the classes
(special s o r t of) class,
G
Indeed,
that is a c o l l e c t i o n of
are c a l l e d sets, w h i c h s a t i s f y
determines
G, w i t h
c o m p o s e d of all the
class is a set,
155
-
since
the
every
subclass
in any
cartesian
G-B
of a s e t is
school
school where G-B
both
universe
which
V
Among
is
be
the
items
that
(x s y E V t s V).
and
satisfy
axioms
axioms
for
is
of
U, w h i l e
mal
subschool
the
and
finally
collection
L
subschool
classes
are
since
of all s e t s N
the
the
include
U the
set
x 6 V)
determines set
of
V
universe;
the u s u a l axiom
axiom schema
Zermelo
implies
Now
the
a set
a normal
the
for r e p l a c e m e n t .
a model
is
a Zermelo-Fraenkel
the u s u a l
Zermelo
but not
the
U
implies
weaker
a set,
determines
of sets w h i c h
theory.
of sets
the s u b sets
of
the
G.
Let lection
product
-
Z-F
axioms
schema
is,
axiom
of r e p l a c e m e n t , The
schema
for
comprehension
a set
V c
U
theory which
such
(t c y 6 V
SU
that
is t r a n s i t i v e
and inclusive a school
a col-
for s e t
of c o m p r e h e n s i o n .
Consider
is the
that
with
collection
implies
item = class
of c l a s s e s
for
= set
a nor-
S U"
A Grothendieck school.
Suppose
axioms
of i n f i n i t y ,
Grothendieck such again
that
so
the that
Z-F
(i.e.,
the e l e m e n t s
x 6 W
that
the
for a n o r m a l To m o t i v a t e
the p o s s i b l e
sizes
of
set
also provide
are
in
transitive satisfy
W
is the
subschool our s t u d y Zermelo
will
universe
there
universes
follows
of items)
that
universe
of
U
satisfies
U
sets
Z-F
collection school
of c o m p l e t e n e s s ,
universes
W
V.
We
sub-
strong which
and i n c l u s i v e
the
the
a normal
are
sets
axioms.
W
It
of c l a s s e s
(and
S Ulet us o b s e r v e use
the u s u a l
156
p o w e r set Re
P(x)
= {YlY c x}.
Within
U
one may define
of all sets of rank less than the o r d i n a l n u m b e r
~
the set b y the
recurs ion R 0 = ~, the l a t t e r for
8
Z e r m e l o universe,
a limit ordinal, Note
set
{~,p~,pZ~,.. 9 }
any
Z-F
> pn ).
contain
then the set
R~+~
is a
t h a t this u n i v e r s e d o e s n o t c o n t a i n the
though the latter set m u s t b e p r e s e n t in
u n i v e r s e , b e c a u s e it can be o b t a i n e d f r o m the s e t
by r e p l a c e m e n t n l
= U Re a<8
R~
Ru+ 1 = P ( R e ) ,
(i~176
as the i m a g e of
~
u n d e r the f u n c t i o n
For the same r e a s o n this Z e r m e l o u n i v e r s e does n o t
the u n i o n
U pn
latter o b s e r v a t i o n
or the c a r t e s i a n p r o d u c t
( n o n - e x i s t e n c e of a product)
c a t i o n of w h a t c o m p l e t e n e s s
Hpn
The
gives an i n d i -
c a n n o t m e a n in the c a t e g o r y of all
sets of a Z e r m e l o universe.
IV
SMALL COMPLETENESS
Consider a fixed normal subschool S, and call things of
N
in
N.
M(C)
(items,
and lar@e w h e n A category and
0 (C)
C
classes)
of m o r p h i s m s
S
and objects
for e a c h p a i r
class
of all m o r p h i s m s
s m a l l c a t e g o r y is then n e c e s s a r i l y versely.
In p a r t i c u l a r , w h e n
N
S
but not necessarily
is s m a l l w h e n b o t h the c l a s s e s
locally s m a l l w h e n homC(c,c')
of a s c h o o l
s m a l l w h e n t h e y are t h i n g s
they are t h i n g s in within
N
c, c' c
are small;
it is
of o b j e c t s of ~ c'
is small.
C, the A
locally small, b u t n o t conis the s u b s c h o o l of all sets
-
in the s c h o o l above,
SG
of some
157
G-B
this u s a g e of "small"
-
universe
G, as d e s c r i b e d
and " l o c a l l y small"
is p r e c i s e l y
the f a m i l i a r one. If S) and F
I
F: I
is any "index"
> C
category
is any functor,
(a c a t e g o r y w i t h i n
the n o t i o n of a limit of
is d e f i n e d by the u s u a l u n i v e r s a l p r o p e r t i e s .
that
F
is a s m a l l
functor when
of the o b j e c t f u n c t i o n of f u n c t i o n of
F
the c a t e g o r y
W e w i l l say
I, the i m a g e
F, and the image of the m a p p i n g
are all three small.
The c a t e g o r y
s m a l l c o m p l e t e if and only if e v e r y s m a l l f u n c t o r has a limit in of
C
C.
In p a r t i c u l a r ,
is a f u n c t i o n
A: I
object
EAi, for
> M(C)
which
of the p r o d u c t . of
C
F: I
a
and
C
morphism
Fi:
a
I
and a p r o d u c t
i m p o s e d on the f u n c t i o n
gives the p r o j e c t i o n s
Similarly,
b
> C
w h e r e b o t h the s e t
are small,
conditions
is a f u n c t i o n
jects
F: I
i 6 I, of such a family is d e f i n e d as usual;
there are n o s m a l l n e s s P: I
A
is
a s m a l l family of objects
~ O(C)
and the image of the f u n c t i o n
C
of > b;
Pi:
~Ai
> Ai
a family of c o t e r m i n a l m o r p h i s m s > M(C)
t o g e t h e r w i t h two ob-
such that for e a c h the e q u a l i z e r
i s I,
Fi
is a
of s u c h a family is
d e f i n e d as usual. This d e f i n i t i o n of s m a l l c o m p l e t e n e s s lated as to include Zermelo universe. universe V
and
Nv
is so formu-
the c a t e g o r y of all s m a l l classes in a Indeed,
let
S
U
be the s c h o o l of a
Z-F
the n o r m a l s u b s c h o o l d e f i n e d f r o m a m o d e l
for the Z e r m e l o axioms,
as d e s c r i b e d above.
T h e n the
158-
-
category
Ens v
of objects classes
of all s m a l l classes has
V
as its class
and the class of all f u n c t i o n s b e t w e e n s m a l l
as its class of m o r p h i s m s ;
is l o c a l l y small. s m a l l family small; h e n c e
Moreover,
A: I
> V
the u n i o n
it is a large c a t e g o r y w h i c h
Ens V
is s m a l l complete.
of o b j e c t s has b o t h U~
of all classes
I A
and
For a Im(A)
is small,
b e c a u s e it is a union of a s m a l l class of s m a l l classes the Z e r m e l o axioms p r o v i d e The p r o d u c t
nA
f: I
9 U Ai
with
fi 6 A i
Since this is also a s m a l l class,
all s m a l l products. equalizers.
of s u c h unions).
is then d e s c r i b e d as u s u a l as the class of
all those f u n c t i o n s i s ~.
for the e x i s t e n c e
(and
Ens v
A s i m p l e r a r g u m e n t shows
Its s m a l l c o m p l e t e n e s s
lowing p r o p o s i t i o n , w h i c h t r a n s l a t e s
for e a c h does h a v e
that it has all
then follows
f r o m the fol-
to our g e n e r a l s i t u a t i o n
s o m e f a m i l i a r facts a b o u t the c o n s t r u c t i o n
of limits.
Prqposition
Let
C
ject is small.
be a c a t e g o r y w i t h i n If
C
has
in w h i c h e v e r y ob-
a p r o d u c t for e v e r y s m a l l family
of its o b j e c t s
and an e q u a l i z e r
its m o r p h i s m s ,
then
C
S
has
for e v e r y
an e q u a l i z e r
c o t e r m i n a l p a i r of for e v e r y s m a l l
family of c o t e r m i n a l m o r p h i s m s .
Proof. morphisms {b}
small.
F. : a 1
If
F: I
> b, t h e n
> M(C) b
is a s m a l l family of
s m a l l by h y p o t h e s i s
The c o n s t a n t f u n c t i o n
I • I
)
{b}
is
implies
-
therefore
1 5 9
-
a s m a l l family of objects
is a p r o d u c t
~b, I x I
times,
of
C.
The=efore,
and two e v i d e n t maps
there a
w h o s e e q u a l i z e r is the e q u a l i z e r of the g i v e n f a m i l y
>
Eb
F.
Proposition 3
Let
S
be a choice school.
If
C
is a c a t e g o r y
w h i c h has a p r o d u c t for e v e r y s m a l l family of its o b j e c t s an e q u a l i z e r then
C
for e v e r y s m a l l family of c o t e r m i n a l m o r p h i s m s ,
is s m a l l complete. Given a category
F. 7
and
I
and a s m a l l f u n c t o r
> C, the u s u a l p r o o f forms f i r s t the p r o d u c t
KF.
over
1
all objects
i
of
gives two maps
KF i
s u c h pair;
I.
Each morphism
> Fk; one chooses
the limit of the given
intersection
F
of these e q u a l i z e r s .
f: j
> k
an e q u a l i z e r
thesis on
C
I
then
for e a c h
is t h e n o b t a i n e d
as the
It is this use of choice
w h i c h r e q u i r e s the a s s u m p t i o n of our p r o p o s i t i o n a choice school.
of
that
This a s s u m p t i o n c o u l d be r e p l a c e d by
S
be
a hypo-
g i v i n g to e a c h p a i r of c o t e r m i n a l m o r p h i s m s
a
fixed equalizer. A functor S
U: C
> X
between
two c a t e g o r i e s w i t h i n
is s a i d to be s m a l l c o n t i n u o u s w h e n e v e r y p r o d u c t in
a s m a l l family of objects
of
C
is m a p p e d by
in
X
of the image family
in
X)
and w h e n e v e r y e q u a l i z e r in
morphisms
of
C
U
C
of
to a p r o d u c t
(which is n o t r e q u i r e d to b e s m a l l
is m a p p e d by
U
C
of a p a i r of c o t e r m i n a l
to an e q u a l i z e r
of the i m a g e
-
pair.
1 6 0
-
The s t a n d a r d a r g u m e n t shows for
that a s m a l l c o n t i n u o u s of any s m a l l f u n c t o r
V
Again, school
functor
F: I
9 C
THE A D J O I N T
N
U
C
small complete
carries
the l i m i t i n
i n t o a limit of
C
UF.
FUNCTOR THEOREM
is a n o r m a l s u b s c h o o l of a c h o i c e
S.
Theorem 4 ! If within
C
S,
a category within
X
continuous
is a s m a l l c o m p l e t e and locally s m a l l c a t e g o r y
functor,
then
U
has
S, and
U: C
9 X
a small-
a left a d j o i n t if and only if
it s a t i s f i e s the s o l u t i o n set condition. The s o l u t i o n set c o n d i t i o n a s s e r t s t h a t to e a c h object
x
of
X
f: I for
i E I
there is a s m a l l class > M(X), b:
with
Ira(f)
morphism
g: x
with
(Ug')f .
g =
9 U(a)
I and
9 O(X), Im(b)
I
and f u n c t i o n s
fi : x
> U(b i)
s m a l l s u c h t h a t to e v e r y
there e x i s t
i E I
and
g': b i
9 a
This is just F r e y d ' s s o l u t i o n set condition,
w i t h the a d d e d r e q u i r e m e n t s
that
Im(f)
and
Im(b)
are small.
The p r o o f of this t h e o r e m is the s t a n d a r d one, w h i c h w e r e p e a t n o w c h i e f l y to p u t down in p r i n t c e r t a i n s i m p l i f i c a tions w e l l k n o w n in the f o l k l o r e others).
(Lawvere,
Kelly,
F r e y d and
The s o l u t i o n set c o n d i t i o n is c l e a r l y n e c e s s a r y ,
since a functor
F
left a d j o i n t to
U
produces
a solution
161
-
set with x
>
just
one
object
b = F(x)
and
one
(universal)
let the s o l u t i o n
set
condition
map
UF(x) . Conversely,
By
-
completeness,
the
small
product
d = Kb.
be
given.
with
projections
UPi:
Ud
i
Pi" is
d
> bi
exists
a product
defined
in
by
X;
in
C.
there
(UPi) f = f
By
continuity,
is t h e r e f o r e for e a c h
a morphism
i 6 I.
Given
>
Ub i
f: x
>
Ud
any
i
g: x ists small,
> g
Ua
as i n
: d the
is small,
>
a
class
K
g =
set
condition,
then
there
Since
is
locally
(Ug*)f.
of a l l e n d o m o r p h i s m s
{klk:
completeness,
class
with
a n d s o is
K = By
the s o l u t i o n
the
d
the o b j e c t
d
in
C
class
>
there
of
C
ex-
d
in
is
of c o t e r m i n a l
C
with
(Uk) f
e:
an e q u a l i z e r Like
morphisms.
=
f}
dO
>
.
d
for t h i s
any e q u a l i z e r ,
e
is
monic.
Lemma
(Kelly)
If
h:
Proof. the e q u a l i z e r
d0
d
of
Both
(Uk) f =
hence
there
g: x
>
for s o m e
Ua v:
f.
id
U
is
But
exists
m:
do
)
eh
are in
since
e
continuous,
f
is
x
>
we now have
U(eh) f = f,
and
K, e h e = e;
Be cause with
has
another Ud 0
a; to s h o w
m
K;
he =
since
is m o n i c , Ue
equalizes
equalizer
with
g = U(g*)f
then
and
therefore
universal
e
he = all
of t h e s e
f = (Ue)m.
1.
is 1. Uk Uk;
To any g = U(v)m
it r e m a i n s
only
-
to show has
this
unique.
U ( w ) m = g.
v, w:
d0
>
so d o e s As
v
Take By
a.
m.
s:
has The
>
continuity,
U(els)f (monic)
m =
(Ue 1 ) m I
g = ml,
eI
equalizer Since
e
= f
dO
dI
>
> dO
of
Uv,
for s o m e
m I: x
do
e >
the e q u a l i z e r
Uw,
and >
U (d l) .
m I = U (s) f
d.
the
a right
also
a
equalizes
and by
thus h a s
was
w:
one t h e n h a s
e] -
U(eels)f
so
el:
Ue I
s .~ dl
d
= m, so
isomorphism. have
dl,
not,
the e q u a l i z e r
Therefore,
d
-
Suppose
for any s u c h m o r p h i s m
some
1 6 2
But then lemma
v
els
else =
inverse, of
for
so is
and
w,
l.
an we
1
v = w. To e a c h
and a u n i v e r s a l x;
for e a c h a functor
object
morphism
then
F
we
m:
x ~
as u s u a l
left
family
of
thus h a v e Ud 0.
Fx = d O
adjoint
In p l a c e that every
x
to
is
an o b j e c t Choose
the
dO
one
object
of
such
C
d0
function
of
U.
the a x i o m
of c o t e r m i n a l
of c h o i c e , morphisms
one m i g h t has
assume
a chosen equali-
zer. This theorem
shows
"abstract" that
interpretations: G-B set
theory,
universe, model ness
of
the
theorem
usual
Zermelo
theory.
can be
limits.
has
The
spelled
It s e e m s
out
adjoint
several
"small"
another where
set
of the
one, w h e r e
another where
and s t i l l
conditions
specified
this
version
functor
different
"small"
means
means within
sorts
a set
of
of
a Grothendieck
"small"
means within
a chosen
leading
idea
complete-
is
as c o m p l e t e n e s s
probable
that
other
that
for c e r t a i n
theorems
of
-
category
1 6 3
-
theory may be u s e f u l l y r a m i f i e d w h e n o v e r a l l
hypothesis
are r e p l a c e d by c o m p l e t e n e s s
VI
"completeness"
for certain s p e c i f i e d
TYPES A N D I T E R A T E D S C H O O L S
The m a j o r p r o b l e m of the f o u n d a t i o n s t h e o r y remains t h a t of h a n d l i n g s u c c e s s f u l l y
of c a t e g o r y
larger c a t e g o r i e s
"of all s o - a n d - s o ' s " .
This can be done e f f e c t i v e l y b y u s i n g a
succession
like a c u m u l a t i v e
let S
U
of schools,
w i t h i t e m = class = set of
S1
w i t h i t e m = set of
in
U, and w i t h class S
are small;
as
S1
of
and w i t h
Categories within
S2
this p r o c e s s y i e l d s
all
(in
for
n = 2
a sequence
arguments.
U2
is reS
S 2)
the c o l l e c t i o n of all any s u b c o l l e c t i o n of
of schools
~.
C o n t i n u a t i o n of
S c S1 c
S 2 c S 3 c...,
The c a t e g o r i e s w i t h i n
"metacategories"
frequently (a type
may be d e s c r i b e d by a s u i t a b l e
The s t a n d a r d p r o o f that
s i s t e n t l y r e l a t i v e to
S1
Now form a still
Such a scale of schools
theory on top of a set theory) a x i o m system.
Thus
Categories within
m i g h t be c a l l e d ~iant.
are e x a c t l y the
u s e d in i n f o r m a l
U0
(suitably constructed)
e a c h a n o r m a l s u b s c h o o l of the next. Sn
of
are large.
S 1 , with class
r e g a r d e d as a s c h o o l
the o r d e r e d pairs those g i v e n
G-B sets.
w i t h items
of classes
these items,
to
First,
C o n s t r u c t a larger s c h o o l
any s u b c o l l e c t i o n
GB-class
S2
U.
U, w i t h
those w i t h i n
larger school n-tuples
type theory.
be some s e t - t h e o r e t i c a l universe,
lated to
limits.
ZF set theory
G-B set theory is con-
(J. B. R o s s e r a n d Hao Wang,
-
164
-
for Formal Logics,"
15; 113-129,
can probably be translated to show that
(1950))
the consistency
J.
SymboZ~o Log~oj
"Non-standard Models
of the original set theory implies the consis-
tency of a suitable
language for any one
S n.
In this way,
the
use of categories within schools can provide a suitable and consistent foundation
for category theory, by providing an explicit
setting for successively
larger types of metacategories.
-
165
-
ON THE DIMENSION OF OBJECTS AND CATEGORIES III HOCHSCHILD DIMENSION*
by Barry Mitchell**
R. Swan has observed that the situations encountered in [7] and
[8] can be considered as special cases of the
following.
Let
K
be a commutative ring, and let
K-algebra.
(All rings have identities and all ring homomor-
phisms preserve identities.) category and let C(A) on
~: K
> c(A)
A
be a
is an additive
be a ring homomorphism where
is the ring of endomorphisms of the identity functor A.
Let
AA
denote the category of left A-objects in
(that is, ring homomorphisms Let
Suppose that
A
AA~
A ---> HomA(A,A)
denote the full subcategory of
AA
with
A
A 6 A).
consisting of
all those A-objects such that the composition A --> HomA(A,A )
K
by
~.
If
A
is the same as the homomorphism induced
is abelian,
then so is
AA~.
As an example, consider a ring homomorphism ~: K ---> c(A)
*
and let
f(x)
be a polynomial with coefficients
Research supported by National Science Foundation Grant No. GP-6024.
** Dept. of Math., Bowdoin College, Brunswick, Maine.
-
in A
K.
Then the category
satisfying AAr
category Another
f(u) where
example
is another obvious
= 0
Af (see
A
in
the category
and taking
A
A@FA of left A|
of
of
A
AA
A
a lower bound for dim
gl. dim.
AA
F
F-operations
lemma
over K).
1.4 of the for the
the H o c h s c h i l d
The latter
is defined
as a
A e = A* @ A
lemma inversely
in certain
as
to
cases where
it is true that
h.d. for all non-zero
AeAr
~hat
considered
A
and
is known.
Sometimes
category
A
(tensor product
we shall be using this
determine
and
to be the
to give an upper bound
of
F
This is the same as
in the case where
dimension
Actually
d = dim A
commute w i t h
objects
where
is the category
A(FA)
(dim A) is known.
the h o m o l o g i c a l
FA
> c(FA)
K
observed
in
K[x]/(f(x))
with
A
the same K-operations.
dimension
module 9
r
u
is the same as the
which have simultaneous
paper can be used
dimension
w
Then
Swan has further
global
[7,
such that A-operations
and both induce
present
of all endomorphisms
is the K-algebra
ring homomorphism.
operations
-
is had by replacing
K-algebra
of all objects
166
objects A ~
A
(i)
A @K A = d + h.d. A A A
in an abelian
is c o n s i d e r e d
category
as an object
A, where of the
-
This q u e s t i o n form
K(~),
question
is of p a r t i c u l a r ~
denoting
167
-
interest when
a finite p a r t i a l l y
is then r e l a t e d to the e x i s t e n c e
for a finite o r d e r e d set or,
A
is of the
o r d e r e d set.
The
of a "dimension"
in other words,
an integer
d'
such that
gl.dim.
~A = d' + gl.dim.
for all a b e l i a n categories of c o v a r i a n t
functors
we shall show that valid.
However
A.
from
d = d'
to
~A
A.
denotes
For
(1) is always
it is now k n o w n that not every d'.
an o r d e r e d set of 15 elements at the U n i v e r s i t y
the category
d' = 0, i, or 2,
and that e q u a t i o n
o r d e r e d set has a d i m e n s i o n
Spears
Here
A
finite p a r t i a l l y
A counter-example
involving
has b e e n p r o d u c e d by W i l l i a m T.
of Florida.
Details will appear
in
his thesis.
Throughout
this paper
A
will denote
an a b e l i a n
category.
i.
Let
A
AN A D J O I N T
be an
RELATION
(abelian)
c a t e g o r y with
indexed by some cardinal
number
M
then the tensor p r o d u c t
is a right
defined admits
A-module,
for all left A-objects an exact sequence
p.
A
of right
If
in
A
A,
coproducts
is a ring and M |
providing
A-modules
A M
is
168
JA
where
I
and
J
-
> IA
have c a r d i n a l numbers
no b i g g e r
than
When u s i n g the above tensor p r o d u c t we shall always that
M
satisfies
this c o n d i t i o n w i t h o u t
We then have a natural e q u i v a l e n c e
Hom A(M, for
M 6 G A, A 6 AA
denotes
the category If
exact,
M
then
M |
M
is a retract
of abelian
A
(see
M = A
s t a t i n g so.
(l)
A,B) Here
[9,w
G
groups.
and the c o p r o d u c t s
is an exact functor and,
then follows
in
are
A
of the A-object
consequently,
for
M
A.
free
That it is true for all
from the fact that any p r o j e c t i v e
of a free.
i. i
Let A
B 6 A
assume
of trifunctors
~ HornA(M |
of the exact coproducts.
projective
Lemma
and
is p r o j e c t i v e
This is trivial for because
I
HornA(A,B))
always
p.
and
F
~: K
> C(A)
be K-algebras.
be a ring h o m o m o r p h i s m
Then there
is a n a t u r a l
and let
equivalence
of t r i f u n c t o r s
HomAsF, (M, Hom A (A,B)) for
M 6 G A|
, A 6 AA~ ,
and
~ Hom F (M 8 A A,B)
B 6 FA~
.
(2)
-
Proof. on
HomA(A,B) ,
A | F*-module, forgetting
Since
it follows
and using naturality
yields
induce the same K-operators
that the latter is a right (2) makes sense.
we have the natural equivalence
in
on either
M
and
B,
Now, (i),
we see that the
side correspond
to each other.
This
1.2 Relative
to K-algebras
natural equivalence HomA|
and
Replace
the induced
A --~
functor
S: AA~
F
B 6 Z| by
ZA
in 1.1.
be a K-algebra homomorphism.
T: FA~ ~
> FA~
right adjoint the functor R(A) = HomA(F,A)
roles of
A
A,B)
1.3 Let
functor
there is a
of trifunctors
A 6 A|
Proof. Corollary
A, F, and Z,
(M, Hom E (A,B)) ~ Homz| r (M |
M 6 G A|
places)
F
(2).
Corollar~
for
and
-
and so the left side of
F-operators,
F-morphisms
A
1 6 9
AA~
given by
has as left adjoint the S(A) = F |
R: AA~ --~
FA~
A,
and as
given by
(symbolic Hom).
Proof.
The dual of 1.1 yields
A
F
and
Then
so as to get
A
and
(interchanging B
the
in the right
-
HomA| Then,
taking
170
-
(M, Hom A (A,B) ~ Hom A (A, Hom F (M,B))
M = F
(with right A-operators
the given algebra homomorphism)
(3)
.
d e t e r m i n e d by
and combining
(2) and
(3),
we obtain
H~ Since that
|
Hom(F,B) S
A,B) ~ HomA(A,
HomF(F,B))
T(B),
as a left A-object is just
is the required left adjoint.
9
R
That
this shows
is the right
adjoint follows by duality. Remark 1.
The u n d e r l y i n g assumption needed to
obtain the left adjoint have exact coproducts
S (right adjoint R) is that
(products)
the cardinal numbers of
I
and
A
indexed by the m a x i m u m of J,
where
F
admits an
exact sequence of right A-modules JA
If the coproducts A-projective,
(products)
then the left
Remark 2. corollary
If
in
r
A
) 0
are exact and
T: FA~
F
is
(right) adjoint is exact.
A = K,
then
AA
= A.
1.3 gives us left and right adjoints
ful functor Lemma 1.4
~ IA--~
Thus for the forget-
9 A.
(Swan) Let
A
be a K-projective
algebra and let
d = dim A.
171
-
If the c o p r o d u c t s
A
in
are e x a c t ,
then
for e a c h
A
AA~
E
we h a v e
h.d.
A _< d + h.d. A A
.
Consequently
gl.
dim.
Proof.
AA
> --- "--> P1
be a p r o j e c t i v e
resolution
is a p r o j e c t i v e
K-module,
a sequence obtain
in
right
of r i g h t
an e x a c t
A
A
> Po
and,
>
A
> 0
as a A e - m o d u l e .
it f o l l o w s
A-modules.
that
Hence,
Since
A e = A* |
consequently, for
(4)
A
A is
(4) s p l i t s
A 6 AA
,
as
we
sequence
) --- P1 |
Pd |
A
AA- .
It s u f f i c e s
A
> Po |
t h e n to p r o v e
for p r o j e c t i v e
Ae-modules
coproducts
A,
But,
for
A-module
0 ~
in
dim.
Let
0 ---> Pd
a projective
-< d + gl.
P.
that
A
> A
h.d.
> 0
P |
A < h.d. A
In v i e w of the e x a c t n e s s
we need only consider
the c a s e
of
P = Ae.
in this case, w e h a v e
Ae |
where,
A =
by r e m a r k
the f o r g e t f u l coproducts
(A |
A) |
2 following
functor
T:
1.3,
AA
and p r o j e c t i v i t y
A = A |
S
) A. of
A
A = S(A)
is the left a d j o i n t Again,
because
as a K - m o d u l e ,
S
for
of e x a c t is
A
-
exact,
Lemma
and so o u r
conclusion
172
-
follows
from
[7,
1.2].
i. 5
A
Let
be a K - p r o j e c t i v e
that
the c o p r o d u c t s
left
A-module
such
in
A
that
are h.d.
K-algebra,
exact. M = m,
M
If
and
suppose
is a K - p r o j e c t i v e
then
i
h.d.
for all
A 6 A,
M |
where
A < m + h.d. A A
is r e g a r d e d
M @K A
as an o b j e c t
of
AA r Proof.
Consider
a projective
resolution
> ~i
---> M
of
left
A-modules
0
Since
A
and
a n d so w e
It s u f f i c e s projective
are
>''"
then
an e x a c t
for
T:
from
[7, 1.2].
r
But
A @
> A
and
P. K
(5) s p l i t s
> 0
.
(5)
as K - m o d u l e s
sequence
--~ P ~ 8K A --~
to p r o v e
A-modules
> P0
K-projective,
A ---~ ---
P = A. AA
M
obtain
0 --> P m |
case
> Pm
that
h.d.
Again
we
A = S (A) so o n c e
P0 |
) M |
P 8K A ~ h.d. A
need only where
more
A
S
A
consider is the
our conclusion
A ---~ 0
for the
left
adjoint
follows
.
-
Corollary
the h y p o t h e s i s
h.d.
A |
for all A 6 A, where Ae in A~ and d = dim
Proof. so the c o n c l u s i o n
of 1.5 w e h a v e
A < d + h.d. A A --
K
A
-
1.6
Under
by
173
A |
is c o n s i d e r e d
A
as an o b j e c t
A.
Since
A
follows
is K - p r o j e c t i v e , by replacing
A
so is
by
A e,
Ae
and
and M
in 1.5.
2.
is a s m a l l
If w e let
K(~)
morphisms
of
necessarily
FUNCTOR
denote ~. with
the
K(~)
CATEGORIES
category
and
free K - m o d u l e
is a ring,
then
on t h e s e t of
c a n be c o n v e r t e d
identity)
K
into a ring
if m u l t i p l i c a t i o n
(not
is d e f i n e d by
the rule
(kx) (ly) =
(k/) (xy) w h e n
range y = domain
x,
= 0 otherwise.
Here xy
kl
denotes
denotes
is a f i n i t e of o b j e c t s morphisms),
the p r o d u c t
the c o m p o s i t i o n category although
of t w o e l e m e n t s
of t w o m o r p h i s m s
(that is, it m a y h a v e
and if its o b j e c t s
~
has o n l y
i, 2,
in
K,
number
..., n,
and
~.
a finite
an i n f i n i t e are
of
If number of then
K(~)
-
has
an i d e n t i t y ,
more,
K
rings
K(~)
Lemma
namely
-
If,
11 + 12 + ... + i n .
is c o m m u t a t i v e have been
174
then
K (~)
considered
further-
is a K - a l g e b r a .
by Lawvere
Such
[6].
2.1
Let
~
be a f i n i t e
be any r i n g h o m o m o r p h i s m
category
where
T h e n we h a v e an e q u i v a l e n c e
K
~: K ---~ C(A)
is a c o m m u t a t i v e
ring.
of c a t e g o r i e s
T: ~A where
a n d let
> AA
A = K(~).
Proof. (in o t h e r w o r d s ,
Given
a covariant
an o b j e c t
of
functor
D:
~
,
A
~A) w e d e f i n e n
T(D)
=
~
D(i)
,
of
~.
i=l
where ~,
1,2,...,n
are the o b j e c t s
then we define
morphism
x
of
the a c t i o n
T(D)
of
x
on
If T(D)
x: j
> k
in
to be the e n d o -
g i v e n by
U. = 0 1
for
i ~ j ,
uj = u k D(X) n where
ui
denotes
the i th c o p r o d u c t
injection
D(i) .
for i=l
This We
action
leave
respect
determines
uniquely
it to the r e a d e r to m o r p h i s m s .
an o b j e c t
to d e f i n e
of
AA~.
the b e h a v i o r
of
T
with
-
Now define S(A) (i) = I.A 1 A).
If
x(l.A) ] and so
x
a functor
(that is,
x: j ~
k,
=
-
leave
respect
becomes
reader
TS AA
of
ii
on
c ikA
) S(A) (k)
an o b j e c t
of
.
~A.
the b e h a v i o r
A g a i n we of
S
with
to m o r p h i s m s .
equivalent
on
of the a c t i o n
(ikx) A = lk(XA)
to d e f i n e
It is now easy
that
by t a k i n g
a morphism
S(A)
it to the
~A
we h a v e
S(A) (x) : S(A) (j)
In this way,
>
S: AA r
the i m a g e
(xl.) A = 3
induces
1 7 5
to see t h a t
to the i d e n t i t y is n a t u r a l l y
functor
equivalent
ST
on
is n a t u r a l l y
~A.
In s h o w i n g
to the i d e n t i t y
functor
one m a k e s
use of the r e l a t i o n s 1 1 = 0 for i 3 n i ~ j, 12i = i i, and Z i. = 1 in K(~) to see t h a t any i=l i K(~)-object A is the c o p r o d u c t of its s u b - o b j e c t s I.A l r
in
A.
If
~
a commutative
and
1
ring,
using
~i x ~2
of
| K(~2)
is the p r o d u c t
the r e l a t i o n
ing a l g e b r a
are f i n i t e
K(~)
categories
and
K
is
t h e n we h a v e
K(~I) where
~2
K(~)*
~ K(~ 1 x ~2)
,
of c a t e g o r i e s .
= K(~*),
is g i v e n by
we see t h a t
In p a r t i c u l a r , the e n v e l o p -
176
-
-
K(~) e = K(~*
Now consider
an o b j e c t
homomorphism
~: K
as an o b j e c t
in
of c a t e g o r i e s which
one
K(~)eA~,
denote
sees
follows.
> C(A).
that
Then
U
U~(A)
(A).
(i,j)
UT(A)
7" x ~
6 7" x ~.
(that is, x:
we are g i v e n
K(~)
|
A,
Using
If
>
considered
of
the f u n c t o r
of
S
explicitly
as
is g i v e n by = Hom(i, j) A
(x,y) : (i,j)
i'
a ring
the e q u i v a l e n c e
under
can be d e f i n e d
UT(A) (i,j) for
.
in 2.1 to an o b j e c t
by
On o b j e c t s ,
where
corresponds
established
we shall
2.1,
A 6 A,
x ~)
i and y:
(1) >
j
(i',j')
in
~ j' in 7) then w e
have U where
uz
denotes
If w e r e g a r d that
the
UT(A)
its v a l u e
(A) (x,y)
at an o b j e c t
is the left a d j o i n t
Tj:
7*A
be c o n s i d e r e d
be u s e d We
as a f u n c t o r
lemmas.
j q ~
[5, 1.2] with
dimension
to g e t u p p e r leave
of
for the c o p r o d u c t ~(7*A),
is just
of the e v a l u a t i o n
In v i e w of
all h a v e h o m o l o g i c a l will
z th i n j e c t i o n
as an o b j e c t
Sj
> A.
u z = U'yzx
bounds
then w e see Sj(A),
where
functor
it f o l l o w s
7 as d o m a i n equal
(i).
that
UT(A)
and w h o s e
values
This
remark
to h.d. AA.
for h . d . U T ( A ) .
it to the r e a d e r
can
to v e r i f y
the f o l l o w i n g
177
Lemma
2.2
If the o b v i o u s
z
and
T
isomorphism
are small
Lemma
U
~
under
object
(~x~)A
T
is an e q u i v a l e n c e
,
to t h e o b j e c t
U
>
~
the i n d u c e d e q u i v a l e n c e
(A)
If object), ~A
is t a k e n
#
t h e n the > A
the c o p r o d u c t A(#)
x
corresponds
U
(A).
TX~
of
of an e l e m e n t
S(A)
copies
y E ~.
S
A.
(morphism)
the r i g h t o p e r a t i o n s
of
~
y
x 6 #
=
u
on
y
=
u
yx
where
by
X,
A(~)
functor denotes
the a c t i o n
on
we have
(1)
xy
A(#)
are g i v e n b y
X*u
the
(A).
Denoting
In this c a s e w e h a v e
the left o p e r a t i o n s
,
for the f o r g e t f u l
= A(~)
of
xu
for all
T*•
(that is, a c a t e g o r y w i t h o n e
left a d j o i n t
~
U
)
categories
MONOIDS
is a m o n o i d
is g i v e n by
of s m a l l
~*x~A
i n t o the o b j e c t
3.
T:
(~•
(U (A)) ~
under
2.3
If then,
T
then,
of c a t e g o r i e s
T*•215 the o b j e c t
categories
u
(A) = A(~)
are g i v e n b y
where
(i) a n d
-
Of c o u r s e w e m u s t a s s u m e ~.
If t h e s e
coproducts
consequently,
by
that
a n d so, A,
9
corollary
restriction
A(~)
T:
x = bt
x 6 ~
with
i n d e x e d by
In this
submonoid
of
one w h o s e
dual
B,
h a v e by
then
~.
A right
If
and
has
an e x a c t
B
representation then
if
A
Z(~)
is
has
a n d is e x a c t .
t 6 9
implies
a left partitionin@
is a left p a r t i t i o n i n g
T
a subset
exists
with
in
for the
t 6 ~,
S
Z(~),
coproducts
a unique
then xt 6
A
Z(T) c
has
Consequently, B,
and
(2)
S
~
~artitionin@
submonoid submonoid
of of
has e x a c t p r o d u c t s
right adjoint.
~
is
~.
If
indexed
In this c a s e w e
[7, 1.2]
h.d.
A < h.d. y
for any
.
then
case w e c a l l
is r i g h t p a r t i t i o n i n g by
~,
admits
t h e n the r e l a t i o n
x 6 T.
i n d e x e d by
is e x a c t and,
of a p p r o p r i a t e
b 6 B
exact coproducts
that
of
HA ---> ~A.
Z(T)-module.
1 E B,
S
1.3 g i v e s us a left a d j o i n t
free as a r i g h t
If
then
= h.d. A A
to the e x i s t e n c e
functor
form
has c o p r o d u c t s
[7, 1.2] w e h a v e
such that each element of the
A
is a s u b m o n o i d
subject
-
are e x a c t
h.d.
If
1 7 8
left
T-object
Examples following:
--
A
(3)
7T
A.
of p a r t i t i o n i n g
submonoids
are t h e
- 179
(i)
If
is b o t h
then
(2)
their with
free
~ left
If
T
is a g r o u p and right
and
ending
with
is a left p a r t i t i o n i n g is r i g h t
If
I
then
the p a r t i a l l y
left
and
right
this
form left
If
right
of
T of
with
(5) monoids
1T ~
•
'IT
I'
and
of t y p e
submonoid
see
that
Similarly,
(See
a special
8
and T x 8
case
w
and
and
and ~2
right and
of
left
respectively,
submonoid
of
right
T
taking
9 x 1 X
is a
8o
sub-
~i x T 2
is a
In p a r t i c u l a r ,
submonoid
partitioning
B
of the
partitioning
then
~i x 72.
partitioning
then,
T =
free
(2).
of all p a i r s
see t h a t
are
is a
definition.)
of e x a m p l e
consisting
subsets,
partially
for the
~2
T1
are
(J, J N I')
are m o n o i d s ,
we
s 6 8,
[7,
J
of t h e
If
is a left o
~e6.
submonoid
~I
is a left
~* x ~
we
9,
of
is
together
1
of
partitioning
left p a r t i t i o n i n g T
9 e8
and
to be
is a set a n d
(I, I').
subset
(1,s) and
B
submonoid
free monoid
is j u s t
(4) to be the
is a s u b g r o u p ,
partitioning.
an e l e m e n t
partitioning
of t y p e
Indeed,
9
partitioning.
(3)
monoid
and
are m o n o i d s
e
product , then taking
all w o r d s
-
of
submonoid
~, of
if then
-
Lemma
180
3.1
If
T
of
#,
and if
by
#,
then
is a left and r i g h t p a r t i t i o n i n g A
has
coproducts
h.d. T,xTA(T) for all
the r e l a t i o n x 6 t.
T
Since
tlxt 2 6 T
It f o l l o w s
that
T* x T - o b j e c t s .
(3) if w e r e p l a c e by
and exact products
indexed
__< h . d "#*x~A(~)
~
is left a n d r i g h t p a r t i t i o n i n g ,
with
tl,t 2 6 T
A(~)
is a r e t r a c t
Therefore, by
implies
#* • ~,
of
the result T
by
that
A(~)
follows
T* • T,
as from
and
A
A(~).
Theorem
3.2
If
A
has e x a c t
is a free m o n o i d
or g r o u p
coproducts
for all n o n - z e r o exact products
A 6 A.
as w e l l
= 1 + h.d. AA
~
is p a r t i a l l y
as e x a c t c o p r o d u c t s
#
then
A(~)
If
and
i n d e x e d by
on one g e n e r a t o r ,
h,d.#,x
then
submonoid
A 6 A.
Proof.
left
-
(4)
free and
indexed by
A
has
#,
(4) is valid.
Proof. Then we have
Let
an e x a c t
~
be p a r t i a l l y
sequence
in
~A
free on
I
generators.
-
0
relative yields,
to any
> IA(~)
left
181
8 > A(~)
~-object
in the case w h e r e
-
A
A
~
> A
[7, 2.1].
has e x a c t
In p a r t i c u l a r ,
replacing
~*,
(2), w e o b t a i n
and u s i n g
A
by
h - d . ~ , x ~ A(~) To p r o v e the c a s e w h e r e case
sequence
is a s s i g n e d generator by
[7,
the o t h e r
~
operators.
w
8
over
~A, A by A(~),
and
~
~,
by
-
inequality we consider
(5) is a s e q u e n c e
the m o r p h i s m
(2) this
-
< 1 + h.d. AA
is g e n e r a t e d
identity
Using
coproducts
h , d . A _< 1 + h.d. AA
(5)
> 0
first
by a single
element.
of
objects where
~* x ~
has m o r e
(If
~* x ~
is n o t a
In t h i s A
t h a n one
morphism.)
Now,
we have
h.d. A = 1 + h.d. AA
for all n o n - z e r o cation
A
with
identity
of this y i e l d s
h.d. ,x A = 2 + h.d. AA This,
A second appli-
operators.
together with
the e x a c t
h - d . ~ , x ~ A(~) The g e n e r a l
case t h e n
of a s i n g l e
element
follows
in e x a m p l e
sequence
(5), y i e l d s
> 1 + h.d. AA
f r o m 3.1 (3)).
.
.
(taking
J
to c o n s i s t
-
Corollar~
~
is the d i r e c t p r o d u c t of
or groups on one generator, ~,
-
3.3
If
over
182
and if
A
n
free m o n o i d s
has e x a c t c o p r o d u c t s
then
h.d.n,x~A(~) for all n o n - z e r o
A 6 A.
p r o d u c t s of p a r t i a l l y
The same is true for d i r e c t
free m o n o i d s
to h a v e e x a c t p r o d u c t s over
Proof.
= n + h.d. AA
if
A
is a s s u m e d f u r t h e r
~.
This follows by an easy induction,
using
2.2.
R e m a r k i. infinite.
Corollary
n
p r o d u c t of p a r t i a l l y integer
free m o n o i d s has a k - f o l d
free m o n o i d s
as a d i r e c t f a c t o r for e a c h
k.
R e m a r k 2.
Corollary
3.3 implies the w e l l - k n o w n
result di m K[X 1 ,.o.,X n ] = n .
4.
If AE
A
is
This follows f r o m 3.1, u s i n g the fact t h a t an
i n f i n i t e p r o d u c t of p a r t i a l l y
positive
3.3 is true e v e n if
~
ABELIAN GROUPS
is a f i n i t e g r o u p of o r d e r
is such that
mA
is an i s o m o r p h i s m ,
m, then
and if
183
h.d.
for any n - o b j e c t again we then
finite
abelian
A
suppose
~*x~ A(~) that
g r o u p of rank F
on
Applying
[7, 3.4].
this
see t h a t
Now,
where
(i)
A = h.d.AA
structure
h.d.
abelian
-
r.
G
(2)
.
is a f i n i t e l y
Then we can write
is a free a b e l i a n group.
= h.d. AA
g r o u p of r a n k
Combining
generated G = F x
r,
(2), 3.3,
and
~
is a
a n d 2.2, w e t h e n
obtain:
Proposition
4.1
Let rank of
be a f i n i t e l y
and suppose
r, G
G
m.
is
If
A
Before without
proof
has e x a c t c o u n t a b l e
Theorem
g r o u p of
torsion
subgroup
coproducts,
then
= r + h.d. AA
A 6 A
such t h a t
looking
at b i g g e r
a generalization
t h e o r e m of I. B e r s t e i n
abelian
t h a t t h e o r d e r of t h e
h.d.G• for all n o n - z e r o
generated
[3].
m
A
is an i s o m o r p h i s m .
abelian
groups we
due t o B. O s o f s k y (See a l s o B a l c e r z y k
state
[i0] of a [2].)
4.2
Let
D
for some i n t e g e r
be a d i r e c t e d n _> 0,
set of c a r d i n a l
a n d let
{R i ' ~J} i
and
number {Mi ,
Mn ~J} i
- 184 -
be d i r e c t
systems
of r i n g s
and abelian
groups
such that
M i
is a left
R.-module 1
for e a c h
~J. (rm)
for
r 6 R. 1
limits,
and
i 6 D,
= ~J(r) i
m 6 M.. 1
If
R
and such that
uJ(m) i and
M
are the d i r e c t
then
M. h . d . R M <__ n + 1 + sup h.d. i R. i 1 As an a p p l i c a t i o n abelian
g r o u p of o r d e r
directed
union
i
If
is a r i n g t h e n
runs t h r o u g h
subrings that in
G K
~
K(G.). 1
for e a c h
generated
p 6
abelian
of its t o r s i o n
P
K
is r e g a r d e d
operators.
that Gi
is i n v e r t i b l e
K(Gi)
as a left
Consequently,
G
be an is t h e
subgroups
Gi
p
Mn"
u n i o n of its
the s e t of p r i m e s
Then each
h.d. where
be
p
such
is i n v e r t i b l e
is a f i n i t e l y
g r o u p of r a n k at m o s t
subgroup
Then
G
s e t of c a r d i n a l i t y
and s u p p o s e
P.
let
is the d i r e c t e d
let
has p - t o r s i o n
r.
generated
a directed K(G)
Now
theorem
and rank
of its f i n i t e l y
where K
of t h i s
in
r,
a n d the o r d e r
K.
Hence,
we have
K < r , -K(Gi)-module
by t h e o r e m
with
trivial
4.2, w e o b t a i n
h . d . K ( G ) K <_ n + r + 1 ,
a n d so t h e r e
is a p r o j e c t i v e
K(G)
resolution
of the f o r m
-
0
Theorem
r
-
> P1 --~ P0
"'"
---~
(3)
K
>
0
of o r d e r
~n
and
such
that
4.3
Let rank
>
> Pn+r+l
185
G
be an a b e l i a n
and let
has p - t o r s i o n .
F
group
be the set of p r i m e s
If
A
has e x a c t
p
coproducts
indexed
G
by
an
then
(4)
h-d- G A _< n + r + 1 + h.d. AA for all G - o b j e c t s all
p 6 P.
A
such t h a t
for all such by
A ~n'
in
whose
A.
If f u r t h e r
A
has e x a c t p r o d u c t s
>_ r + h . d . A A
(6)
A 6 A.
The a s s u m p t i o n
structure
where
denominators
is a p r o j e c t i v e as a G - o b j e c t k n o w by
(5)
then
Proof. K-object
for
<_ n + r + 1 + h.d. AA
h.d.G• for all
is an i s o m o r p h i s m
Consequently
h.d.G•
indexed
PA
in
[5, 3.1]
K
on
of p r i m e s
left K ( G ) - m o d u l e , with
gives
it a u n i q u e
is the r i n g of r a t i o n a l
are p r o d u c t s
A
A
x
in
and if
acting
as
that
h.d.GP KA _< h.d. AA
P.
P| A X
|
X t
Now
numbers if
P
is r e g a r d e d then we
186
-
Consequently, A.
(4) f o l l o w s
Replacing
Inequality that
G
A
by
has
From
Let
dimension
in
K.
w e k n o w by
non-zero
i.
~
that
h.d.
using
3.1, w e see t h a t
which
has an e l e m e n t
Remark
[2] for c a t e g o r i e s
Hom
(i,j)
case we w r i t e with
FINITE
this
pair
empty
(i,j)
i < j.
the p r o p e r t y
r i n g of
group
where
is n o t
of o r d e r A
is any
such that
P A = 0.
(7)
= = .
(7) h o l d s
for any g r o u p
p.
(4) w a s
ORDERED
section
in o t h e r w o r d s ,
is e i t h e r
each ordered
r.
proved by Balcerzyk
of m o d u l e s .
Throughout set or,
= ~
(5).
the f a c t
be a p r i m e w h i c h
A
with
that
Inequality
5.
ordered
of r a n k
T-operators
of o r d e r
2.
K
we obtain
to b e a c y c l i c
trivial follows
p
d i m K(~)
Then,
A(G)
be a c o m m u t a t i v e
a n d let
[7, 3.4]
1.4 it t h e r e f o r e
by
subgroup
K
Taking
K-module with
A
(3) o v e r
of 3.1 a n d 3.3 u s i n g
a free a b e l i a n
global
invertible p,
and
(6) is a r e s u l t
Remark finite
GA
from tensoring
Any
t h a t if
n
SETS
will
a finite
or c o n s i s t s of e l e m e n t s
and
category
a finite such
of one e l e m e n t of
such category i ~ j
denote
7.
In the
for latter
is e q u i v a l e n t
j ~ i,
then
that
to one
i = j.
-
In v i e w this
of
lemma
property.
2.3 w e Our
shall
187
-
always
terminology
assume
will
then
that
n
follow
has
t h a t of
[8].
Recall i
not
in s o m e
dimension #'.
of
D
11"i .
with
this
particular,
if the
functor
A
at
as a c o n s t a n t
of
for
every
T (~) all
those
to the
rest
~*
to
we
T(~)
shall a l s o by
(i,j)
in
cover
for
T(~) j,
at
abelian then
is
to
ordered (i,j)
A
has at
~'.
for to
In
is the
shall
a
i
not bother to
to
groups
D
zero
~'
U
(A)
constant
refer
of
that
to
D
x in
seen
extended
is a f i n a l
the
~* i < j
is e a s i l y
T(~)
T(~)
denote
subset
such
over
Since
to
by
O's
subset
restriction
of
of
U
(A)
(A).
(i',j')
if a n d o n l y or
D
then we
object
sometimes
An e l e m e n t
the
pairs
• ~.
U
of
sometimes
of
the h o m o l o g i c a l
restriction
restriction
A 6 A,
the
diagram
• ~
of its
for all
(or d i a g r a m ) .
constant
~*
its
= 0
then
category
shall
and
be the
of
~,
t e r m of w h i c h
denote
A 6 A
of
D(i)
projectives),
restriction
functor
Let consisting
is the
for s o m e
and
as t h a t
enough
D
A ~'
reason we
between
Then
A
resolution
For
D E
subset
if
distinguish
~.
if
is the s a m e
category
projective ~
final
Furthermore,
(or any
i
that
j = j'
is a c o v e r if e i t h e r
and
i'
for an e l e m e n t
i = i'
and
is a c o - c o v e r
for
j'
is i.
a
-
An element
(i,j)
is minimal
and
is terminal
j
is maximal
in
T(~)
is maximal
in
~.
with
of
if and only if In particular
i
T(~)
is both initial and terminal.
if and only if
The minimal elements (i,i)
188
T(~)
are the elements
of the form
i 6 ~.
Some examples
of
T (~)
are given in the following
table.
(2)
(4)
(3)
(6)
0
(5)
Before homological results
turning to the problem of computing
dimension
from
[8].
of
U~(A),
The following
the
we recall and apply two is theorem 3.4 of
[8].
Lemma 5.1 Let
D
(a)
If
be pointwise h.d.
D = m 9n"
for all
AEA.
free in t~en
~G. A|
h.d. 71
< m + h.d. A A
-
(b) 0
with then
~ pm-2
projective
h.d.
A|
The object
A|
D:
with
> G
_.> ... ~
and if
K m-1
Part which
of categories
A|
T
pl
in
~G
> p0
> D ~
(a) is actually
a special
> A, corresponds
established
case of 1.5.
under
in 2.1 to the
of
the e q u i v a l e n c e
Z(~)-object
In view of lemma 5.1 we say that a p o i n t w i s e object m,
D
in
~G
is strong if it has h o m o l o g i c a l
and if it admits
projective
and
Km-I
an exact split.
sequence
simple
objects
application
general
case.
of Schanuel's
For such objects
for all
of this n o t i o n
applies
in
[8]
(which is a
to the
we then have
= m + h.d.AA
A 6 A.
The f o l l o w i n g Lemma
A|
D
free
pi
but the proof
lemma)
A|
dimension
(7) was e s t a b l i s h e d
called muscles,
h.d.
(7) w i t h
The i n d e p e n d e n c e
from the choice of the r e s o l u t i o n for special
is lemma 3.7 of
[8].
5.2
If
K1
is the constant
there is an exact sequence
(7)
A 6 A.
is d e f i n e d as the c o m p o s i t i o n G
0
is split but not projective,
for all n o n t r i v i a l
_> m + h . d . A A
Remark.
~
-
If there is an exact sequence
> Km-i
pi
189
d i a g r a m at
A
in
nG
then
-
190-
0 ----> K 2 ---~ pl
where may
p1
take
is p r o j e c t i v e pl
the left a d j o i n t
and
denotes
For object have
of
is split.
In fact, w e
with
an e p i m o r p h i s m
is the c o n s t a n t
~ Sm(Z) m6M for the e v a l u a t i o n
the set of m i n i m a l
i 6 ~
~A
strictly
K2
> 0
to be the d i a g r a m
denotes M
and
r K1
and A
i
at
and
A
at
than
we
let 0
i.
S. i
functor
elements
r Li(A )
Si(A)
diagram
greater
A 6 A
where
of
~.
L. (A) 1
denote
elsewhere. in
~A
Ti
We then
whose
kernel
o v e r the s e t of e l e m e n t s Combining
this w i t h
the
of
5.2 w e
obtain:
Corollary
5.3
F o r any 0 ~
where
pl
and
p0
i 6 ~
K 2 --~ p1
an e x a c t
) p0
are p r o j e c t i v e
For a finite
d(~)
Lemma
we have
ordered
set
= m a x h.d. i6~
> Li(Z) K2
and
~
D 6 ~A
h.d.
Li(Z)
.
we h a v e
D < d(~)
+ m
--~
,
in
0
is split.
we define
5.4
F o r any
sequence
~G
-
where
m = sup i6~
h.d.D.
~
.
If
Otherwise exact
.
By
~ has let
induction
only
one
sequence
in
K
, D
= D 1
h.d.Lk(D)
sequence
h.d.
(8) w e
sidered the
if
as
see
for
D
element
of
, L
(D k) ~
k i @ k.
is t r i v i a l .
~ , and
form
0
the
Now b y
(8)
.
5.1
(a)
< h . d . AD k + h . d . L k ( Z )
> d(~)
h.d.
a diagram
we
have
+ m,
then
K > m + d(~).
over
~ - {k}
from But
, and
< d(~)
the
K can
so t h i s
+ m
exact be Gon-
contradicts
5.5
If d(~)
= 0, gl.
all
directly 5.3
this
elements
induction.
Corollary
for
then
of
1
= h.d.D k | Lk(Z)
Consequently
number
~A ~ K
K k = 0, a n d
on t h e
element,
k be a minimal
0 Then
-
l
Proof. of
191
nontrivial
A.
i, dim.
dim.~A
Proof.
The
applied
to
5.4
The
5.1(b).
or
3,
then
~A = d(~)
If d(~)
3 + gl.
from
2,
< gl.
right other
> 3,
+ gl. then
dim.hA
hand
dim.
we
have
< d(~)
inequality
inequalities
A
+ gl.
in
are
(9)
dim.A
(9)
follows
consequences
of
.
-
192
-
C o r o l l a r ~ 5.6 For any
A 6 A
we have
U~(A) This
Proof. that U
(A), r e g a r d e d
~ d(~)
follows
+ h.d.AA
from
5.4
in
. view
as an object of ~(~*A),
of
the
has S
fact,
(A) at 1
the i th vertex.
Corollary
5.7
If ~ has an e l e m e n t
i such that h . d . L .
(Z) = d(~) i
and Li(Z)
is strong,
then
g l . dim~ for all A.
In this case
global dimension,
Proof. (b).
other
if K is a c o m m u t a t i v e
ring of finite
= d(~)
The first statement
follows
from 5.4 and
Then t a k i n g A to be the c a t e g o r y of K-modules,
statement
follows
Remark hypothesis
+ glo dim,A
then
dim K(~)
5.1
~A = d(~)
of 5.7
Remark
i.
2.
d(~) This follows
from 5.6 and 1.4.
Any g e n e r a l i z e d m - b r a i d
(See
the
[8,
satisfies
w
For any set ~ we have - i < dim Z(~)
< d(~)
from 5.6 and 1.4 in v i e w of the fact that gl. d i m . G = 1
.
the
-
Theorem
-
5.8
If d(~)
= 0, l, or 2, t h e n
h.d.U~(A) for all n o n t r i v i a l
Proof. a n y case.
for d(~)
= d(~)
Corollary
N o w by c o r o l l a r y < dim
By
5.5 and
lemma
= h.d.U
Therefore
inequality
1.4 w e h a v e
(Z)
for s u c h ~ w e h a v e
(Z) is s t r o n g w h e n d(~)
our conclusion
[8, 4.6]
follows
= 0,
f r o m 5.1.
we know that
(10)
gl. d i m . A > 3 + gl. dim. A if and o n l y ordered
if ~ c o n t a i n s
subset.
(See
[8,
it is e a s y to see t h a t crown,
where
then
n > 2.
same as
it c o n t a i n s
a suspended w
(C2) , o n l y w i d e r .
~(c2)
crown
as an u n c r o s s e d
for the d e f i n i t i o n s . )
if ~ c o n t a i n s
an u n c r o s s e d
Now suspended
o n e of the f o r m
L e t us d e n o t e
(5), and
in
(Z) = d(~) .
5.2 w e t h e n see t h a t U Therefore
5.6 g i v e s us o n e
Z(~)
= 0, i, 2, or 3. h.d.U
l, or 2.
+ h.d.AA
A 6 A.
d(~)
From
1 9 3
is
this
(6)
$
One can write
set b y dd. The
set T(c n)
T h u s c 2 is t h e is s i m i l a r
down a projective
to
resolution
-
194
-
of the form 0
,pS
~p~
,p~
~P0
~U
(Z)
....~0
C n
w h e r e the kernel of The m a t r i c e s
pl
~p0 is split but not projective.
involved are rather complicated,
not give any d e t a i l s
here.
h,d.U c
(A) = 3 + h . d . A A n
for all n o n t r i v i a l A.
It then follows
and we shall
from 5.1 that
-
195
-
REFERENCES
[i]
Balcerzyk,
S., "The Global Dimension of The Group Rings
of Abelian Groups II" [2]
Balcerzyk,
8
Fund. Math
S., "On Projective Dimension
of Modules".
Berstein,
Cartan,
(1966)
of Direct Limit
(1966)9
I., "On the Dimension of Modules
IX", Nagoya Math. J. 13; 83-84 [4]
58; 67-73,
Bull. Acad. Polon. Sci. Set. Sci.
Math. Astron. Phys. 14; 241-244 [3]
9
H. and S. Eilenberg,
and Algebras
(1958).
Homological Algebra.
Princeton University Press,
Princeton,
New Jersey,
1956.
[5]
Eilenberg,
S., A. Rosenberg,
and D. Zelinsky,
Dimension of Modules and Algebras,
Math. J. 12; 71-93, [6]
Lawvere,
Mitchell,
(1957).
B.,
10,
(April,
"On the Dimension
1963). of Objects and Categories,
I". J. of Algebra 9; 314-340,
[8]
Mitchell, II".
Nagoya
F. W., "The Group Ring of A Small Category".
Notice8 A.M.S. [7]
VIII".
"On the
B.,
(1968)9
"On the Dimension of Objects and Categories,
J. of Algebra 9; 341-368,
(1968).
9
-
[9]
196
-
Mitchell, B., "On Characterizing Functors and Categories". Mimeographed, Bowdoin College,
[lO]
(1967).
Osofsky, B., "Upper Bounds on Homological Dimensions". (to appear.)
-
197-
LOCALLY NOETHERIAN CATEGORIES AND GENERALIZED STRICTLY LINEARLY COMPACT RINGS.
APPLICATIONS. by
JAN-ERIK ROOS (Department of Mathematics, University of Lund, SWEDEN.) CONTENTS w O.
Introduction
w i.
Characterizations of locally noetherian categories
w 2.
Locally noetherian and locally coherent categories. The
..............................................
conjugate category w S.
1
.......
3
.......................................
7
Coperfect categories. The conjugate category of a locally noetherian category, grull dimension of G~othendieck categories
...............................................
w 4.
Structure of endomorphism rings of injective objects in
w 5.
Explicit realization of the dual of a locally noetherian
locally noetherian categories
............................
9
12
category .................................................
19
w 6.
TopoloEically coherently completed tensor products
39
w 7.
Explicit study of the dual and the conjugate of the Gabriel
.......
filtration of a locally noetherian category .............. w 8.
42
Generalized triangular matrix rings with a linear topology, and classification of stable extensions of locally noetherian categories
w 9.
46
Change of Krull dimension in stable extensions of locally noetherian categories
w i0.
....................................
....................................
63
Application i: The structure of right perfect, left coherent, stable PinEs ...................................
64
w ii.
"Application" 2: Quasi-Frobenius categories
66
w 12.
Final remarks BIBLIOGRAPHY
w O.
..............
............................................
68
.............................................
70
Introduction. The principal aim of this paper is to pmove (and to give applications
and precisions of) a structure theorem (cf. notably Theorem 6 and its corollaries below) for those categories
~
that satisfy the following two
-
198
-
axioms: A)
C
is an abelian category, that has direct limits that are exact
functors when taken over directed sets (this is the axiom AB 5 of [30] ); B)
C
has a set of generators
object of
C
(i.e.
Na
{Na}
, where each
Na
is a noetherian
satisfies the ascending chain condition on sub-
obj acts ). Following Gabriel [24], [25], we will say that
C
is a locally
noetherian (abelian) category if A) and B) are verified. Here are some examples of locally noetherian categories: i) ring 2)
Mod(A) = the category of unitary left modules over a left noetherian A; The category
prescheme X 3)
Qcoh(X)
of quasi-coherent sheaves over a noetherian
[25_];
The category
Mod(0 M)
of sheaves of 0_x-mOdules, where
noetherian presche e w i t h structure sheaf ~)
The categoz~
Tots(Z)
is a locally
[32];
of abelian torsion groups, and more generally
any closed subcategory [25] of 5)
X
Mod(A),
where
A
is as in 1);
The category of topological discrete G-modules, where
G
is a pro-
finite group [583 ; 6)
Let
be a graded r i n g ,
A = (An)n 9 0
ring, and where eac~ natural left A~
An
where
A~
is a left
is a finitely generated left A~
structume.
Then the category
category
Modgr+(A)
if and only if
A
for its
Modgm-(A)
negatively graded left A-modules is locally noetherian.
noetherian
of
However, the
of positively graded ~modules is locally noetherian is noetherian in the graded sense, and this condition
is strictly stronger than the one mentioned above (example: the Steenrod algebra, which is coherent, but not noetherlan [15~); 7)
The dual category of the category of pro-algebraic commutative g~oup
schemes over an algebraically closed field [4~]. It should be remarked that the cases 2) and 3) are quite different and that in the examples 2)-7) the category
~
is not in general equivalent to
9-
1 9 9
-
a module categor7 (i.e. a category of type i)). We make no attempt to summarize the results of this paper here, and we only remark that the principal ~esult is that the study of locally noetherian categories is entirely equivalent to the study of a certain class of toRological rings with a linear topology that is exRlicitly described.
Thus, problems and results that seem to belong to "pume cate-
gory theory" are in fact equivalent to "purely ring-theoPetlcal" ones~ and this gives new results in both directions.
More details about this are in
particular given in w 5, where it is also explained how our results ape inspired by and related to earlier results of Kaplansky [3~, Matlis [40_], Gmothendieck ~31], SePre [56], Gabriel [2~,
[2~ and Leptin ~(I,
~"
Several ring-theoretical and "categorical" applications as well as several open problems ape scattered throughout the paper.
Among the "categorical"
applications, we wish particularly to stress those given in w 8, where we ape able to classify all "exact sequences of categories"
o
,A
,s
,o
where
D
is a localizing subcategory of
where
D
is supposed to be stable under the fo1~nation of injective
envelopes in
Q
(locally noetherian), and
C.
Some results of this paper have been su~aDized in [54],
and the
pmesent paper will be a pamt of a systematic study of Grothendieck categories, that we hope to publish soon (hopefully together with a solution of at least some of the problems mentioned above). In omdem to facilitate the reading of this paper and in ordem to give some general hackgmound, we will first briefly review and complete some well-known results about locally noetherian categories.
w i.
Characterizations of locally noetherian categories. A categomy satisfying the axiom A) above, as well as the following
weakened form of B:
-
Bg)
~
200
-
has a set of generators
{G ) , a~ I is generally called a Grothendieck category [2~ , and Grothendieck proved in [3ql that such a category has sufficiently many injective objects.
now
~
is also locally noetherian, then we have much mope precise results.
In fact, Matlis proved in [4~, that if A
If
~
is of the form
is a left noetherian ring, then every injective in
~
Mod(A),
where
is isomorphic to
a sum of indecomposable injective objects (which in the commutative case cormespond
to the prime ideals of
isomorphic in a natural way. generally every directed
A), and any two such decompositions are
Furthermore, in this case every sum and more
lim
of injectives in
Mod(A)
is again injective
>
[l~.
These results were extended by Gabriel [23~, C2~,
general locally noetherian case.
[2~I , to the
However, there ape also several con-
verses of these results, which in the module case are due to Bass [4], Papp [45] and Faith-Walker [213 and which together with the Matlis results can be formulated as follows: PROPOSITION I.-
The following conditions on a category
Mod(A)
are
equivalent: (i)
Mod(A)
is locally noetherian (i.e.
(ii)
Ever~ injective object of
Mod(A)
A
is left noethePian);
is isomorphic to a direct sum of
indecomposable in~ectives~ (ill)
Any sum of injectives in
(iv)
Any (directed) direct limit of invectives in
(v)
[Bounded decompositions of injectives. 3
such that every in~ective in each havin K (vl)
~ s
Mod(A)
is injective; Mod(A)
is in~ective;
There exists a cardinal
c__
is isomorphic to a sum of injectives,
Kenerators!
[Strict cogenerator.]
that every, object sum of copies of
Mod(A)
in
There exists an object
Mod(A)
C
of
Mod(A)
such
is isomorphic to a subob~ect of a direct
C.
If we now replace
Mod(A)
by a general Grothendieck category
then the conditions (i)-(vi) above are not all equivalent in general
~,
-
201
-
(as for the general formulation of (v), cf. below), a counter-example to the implications (iii) => (i), (iv) => (i), (vi) => (i) is for example given by a non-discrete spectral category i).
The results of Gabriel are
exactly the implications (i) => (ii), (1) => (iii) and (i) => (iv). Also the implication (ii) => (vi) is universally valid but we do not know for example whether (ii) => (i)
is always true for all Grothendieck categories.
As for the condition (v) as it stands it has no meaning for Grothendieck categories.
However, if
that an A-module HomA(M,.)
M
s
is an infinite cardinal, it is easy to see
has ~
generators if and only if the functor
commutes with c_-directed unions.
[Foc more details, and for
relations with the notion of object of (abstract) finite type, we refer the reader to [53J( I Thus, if we say that an object category
s
has
~ ~
generators [~
cardinal, cf. ~33J if the functor
C
in a Grothendieck
infinite cardinal, or the finite
Home(C,')
commutes with c--directed
unions, then (v) makes sense, and we can in fact prove the following THEOREM i.-
The followin~ conditions on a Grothendieck category
C
are
equivalent: (i)
~'Bounded decompositions of injectives.'~ There exists a fixed
cardinal
c
composition (i)'
such that every inject ive
I
I =~Ii,
has
where each
Ii
There exists an injective object
is a direct sum of direct factors of (il)
Eyery sum of injectives in
(iii)
["Strict cogenerator.'~
every object of
~
~
in
I
C
< c --_
admits a direct de~enerators; --
such that ever,j injective .in
I;
is still injective;
There is an ob~gct
D
of
~,
such that
i_.s a subobject of a suitable direct sum of copies of
D.
I)
A spectral cateKo~ is a Gmothendieck category, where every object is
injective;
such a category is locally noetherian if and only if it is
discrete in the sense of [27].
For the general theory of spectral categories,
and examples of non-discrete ones, we refer the reader to ~ i 3 and the literature cited there.
-
2 0 2
-
This result will not be needed in what follows, so we give no proof. Indications of the proof together with the necessary generalizations to Grothendieck categories of results of the Kaplansky type about direct decompositions of modules can be found in [55 . As we mentioned above~ the example of the non-discrete spectral categories shows that the categories satisfying the equivalent conditions of Theorem 1 are not necessarily locally noetherian.
The Theorem 1 suggests
the introduction and study of a continuous variant of locally noetherian categories.
However, several problems remain unsolved.
It is for example
not known (except in special cases) whether the condition (ii) of Theorem 1 implies that (directed) direct limits of injective objects are still injective [53].
In what follows, we will however stick to the (discrete)
locally noetherian case.
This case is easily isolated from the continuous
case, if we require the axiom AB 6 [503 instead of axiom AB 5 (the axiom A above) as the ground axiom on our categories. THEOREM 2.-
We have in fact the following
[Characterizations of locally noetherian categories.]
following conditions on an abelian category
C
The
with a set of generators are
equivalent: (i) (ii)
~
is locally noetherian;
[resp. (ii'~ ~
satisfies the axiom
AB 6,
and every sum (resp.
directed
lim ) of injectives is injective;
(iii)
~
satisfies
AB 6
and has bounded decompositions of injectives;
(iv)
C
satisfies
AB 6
and has a strict cogenerator;
(v)
C
satisfies
AB 6
(AB 5
>
might be sufficient here),
and every
in~ective is a direct sum of indecomposable invectives| (vi)
C
satisfies
AB 6
and evePy direct limit of a direct system of
essential monomorphisms is an essential monomorphism. For indications of the proof of this theorem we refer the reader to ~3] and to ~0] where the condition
AB 6
and its variants are discussed.
This Theorem 2 contains both the Proposition 1 above as well as the theorem of
[43] since a module category, and mope generally a category that is
-
203
-
l o c a l l y of f i n i t e type [~33, both automatically satisfy the axiom AB 6 [507.
w 2.
Locally noetherian and locally coherent categories.
The conjugate
category. Gabriel has observed [24], [25] ~f. also [31], [561~, that if locally noetherian and then and
N(C) C
N(C)
C
the category of noetherian objects of
is C,
is an abelian category, that is equivalent to a small category,
is equivalent to the category
contravariant functors from Conversely, if
N
N(C)
to
Lex(N(C) ~
Ab) i) of left exact
Ab (= the category of abelian groups).
is a noetherian abelian category (i.e. an abelian care-
gory, where every object is noetherian) that is equivalent to a small category, then
Lex(N ~
Ab)
is locally noetherian, and its category of
noetherian objects is naturally equivalent to
(1)
N
~
>
Lex(N ~
N.
Thus the map
Ab)
defines a one-one correspondence between the (equivalence classes of) small abelian noetherian categories and the (equivalence classes of) locally noetherian categories. It is natural to ask, what kind of Grothendieck categories we obtain to the right in (i), when category
D.
N
is replaced by an arbitrary small abelian
To answer this, we first need the definition below.
first that an object
C
in a Grothendieck category
finite type [43] if
HOmc(C,')
DEFINITION i.-
C
that
C
Let
where
C'
is said to be of
commutes with directed unions.
be an object in a Grothendieck c@tegory
is coherent, i~f C
C' ~ f > C,
~
Recall
~. We saZ
is of finite type~ and if for any morphism
is of finite type,
Kerf
is so too.
If
C
family of generators~ formed by coherent objects, then we say t h a t
has a C
is
a locally coherent category. The following result can be found with small modifications in [22] and
l)
If
~
is a category,
C_~
denotes the dual category of
~
[34.
-
204
-
its proof is essentially an adaption to the locally coherent case, of the corresponding proof of Gabriel for the special case of the locally noetherian categories. PROPOSITION 2.-
The map
D
~"
> Lex(DO, Ab)
defines a i-i correspondence between the (equivalence classes of) small abelian categories and the (equivalence classes of) locally coherent Grothendieck categories.
Furthemmore
category of coherent objects of into
Lex(D_~
~
Lex(D ~
is naturally equivalent to th? and
Ab),
A_~b), bj means of the functor
D
is naturally embedded
D P--> HOmD(" , D) = h D.
Examples of locally coherent categories: i)
Mod(.A)
is locally coherent if and only if the ring
A
is left coherent
in the sense of [9], p. 63 (cf. also [13]), i.e. if and only if every finitely generated left ideal of 2)
A
J. Cohen has proved [15] that
algebra, is locally coherent. noetherian. Now let
is finitely presented. ModgT+(A),
A
is the Steenmod
This category is however
not locally
[For the notation, see example 6 in the introduction.] ~
be a locally coherent categoz,y, and let
categomy of coherent objects of
~.
Ab).]
Put
~=
Coh(~)
be the
[Then, as we have seen, Coh(C)
abelian, equivalent to a small category, and Lex(Coh(~) ~
where
Lex(Coh(C), Ab).
~
is equivalent to This category (which is
evidently locally coherent) will be called the conjugate category of what follows.
It is easy to see that
the category of left exact functors lim . FurthePmore -
~
~ ~m>
is equivalent to Ab
is
~
in
L e ~ l i m ( ~, A_b_) =
that cormnute with directed
is natumally equivalent to
C.
Oum aim in the next
>
section is to characterize completely the categories locally noetherian category.
Since every object of
~,
where
Coh(~) ~
~
is a
is artinfan
(i.e. satisfies the descending chain condition on subobjects) if
Coh(~)
is noetherian, one might think that the conjugate categories of locally noetherian ones, would be exactly the locally coherent and locally artinian
9 -
205
categories (locally artinian means for us:
-
there exists a family of
generators that ame artinian - this is different from 0ort's ter~ninology [44]).
However, these categories form only a special case of the class of
conjugates of locally noetherian categories, and
Mod(Z)
is not of this
special form (cf. Example 1 following Corollary g of Theorem 13 in w 8). Thus we must study a descending chain condition, that is weaker than the usual one:
w 3.
Coperfect categories. The conjugate category of a locally noetherian
categoz~.
Krull dimension of Grothendieck categories.
As is well
[25], not only does every object
C
in a
G~othendieck category admit a monomorphism into an injective object, but it also admits a minimal one, that is essentially unique, and that is called the injective envelope of
C.
The dual of this notion is the projective
envelope (also called projective cover). jective envelopes
However the existence of pro-
is a rare phenomenon, even if the category is a module
category (so that we have sufficiently many projectives). a ring
R
Bass [3] called
a right perfect ring if every right R-module has a projective
cover, and he proved the following result: THEOREM OF BASS.
-
R
The followin~ conditions on a ring
are equivalent:
(i)
R
is right perfect;
(ii)
R
satisfies minimum condition on pFincipal left ideals;
(iii)
R/radjR
is an artinian semi-simple ring, and
radjR
is right T-
nilpotent [3] (radj = the Jacobson radical); (iv) e
Every direct limit of projective right R-modules is projective;
oe
Bass left as an open problem, whether (ii) is in fact equivalent to: (ii)'
R
satisfies minimum condition on finitely generated left ideals;
but J.-E. BJORK proved recently [6 ] that (ii) and (ii)' equivalent.
are in fact
This generalizes an earlier result of S.U. CHASE (cf. Appendix
to [20]) which says that (ii)' is verified for semi-primary rings (a special case of the perfect ones).
In view of the BASS-BJ~RK result, it is natural
-
to call a module
M
coperfect, if
finitely generated submodules. DEFINITION 2.-
perfect if ~
206
M
-
satisfies the minimum condition on
More generally:
A Grothendieek category
~
is said to be (locally) co-
is locally of finite type (cf. [4~),
family of generators
{Ga } ' where each
G~
and if
C
has a
is coperfect~ i.e. satisfies..
minimum condition on finitely senerated subob~ects. Remark:
If
~
is locally copemfect, then every object of
~
is in fact
copemfect, so we can and will omit the womd "locally" in what follows. THEOREM 3.-
The map
~
I
>~
(cf. w 2) defines a one-one correspondence
between the (equivalence classes of) locally noetherlan categories and the (equivalence classes of) locally coherent and coperfect categories. PROOF:
Suppose first that
prove that hN
~
~ = Lex(N(~), Ab)
HOmc(N,') , N q N ( ~ )
is locally noetherian. is coperfeet (el. w 2).
hN
is copePfect.
is coherent, every finitely generated subobject of
thus of the form by
h N'.
N ~--> h N, N'
the fact that
N(~)
Since
N(~)o
is a quotient of
N.
copePfect if and only if
R
But since
is coherent,
Thus the result follows from
is a noetherian category.
A module category
hN
~,
is fully and faithfully embedded in
Conversely, given a locally
coherent and coperfect category ~, it follows that cu abelian category, and so D is locally noetherian. Remark and example:
The set of objects
is exactly the set of coherent generators for
and so it is sufficient to prove that every hN
We only have to
Mod(.R)
Coh(D)
is an artin~an
is locally coherent and
is left coherent and riKht perfect.
This class
of rings was first introduced by Chase [13], who proved that it is equiva!~nt to say, that every product of right projective R-modules is again projective. In the commutative case, these Pings coincide with the artin~an ones [IS], but in the general case they need not be either left or rlght artlnian, as the following example (which was communicated to me by L.W. Small) shows: R = (~
~>.
(For more details~ and for a general theory of generalized
triangular matrix Pings, cf. w 8 and w 9.)
-
The correspondence
s
2 0 7
-
wili be studied in more detail below,
using the endomorphism pings of injectlve objects, and we will just conclude this section with a few elementary remarks, that are easy consequences of the theory of Krull dimension of Grothendieck categories [25], that we will first recall briefly: Suppose
~
is a Grothendieck category.
Then a full subcategory
is said to be a localizing subcategory [25] if
~
_L
of
is closed under
formation of subobjects, quotient objects, extensions and
(in ~) > In this paper every subcategory will be a strict subcategory, i.e.
~Note)
lim
the subcategoPy should contain all isomorphic copies (in the big category) of its objects.] If
~
is a localizing subcategory, then it is also a
Grothendieck categor,j, the quotient category
~/~
can be for~ned, and the
.M
natural functor
~
3
> ~/~
is exact, and has a might adjoint
is full and faithful [25]. Thus Now let
~/~
C be the smallest (exists!) localizing subcategory of --o
subobjects of
S)
of
s
Form C/Co ,
localizing subcate6ory of C/C etc... ----o
aM
and l e t
containing those
s
C
x.
example:
~,
and
that S
ape
be the smallest
C ~,
that ape simple or 0 ...C...
fop a limit ordinal is evident), and theme exists
such that
C
= C
--M
= ...
If
said to have a Krull dimension defined, and we put such
0
We get a transfinite f i l t r a t i o n l ) C C ~ l C
(the definition of an ordinal
~,
that
is also a Grcthendieck category.
contains the simple objects (S simple <--> S # 0, and only
in
JM
C
--M
= ~,
then
C
--
is
dim ~ = the smallest
Not every Grothendieck category has a Krull dimension. a continuous (non-discrete) spectral category.
Counter-
One can however
introduce a continuous analogue of Krull dimension, and then this last category has dimension
0.
However,
Mod(A, 2) ~8], [49~,
non-discrete aPchimedean valuation Ping
and
~
where
A
is a
the maximal idealjdoes
not even have a K~ull dimension in this mope general sense. It is easy to see that 0
if and only if every
~
C # 0
has Kmull dimension defined and equal to contains a simple object o A
i) This filtmatlon will be called the Gabriel filtration ef
locally ~.
-
noethcrian category
~
is zero if and only if
category
~
2 0 8 -
always has a Krull dimension and this dimension ~
is locally finite [25]. [The Grothendieck
is said to be locally finite if
~
has a family of generators,
formed by objects of finite length, i.e. objects that are both artinian and noetherian [253.~ Finally, the Krull dimension of
Mod(A),
where
is a commutative noetherian ring, coincides with the dimension of fined by means of chains of prime ideals (Krull) ~5].
A
A de-
This last descrip-
tion is valid in some non-commutative cases too [25]. For another definition of Krull dimension (when it is ~ o Now if ~ dimension
0,
), see ~8].
is arbitrary locally noetherian, then T for if
C ~ 0
has Krull
is an object of _~, then a minimal element
in the family of finitely generated non-zero subobjects of simple object.
Thus in particular
~
see thus that
n the module case R
~ = Mod(R),
is artinian, if and only if
and one can prove that
Mod((~ ~))
has finite Krull dimension for
~
study of a category ~
Mod(R)
is locally finite, More detailed
In particular we will see that R
and that every dimension can occur.] Thus: noetherian category
itself is
considered above, we
has Krull dimension I.
results are to be found in w 8 and w 9. Mod(~)
must be a
is locally noetherian if and only
if it is locally finite, which is the case if and only if ~ locally finite,
C
left coherent and right perfect, The study of a locally
of arbitrary Krull dimension is equivalent to the of Krull dimension
0
of a special type (locally
coherent and coperfect), this last category being locally finite, if and only if ~
is so (i.e. if and only if ~
is of Krull dimension
0). This
gives an indication of the enormous difference between the cate~omies of Krull dimension
w 4.
0
@nd the locally finite categories.
Structure of endomorphism rings of injective objects in locally
noetherian categories. In [40] Marlin pmoved, that if
A
is a commutative noetherian ring,
-
a prime ideal in in
Mod(A),
A,
and
209
E = E(A/~)
then the endomorphism ming of
completion of the local ring
A .
-
the injective envelope of E
in
Mod(A)
noetherian category
I
~,
is exactly the
For complete (noetherian) local rings
there are structure theorems, due to I.S. Cohen ~8~, one to expect, that if
A/~o
[82]. This leads
is an injeetive object in an arbitrary locally then
Homc(I,I)
should have a natural topology
and that there should be some explicit structure theory for this topological ring.
We will show below, that this is indeed the case (to a certain degree)
and that, furthermore, the (topological) endomorphism ring of a big injective (for a definition of that see Theorem 4 below) in determines
C.
zero (i.e.
C
~,
completely
In the special ease, when the Kmull dimension of
~
is
locally finite) the above results are due to Gabriel [24],
[25], who also had some indications about the general case, but as we will see later, quite new phenomena occur when we pass to the complete study of the general case. The following well-known proposition about linearly topologized rings will be useful in what follows. Grothendieck category
~
Recall ti~t a full subcategory
is said to be a closed subcategor,] of
~
of a
~,
if
is closed under formation of subobject, quotient objects and (directed) lim
in
C
(such a
D
is then necessarily also a Grothendieck category)
>
and that a topology on a ringl)R on
R,
is said to be a (left) linear topology
if there is a fundamental system of neighbourhoods of
0
consisting
of left ideals (these ideals are then open). PROPOSITION 8.topology on
A
Let
A
be a ring.
The map Zha~ tc each (left) linear
associates the full subcategory
Dis(A)
of
Mod(A)
whose
objects are the discrete topological A-modules, defines a one-one corres~ondence betweQn the (left) linear topologies on gories of
Mod(A).
T
on
A,
and the closed subcate-
Furthermore, the open left ideals
those left ideals such that topoloK~
A,
A/!~Dis(A),
1
of
A
are exactly
and for a ~iven left linear
the~o~respendin~ s @ t o f open left ideals
J
l) In this paper all topologies considered on rings are supposed to be compatible with the ring structume.
-
210
-
satisfies :
~,
(il)
(iii)
~
~
e JT => O % ( ~ e d y
s JT'
a
;
aFbitraz ~ in
Conversely, glven a set a unique lineam top01ogy
J
A => (@:a) = { x l x a s
e J~.
of left ideals satisfying (i)-(iii), there is T
on
A
such that
J = JT"
(Cf. Gabriel [25], p. 411-412 and Bott~baki [9], exemc. 16, p. 157.) Now, let C,
and
of
C,
C
be a G~othendieck categoz~,
A = Homc(I , I). then
HOmc(C , I)
the composite map C C I
It is clear that if
C
f > I
A,
an injective object of
C
is an ambi%mary object
is in a natural way a left A-module [we write
is a subobjec% of
way a left ideal of
I
~ 9 I
I,
then
as l'f].
Thus in particular, if
I(C) = HOmC(I/C, I)
and the exact sequence
(I
is in a natumal
is injective):
0 - - > HOmC(I/C , I ) - - > Homc(I, I ) - - > HOmc(C, I)--> 0 shows that the quotient module Considem now the ease when left ideals of
A
C
A/I(C)
can be identified with
is locally noethemian.
Homc(C,I).
Then the set
that contain an ideal of the folln I(N),
where
J
of
N
is
a noetherian subobject of I satisfies the conditions (i), (ii) and (iii) of Proposition 3 as is easily seen. i = 1,2,
[We have for example that if
Ni~I,
then l(N1) f~l(N 2) = HOmc(I/(NI+N2) , I), m
where
Nl+N 2
is the image if the natural map
image is clearly noetherian if = HOmc(I , I)
Ni, i = 1,2
NlJ.LN 2 m > a~e so.~
A
(2)
0.
I(N)
A~>
lim
N~I 9
A/I(N)
N noeth.
A =
forth a
Fu~thez~nome, for
is complete (by that we mean Hausdorff too).
to see this we have to prove that the natural map
and this
Thus the ring
has a natural linear topology, for which the
fundamental system of (idft) ideal neighborhoods of this topology
I,
In fact,
-
is an isomorphism.
211
-
But the map (2) can be identified with
Home(I, I) --->
lim
Homc(N , I)
N cI N noeth. and lim HOmc(N , I) = Homc(lim N, I) = Homc(Ii, I) <-... N~I . . . . " -where the last "equality" follows from the fact that noetherian.
Thus
a natural way. for our
A
is locally
is a (left) linearly topologized complete ring in
However, not every such ring is obtained in this manner,
has more special properties.
DEFINITION 3.tha___~t A
A
C
Let
A
First a definition:
be a (left) lSnea~ly t~olo[ized ring.
We say
is (left) topologicallE co~,erfect (mesp. topologically coherent,
rasp. topologically artir~an, ...) if the Grothendieck cateKomy
Dis(A)
r
is(locally) coperfect (rasp. locally coherent, rasp. locally arti~ian,
...). Remark. -
If the topology on
A
is discrete, then
Dis(A) = Mod(A) I
and the topological notions of Definition 3 coincide with ~he co~esponding usual (discrete) algebraic notions as they should.
In the general linear
topological case, our conditions can also be expressed in ~erlns of ideals of
A
as follows:
i
l)
A
is (left) topologically coperfect if and only if
I
mental system of (left) ideal neighbourhoods of
AIc~
A
O, {O~},
has a fundasuch that every
satisfies minimum condition on finitely generated submodules (then satisfies this minimum condition for all open left i d e a l s % ) .
2)
is (left) topologically coherent, if and only if
A
mental system of (left) ideal neighborhoods of v
0,
J
Aj has a fundasuch that fom every
%
O~ ~ J, A)
the kernel of eve~-y map ~_IA/2[~i ) A/OZ (~i upen ideals of 1 is of finite type. (It suffices even to suppose the ~ i equal and
belonging to
J.).
I
!
-
3)
A
-
is left topologically arti~ian, if and only if for every open left
ideal O%
of
A,
A/OL
PROPOSITION ~.tire of A
2 1 2
~,
Let
and
is an artinian A-module ....
~
be a locally noetherian category,
A = HOmC(I, I)
Since the
generators for
A/I(N)
Dis(A),
coperfect.
(N~I, and since
N
noetherian) form a system of A/l(N) = HOmc(N, I),
sufficient to prove that for any noetherian object is a coperfect left A-module.
an injec-
with its natural linear topology. Then
i~s (complete and) t _ o ~ x
PROOF:
I
But if
finitely generated left A-submodule
N
it is more than
in
~,
HOmc(N , I)
Afl~...+AfnCHomc(N , I) (fi ~ H~
I)),
is a
then it is easy
to see that (f.)
n
Afl+...+Af n = Homc(N/Ker(N--~l >_[1_ I), I) -1 and from this the result follows, since Remark.-
N
is noetherian.
We do not know if conversely every linearly topologized,
topologically coperfect, complete ring can be obtained in this way. [We would also like to know whether ever-] right perfect ring can be obtained as an endomorphism ring of a noethemlan injective object in an abel!an category.]
It is however well-known that the more restricted class of
linearly topologized, co.._mple!e, topologically arti~ia__nrings [these rings are also called strictly linearly compact rings ( ~ 7 , S.l.k. rings ~8],
[39] or Leptln rings ~4]~
manner [24], [25] (el. also w 5). Mod(Z)
P. lll), or i.e.
can be obtained in this
The endomorphism ring of
~l_i~/~
in
is however not topologically artinian (el. Example 1 following
Corollaz~ 3 of Theorem I~ in w 8). However, if
I
is a hie injective (see Theorem # below) then the
endomorphism ring can he characterized completely: THEOREM 4.of
C
Let
C
be a locally noetherian category,
I
a big injective
(i.e. an injective that contains at least one indecomposable in-
jective of each type), and
A
the endomorphism ring of
I
with its
213
natural topology.
Then
A
-
is completer topologically coherent and
topologically coperfect, and conversely every such ring can be obtained in this manner i). PROOF (First part):
We know already that the ring
complete and topologically coperfect. topologically coherent.
A
in Theorem 4 is
We prove here that
A
is also
(The last assertion of Theorem 4 will follow
f-Dom Corollary 1 of Theorem 6 in w 5.) Let us prove more generally, that if ~,
then the discrete left A-module
injective~) as an object of Homc(N , I)
N
is any noetherian object of
HOmc(N , I)
Dis(A).
is coherent (I
First of all, it is clear that
is finitely generated as a left A-module, for if
injectlve envelope of
N,
then since
N
I(N)
is the
is noetherian, only a finite
number of indecomposables occur in the decomposition of exists an integer
big
I(N).
Thus there
v
such that I(N) is a direct factor of ~ ~I, and so v 1 HOmc(N, I) is a quotient of ~ A . Now let T f > HOmc(N , I) be any map -,topologiEj~l -from a finitely ge~gg~Q-d~discrete left A-module T to HOmc(N, I). We have to prove that (N'~I, T
N'
Kerf
is finitely generated.
Since the
noetherian) form a set of generators for
Dis(A)~
and since
is finitely generated, it must be a quotient of a finite sum of such
objects, thus a quotient of an object of the form N1
A/I(N')
is noetherian, and by composition with
Homc(N1, I)
N
F > N1
and
Nl/Im~
such that
I
Ker ~
is f i n i t e l y generated.
Homc(,I) = "~ f.
Thus
Thus
Ker{~ : HomC(Nl/im ~ ,I)
Ker ~ m u s t be finitely generated as
HOmc(N , I)
is coherent and so in
i) We will see in w 5 that the equivalence class of deterlnined by
A.
Now observe
is a big injectlve, there exists a unique
is noetherian, and so
we have just seen above.
we get a map of A-modules:
a
I t i s s u f f i c i e n t to prove t h a t
map
where
f > HOmc(N, I) .
- -
(Lemma 1 below) that since
f
HOmc(NI, I),
~
is uniquely
-
par~iculam
A
214
-
is topologically coherent, and therefore the first part of
Theorem 4 is proved modulo the following lemma: LEMMA I.i_~n ~,
Let
~
be a locally noetherian category,
A = Homc(I , I),
and
I
a big in~ective
Homc(N1, I) -~--> HOmc(N2, I)
a map of left
m
A-modules; where
N1
ly) unique morphism
and
N2
ape noetherian.
N2 G > N1
such that
Then there is a (necessar~
HOmc(G,I) = g.
PROOF:
Let ~ I and ~ I be injectives containing N 1 and N 2. sI s2 (Heme we can suppose that sI and s 2 ape finite sets, cf. proof of
TheoPem 4 above.)
Thus we have two exact sequences:
0 - - > N i - - > II I - - > K i - - > 0
(i = 1,2).
s.
1
Take
Homc(. ,
I)
of these two sequences, and considem the diasTam:
m
0 - - > Hom~(K1, I ) - - >
Homc([lI , I) --> HOmc(N1, I ) - - >
0
0---> HOmc(K2, I) --> Homc([i I, I) --> HOmc(N2, I) --> 0 -Since
_
HOmc(!II , I) = j f A -- s I
modules
?
--
is projective, there exists a map of
A-
s1
making the right square of (3) commute, and this map is
necessarily induced by a map using (3) and the fact that zag map from
N2
to
0 - - > NI - > ~ (4)
5
~~ I
r >.I I I. It is now easy to see, s2 s1 I is a cogeneratom, that the composed zig-
in the diagram
I
--> K1 -->
0
si t r 0 --> N2---->LI ~ I s2
is zemo.
s2
---> K2---> 0
Thus fmom (4) we get an induced map
lies immediately that the fact that
I
N2
G > N1 ' and one veri-
HOmc(G , I) = ~ . Uniqueness of
G
follows from
is a cogenerator, and so in pamticulam the lemma is
proved, and thus also the fimst pamt of Theorem 4.
-
w 5.-
215
-
Explicit realization of the dual of a locally noetherian category. Recall that if
T
is an arbitrary small (or equivalent to a small)
category, then the category of pro-objects of T, denoted by defined as follows (cf. ~i], The ~ o f
Pmo(~)
are the inverse systems
Hompro(T)({T e}
, {T;}s& 6I
{T }
)=
-J
lim lira HomT(T, T;) - --> 8
are epimorphisms.
(or equivalent to a s ~ l l ) category, then (cf. w 2).
every pro-object of [31], [44], [561.
I
<
It follows in particular f-mom ~7], that if
Lex(T,AD) ~
, with I
is called strict ~I] if all transition morphisms
Ta, --> Ts (a' > s)
to
iS
~6]):
{T } ~ a directed set, and the morphisms ape given by the formula:
A pro-object
PrO(T),
T
T
Pro(z)
Furthermore, if
T
is an abelian small is naturally equivalent
is artinian (abelian) then
is isomorphic to a st1?ict pro-object [24], [25],
We will now prove that if
t
(abelian and artinian)
can be realized as a full subcategory of a module category
Mod(R), stable
under the formation of kernels and cokernels (thus also under the formation of
Im, Coim ...) in
Mod(R) i) , then the category
Lex(~, Ab) ~ = Pro(T)
can be given a much more precise Pealizatlon as a category of lineartopological R-modules.
Examples for the possibility of such realizations
which go back to the topological duality theory for abelian groups and vector spaces (see notably p. 79-80 of [3~ and ~
[2,]. [25]. [31]. THEOREM 5.-
Let
[40]. R
be a pinK,
a full subeatego_~ of
l)
q
can be found in
2). T
an@belian artinian category, that is
Mod(R), that is stable under the foz~_ation of
We will see later that such a realization can always be constructed,
and this even in a canonical way. But there are also other interesting eases, besides the canonical one ...
2)
In this paper we restrict ourselves to the abelian case, but there should
evidently also exist a corresponding theo_~y of locally noetherian tcposes
~0],
-
kernels and cokernels in categor~
Lex(T, Ab)
Mod(R).
2 1 6
-
Then the dual of the locally noethemian
is naturally equivalent t o the categor,] Mod (R), "
T
whose ?b~ects ape the linearly to~ologized~ complete and separated Rmodules
M
(R
nei~hbs
considered as discrete) that have a fundamental system of of
0
formed by submodules
M
such that
whose mor~hisms ape the R-linear continuous maps. cokernels, imaKes, coimages, products,
lim
in
and
Furtherm0re ~ the kerns Mod (R)
--
<
M/M -~T,
ape the aiKebralc
T
k ePnelst cokernels, ... equipped with the inducedt quotient t ..., product, ...
topoloKy. i)
Caution:
When
T
is realized as in Theorem 5, then the objects of
although artinian in
T,
are not necessarily so in
clearly see in the examples considered later.
Mod(R)
T,
as we will
Thus we have to be careful
in the proofs below. START OF THE PROOF OF THEOREM 5: ModT(R)
We will first study the category
in detail, and notably prove the last assertion about kernels,
cokernels, ..., lim.
The proof of this is lonE, and is based upon eiEht
<
propositions, some of them of independent interest.
The rest of the
theorem will then follow almost automatically from these propositions. PROPOSITION 5.Mod(R),
Let
R
be a ring,
T
a full abelian subcategory of
stable under the for~m~tion of kernels and cokernels (not
necessarily artini~an!), and let
M
be a T-linearly topologized R-module
(defined as in Theorem 5, but no completeness orHausdorff condition is reguired. Then an open submodule
U
topology if and only if PROOF:
If
M/UET,
topology.
since
l) The
M
M,
M
is T-linearly topologized for the induced
M/UET.
then
In fact:
submodules of
of
U
is T-linearly topologlzed fop the induced
the induced topoloEy on contained in
U.
If
is T-linearly topologized , V lim
in
Mod (R) T
V
U
is defined by the open
is such a submodule, then
contains an M- (thus U-) open
ape however much mope complicated.
-
submodule
V'
such that
M/V' ~ T.
217
-
Thus we have an exact sequence in
Mod(R) 0 --> U/V' --> M/V' --> M / U - - > 0, where
M/V'
and
M/U
are in
T.
Thus
U/V'
is closed under the formation of kernels in Conversely, suppose that module of
M.
Then
U
V
is T-lineamly topologized).
T,
since
T
Mod(R).
is a T-linearly topologized open sub-
contains an open
z-linearly topologized) and (U
U
is also in
V
such that
contains an open
V'
M/V ~
(M
such that
is
U/V'~T
The natumal exact sequence
U/V' --> M / V - - > M / U - - > 0 shows that
M/U
is in
3,
for
T
is closed unoerycoKerne•
in
Mod(R)
and so the Proposition 5 is proved. COROLLARY.-
Let
M
be as in the Proposition 5.
Then the intersection
and sum of two open T-linearly topologized submodules of
M
is again of
the same type. PROOF.
Let
U
and
V
be the submodules in question.
We have an
exact sequence
(5)
0 - - > U/U ~ V - - >
Here
U~V
Thus
U/U ~ V
M/V ~ T . that
M / V - - > M/(U+V) --> 0
is an open submodule of
is a quotient of an object in
Thus since
U/U~V
'~at
is in
T T,
T
and a subobject of
is closed under images in i.e.
U C~V
Mod(R),
we obtain
is a T-linearly topologized module.
Finally, the exact sequence (5) gives that UeV
is T-linearly topologized.
M/(U+V) E 3,
and since
is open, it is T-lineamly topologized too.
PROPOSITION 6.-
Let
T
be as in Proposition 5, and let
b$ two T-lineamly topologized R-modules, and R-linear map.
Suppose also that
M2
M1
f > M2
is Hausdomff.
(directed) intersection of those open submodules of
Then
M1
and
M2
a continuous Ker f
MI, that are
is the
- 218
T-linearly topologlzed and that contain PROOF:
Let
{M2,i}i~ I
-
Ker f.
he the (directed) set of all
topologized open R-submodules of diagram:
M2,
r-linearly
and consider the commutative exact
O
O
f MI/f-I(M2, i) ~
o
(6)
f
> Kern f------> M 1
> Kem f
S2/M2,i
9 > f'l(M2, i)
~
M2
.. >
M2, i
t 0
0
It follows that the first line iS a monomorphism, and since open, it contains an open submodule
V
such that
f-l(M2, i)
MI/VET.
is
Thus we have
two maps MI/V onto> Ml/f-l(M2,1) mono> M2/M2, i where the two extreme objects are in object
is in
T,
and so
f-l(M2, i)
T.
It follows that the middle
is T-linearly topologlzed.
pass to the inverse limit of the last row of (6), and use that Hausdorff.
Now we M2
is
This gives the exact sequence:
0 --> Ker f--> f'~I 1 f-l(M2, i) --> 0 i and so the Proposition 6 is proved. From now on we have to suppose that
T
is artlnian too, and our
results depend heavily on this assumption. PROPOSITION 7.-
(7)
0-->
With the notations and the hypotheses of Theorem 5, let
(ca)-->
(D a)
(fa) .... 9 (E a) --> 0
be an exact.sequence of directed inverse systems in that each
Ca
is. in
T.
Mod(R).
Then the sequence of R-modules
Suppose
219
-
0-->
lim C
-->
lim D
obtained from (I) ~
-->
lim,
-
lim ~
--> 0,
is exact.
PROOF:
The only thing to verify is that
lim D <
So let
{~ } be an element o f
and c o n s i d e r t h e s e t s
l i m K~,
--> llm E 9 .
is onto.
X~ =
<
= f-l(~ ). if
We have
D 8 ~8>
qa~(Xs) C X (X~
+ Let
qaS)
+ C ,
where
For this we will use Bourbaki
We will choose as ~ u
objects of
T.
Mod(R)
in the same way. q~stna
qas(q) = ~ q~8
.
(cf. loc. cir.) the empty
X ~
C8-->
of
of two subobjects in
)
Let us prove (iii). is empty (thus in ~
But then
T. Let
),
(Ca) ,
thus a map between
na ~
T.
Ca
is still in ~ ,
X
= xe + C .
Then
and
Ker q~8
n
with
is in T ,
Thus (iii) is verified.
As
one uses the fact that the
of a map between objects of
can apply theor~me I, loc.cit.,
of
The condition (ii) follows
~:~(n ) = ~ + Ker(q~8),
is a map between objects of
Mod(R)
T
or we have an element
to (iv)~ it is verified in the same way~
that
whose associated linear
C
and then use the artinian hypothesis on
v-l.
[83, Chap. 1,
To verify (i), it is thus sufficient to observe that the
intersection in
PROBLEM:
Then
Let us prove that the conditions (i)-(iv) of loc. cir. %J Observe flrst, that the associated linear map qu8 of
is the transition morphism
image in
q~8"
T.
ame verified.
%2
of
then
Everything will follow, if we can prove that
is non-empty.
variety is in
since
It is cleam that
(D),
be t h e r e s t r i c t i o n
set, and those affine subvarieties of
either
= ~a"
is an inverse system of sets~ and it is even an inverse system
theor~me i, p. 138.
v qu8
f (x)
are the transition maps of
qV 8 : X S - - > X
of affine vamieties. lim X
= x
(8 > ~)
Da
v
X
T
is still in
T.
Thus we
and the Proposition 7 follows.
It is easy to see that Proposition 7 can be formulated as saying
lim(1)Ca = 0 [46].
the same hypotheses?
Is it true that
lim(i)c ~ = 0, i _> i,
under
[This is true at least in some special cases [47l
Now that we have Proposition 7 we can easily continue to develop
-
the theory of
Mod (R). T
2 2 0
-
In all that follows, we suppose that we ape
under the hypotheses of Theorem 5 (last time we recall it). PROPOSITION 8.-
Let
M
be an object of
Mod (R),
M.
T
a filtered
1
decreasing family of open T-linea~l Z topoloKized submodules (they ame then in
Mod (R)). T 0-->
Then the sequence of R-modules
O
Mi
9 M
9
1
9 0
lim M/M i < .
i)
i
is exact PROOF:
The proof is analogous to that of prop. 16, p. 391 of Gabriel
[25], but we have to make
T
appear explicitly ever-jwhePe. Let
{U } s
be the directed (cf. The Comollar T of Proposition 5) family of open Tlinearly topologized submodules of
M.
Considem the commutative exact
diaETam 0
> N
0
>
Us
>
M
>
lim
M/U u
(8) <
lim(Mi+Os)/M i -->
i,s Here
T,
lim M/M i
9
i
(Mi+Us)/Mi ~-~> U s / M I N U s
this object is in
<
<
lim M/(Mi+U s)
i,~
and by Pmopositlon 5 and its Corollary
and so by PPopositlon 7, the last morphism of
the last llne of (8) is onto, and in the same way we see that the vertical morphism to the right of (8) is onto.
Since
M-->
lim M/U s <
is onto (it is even an isomorphism), the Pmoposition 8 now follows f-eom the diagram (8) if we can show that it is sufficient to prove that since theme is an
s
such that
~im(Mi+Us)/M i = 0. For this IgS lu~(Mi+U )/M i = 0, and this is clear U s = Mi,
and so the proposition is
pmoved.
l)
and
lim <
ape here taken in
Mod(R), but we will see later that
,
they coincide with the corresponding
~
and
lim <
in
Mod (R). T
-
Remark:
221
-
Pmoposition 8 as well as the following one, will be made more
precise below (the axiom PROPOSITION
9.
AB 5" [301 is valid in
Le__~% M EModT(R)
-
topologized p~en submodule and
and let
{Mi}
= I
i
PROOF (~ la Gabriel):
-->
0
be a ~-linearly
a directed decreasin~ family of
T-lineamly topolo~ized open submod.ules of
u +O"i
U
SodT(R) ...).
M.
Then
§ "i )"
i
Consider the exact conl,utative diagram
U
>
M
0 ---> lim U/U (~M i ~ > <
------>
llm M/M i ~ >
0
'>
lim M/(M.+U)I - - *
<
Mod (R)
MIU
0
<
Since
U
is in
for the induced topology, and since
is in
9
(Corollar-] to Proposition 5) we get by Proposition 8 that
is onto. (The same pr~position shows that
j
lemma ([9], Chap. i, w i, n ~ 4) shows that = in
But
and the image of
U is thus O
Ker h
PROPOSITION i0.-
Let
kernel of
Mod(R),
f
t.~o ModT(R) PROOF:
1
Thus the snake
Ker h = Im(Ker j),
9 Ker j = ~ i Mi,
x l(~r j
is onto.)
U/U (AM i
in
M1
Mi + U U f > M2
and so the proposition follows.
be a morphism in
ModT(R).
Then the
equipped with the induced topology, belonKs
[so this is also the kernel of
f
in
ModT(R) ~.
By Proposition 6 we have
(9)
Kerf = KerfcM l,iC M 1
Ml, i
(directed intersection)
MI, i open, MI/MI, i E T To prove that
Ker f
with the induced topology is in
sufficient to prove that every open submodule
U C MI
open submodule
is in
V
such that
Ker f/VC%Ker f
Mod (R),
it is
contains a smaller T.
(The
-
completeness of
Ker f
suppose that
is such that
U
can in fact take
222
-
for the induced topology is trivial.).
V = U.
M1/U q T,
We may
and we will prove that then we
Consider the diagram
= Ker f C_->M 1
$, MI/U
0
It is clear that
E(MI, i)
is in
T,
for
g(Ml, i) = Ml~i U (~MI, i
and we
can use Proposition 5 and its Corollary. Thus the
g(Ml, i)
MI/U ~ T.
Since
family, say
T
is amtinian, there is a minimum element of this
g(Ml,i,).
(i0)
of
form a filtered decreasing set of subobjects in
Ml,i M
Thus + U = MI, i + U,
i ~i x .
But according to Proposition 9 we have
. / • . (MI,i+U) M
I
> 1
1
= i>~i(Ml,i
)
+U
and this together with (i0) and (9) gives
Ml,iX Thus to
Kem f / U ~ K e r T
+ U = Ker f + U.
f = (Ker f + U)/U = (Ml,i M
+ U)/U = g(Ml,i K)
belongs
and the pmopositlon is proved.
COROLLARY.-
Under the hypothesis of Pmoposition i0, the quotient in
Mod(R), MI/KeP f,
equipped with the quotient topolosy, belongs to
ModT(R). PROOF:
Let
U
be an open submodule of
M1
such that
MI/UET.
by the proof of Proposition i0 we know that rv
(Ker f + U)/U-->
Ker f / V N K e r
f
belongs to
T,
and so the exact
Then
-
2 2 3
-
sequence
(ii)
0 - - > Ker f / V ~ K e r
shows that
f - - > MI/U--> Ml/(Ker f + U) ---> 0
Ml/(Ker f + U) E T,
and so
Ml/Ker f
topologized for the quotient topology.
is T-linearly
If we pass to the inverse limit
in (ll), and use Proposition 7, and the fact that complete i )
M1
and
Ker f
ape
we obtain that
M1/Ker f
is also complete for the quotient
topology , thus an object of
ModT(R),
and the corollax,] follows.
Now we wish to compare
M1/Ker f
with its quotient topology, and
Im f
Mod(R~ with the induced topology from
[image in
M 2.
With the notations and h>-pothesis of Proposition i0, the
PROPOSITION ll.-
natural alsebraic isomorphism isomorphlsm, when
Ml/Ker f
h
MI/KeP f and
> Im f
is a topological
are ~iven the quotient and the
Im f
induced topology, respectively. PROOF:
h
Since the algebraic isomorphism
map, it is clearly continuous. ever7 open submodule of Ml/Im f
and
Im f
submodules of
Ml/Ker f
such that
is also open in Now let
M2/ViqT,
(V i)
Im f,
Im f).
The
V i (~Im f
= (Im f + V.)/V. 1 1 Im f / V i O I m
be the set of open
0
(V i (~Im f) for the induced
Since also
is a subobject of
M2/Vi~T,
Im f/V i 6Aim f
Im f/Vi ~ I m
is
f =
we obtain that
f q T.
Now let such that
T.
(we will identify
are open for the induced topology on
thus open for the quotient topology, and so
the quotient of an object of
Im f
and consider
(this is a fundamental system of neighbourhoods of topology on
continuous
Thus it is sufficient to prove that
algebralcally).
M2,
is induced b y a
P
be an open submodule of
Im f / P C
T
for the quotient topology,
~ecall that by Corollary of Pr.oposition I0, Im f
with the quotient topology belongs to
i) The completeness of
Im f
Kerf
ModT(R ~ , and consider the
follows from Proposition i0.
-
224
-
diagram 0
~> V i N I m f ----> Im f
Im f/P
0
I claim That
k(V i N Im f) ~ T .
k(Vi6]Im f) =
But
Vi6~Im f
We have (V i ~ Im f) + P p
V. ~ I m 1
f
P {~V i f~Im f "
is open z-lineamly topoloEized submodule of
quotient topology, and since
P
is also such a submodule,
Im f
for The
P f~Im f6~V. 1
is SO tOo (Corollary to Proposition 5) and so by PmoposiTion 5. in
T
Since
{k(Vi(~Im f)}i
T
(Vi 6Aim f)/~C~V i /Aim ~ ET
is amTinian, The decreasing family of objects
has a minimum element, say
k(Vi,(~Im f),
i.e.
(Vi" ~AIm f) + P = (VifAIm f) + P, i ~ iN. Now apply the Proposition 9 (quotient Topology).
(ViM~Im
~ he
f) + P = ( ~--~M. (Vi ~ I m i>i
last equality follows fmom the fact That
V.M N Im f ~ P . 1
We obtain
We have thus proved That
P
M2
f)) + P = P
is Hausdorf~
Thus
that is open fop The
quotient Topology, is also open fop the induced Topology, and so These two topologies coincide and the Proposition ii is pmoved. PROPOSITION 12.M,
Let
M ~ M o d T ( R ).
In order that a R-submodule
equipped with The induced Topology, belongs To
N
ModT(R) , it is
necessarTand sufficient that
(12)
N =
1 % NCUCM
t)
(This is automatically a directed intersection, cf. PmoposiTion 5.)
U open, M / U & T PROOF:
I f (12) is verified, then we have an exact sequence
of
-
f>
0 ---> N ---> M
225
-
F. I(pPoduct over the U:s of (12)I %. .L
77MIu U
II M2 If we equip
M2
with the product topology (every factor is given the
discrete topology), then we get an object of
Mod (R)
as is easily seen,
T
and
f
is continuous.
Thus
N = Ker f,
belongs to
Mod (R)
(Proposition
T
i0).
C onvemsely suppose that
Mod (R),
and let
T
Then of
U ~N T.
U
N
with the induced topology belongs to
be an open suhmodule of
is open in
N,
and thus
M
N/U /%N
with quotient in
T.
is a quotient of an object
But the exact sequence
(13)
0---> N / U ~
shows that
N-->
M/U--> M/(U+N)--> 0
N/Ug~N
is a subobject of an object of
T,
M/(U+N)~ T
an object of
and so also
If we pass to the
lim
T
too, thus itself
by (13).
in (13), using the Proposition 7, we get an
<,
exact commutative diagram
0 -->
0
lira N/U 6~N -->
>
N
lira M/U -->
~>
M
lira M/(U+N) --> 0
I~
M/N
>
where the first two vertical morphisms are isomorphisms.
0
Thus the third
one is so too, and this gives
N=
/ IV NCVcM V open, M / V q T
and so the Proposition 12 is proved. END O F T H E that
PROOF OF THEOREM 5:
Mod(R)
It now follows from what we have done,
is an abelian categor,], where the kernels, cokez~els,
images etc. are the algebraic kernels, cokernels, induced, quotient, has ambitrary
llm, <--
... topology.
... equipped with the
It is also easy to see that
and that they ape the algebmaic
llm <
ModT(R)
equipped with
-
226
-
the
lim topology. To show that the lim over directed sets are exact < < functors in ModT(R) , essentially according to the dual of proposition 1.8 in [30] [a complete proof of what we need can be found in Chapter III of Mitchell's book [41] (use proposition 1.2, lemma 1.5 and theorem 1.9 of loc.cit. ~ ,
it is sufficient to prove that if
directed decreasing family of
{Mu}
ModT(R)-subobjects of
is a
M E ModT(R),
then the sequence
0--> ~M
lim M/M s --> 0
--> M - - > <
is exact [in seen].
Mod(R)
or
ModT(R) - it is the same thing as we have just
But we know (Proposition 12) that every
M
M CVsj~M M/V
. E s3
, Vej open ~C
Thus we get an exact sequence of inverse systems
0-->
{V j } - - > {M}---> {M/Vsj}--> 0.
But the set of these
V . (s and j vary) ordered by inclusion is directed. s]
Thus Proposition 8 gives an exact sequence
(i~)
0-->/~Vaj ---> M-->
lim M/Vaj --> 0 <
and by the same proposition and the associativ~ty of
lim
we get from
<
(14) if we first pass to the limit over each
0-->~Ms-->
M-->
j
lim M/Ms--> 0 <
is exact, and so we have verified the axiom It is clear that the objects of
T~
AB 5x
Mod(R).
quotient sequence
M/M s
Mod (R), in
ModT(R)
Fumthermore, they form a family
to) a small catego~y~ of cogenerators for an object of
for
T.
and let
{Ms}
ModT(R).
considered as modules with the
discrete topology, are the amtinian objects in embedded in
that
ModT(R).
and
T
is fully
IT is (equivalent t_ In fact, let M be
be the open submodules
with
By hypothesis (M Hausdorff) we have the exact
-
0---> M - - - > ~ M / M
227
-
a
and this proves the assertion. We have thus proved that the dual categomy of
Mod (R)
is a
Y
locally noetherian catego~. Finally, consider the functor
F : Pro(T)
> ModT(R)
that is defined on objects by: {Ca } J
> lim C
(with the lim
<
topology)
<
and that is given on morphisms in the natural way. This functor is an equivalence of categories: of
Since every object
Mod (R) is clearly isomorphic to an object of the form
(where
{C }
F({Ca})
is even a strict pro-object), it is sufficient to prove
that the natural map
lim lim HomT(Ca, C~) - - > <
.J
8
~
<
-
<
..
a
is an ison~rphism. T~T,
HOmMod (R) ( lim Ca, lim C~)
>
But for this it is sufficient to prove that for any
the natural map
(15)
lim
Hom (Ca, T)
> H~
>
(R) ( lim Ca, T) ~
<
,..
a
is an isomorphism.
But since we now know that the dual of
ModT(R)
is
locally noetherian, the assertion that (15) is an isomorphism, follows from the dual of the last part of eoroilary I, p. 858 of [25~. Thus the Theorem 5 is completely proved. We now pass to some applications of Theorem 5. case for us here will be the case when category, where the map
R
C
The most interesting
is a locally noetherian
the endomorphism ring of a big injective
T CMod(.R)
will now be defined.
I
in
~
and
We know by Lemma 1 of w ~ that
-
(16)
N(~)~
~
2 2 8
-
> HOmc(C , I ) ~ M o d ( R )
(N(~) = the category of noetherian objects of of
is a full embedding.
Furthermore
R
C)
has a natural topology, and it is
easy to see that (16) defines a natural equivalence between the category T
Coh(Dis(R))
of coherent objects in
= Coh(Dis(R)) E~'~> N(~)~
ModT(R).
Dis(R).
C_.~
Lex(N(~) ~
and
Now if we take
then all the conditions of Theorem
easily seen to be verified, and so equivalent to
N(~) ~
Ab) ~
5 a~e
is naturally
However, due to the special form of
T,
this
last category can be intePpreted as the category of topologically coherent, complete R-modules, denoted by topologized R-modules
M,
TC(R),
that are complete, and that have a fundamental
system of open neighbourhoods of M/M
O,
formed by submodules
is coherent considered as an object of
ape the continuous linear map. is an object of of
whose objects ape the linearly
TC(R)
TC(R),
FurthercBore
Dis(R) R
M
and whose morphisms
with its natural topology
considered as a left R-module, and the objects
aloe automatically topological left R-modules, when
this topology that is coarser than the discrete one. HOmc(C, I)
R
is given
Finally every
with its natural linear topology [defined by the
HOmc(C/N , I)
(N~C,
C.~
is given by
TC(R)
noetherian~, belongs to
TC(R)
and the equivalence
C ~--> HOmc(C , I).
In the reasoning above, it is not necessary to start with fact, if
such that
R
~.
In
is an arbitrary left topologically coherent and topologically
coperfect ring (R
not necessarily complete), then
az-tinean abelian category, and so locally noetherian although conjugate category
R
TC(R)~
is not in
Lex(Coh(Dis(R)) ~
Ab)
Coh(Dis(R))
Lex(Coh(Dis(R), Ab) TC(R)
is an is
in this case, and its
is equivalent to Dis(R). Let us
summarize and complete the results obtained: THEOREM 6.in ~,
Let
C
be a locally, noethezian categoPy,
R = Homc(I, I)
the endomorphism ring of
I
I
a bi~ in~ective
with its natural
-
I)
llnear topology
, and
TC(R)
229
-
the category of topologically coherent
complete linear-topological R-modules (they are then automatically topologically coperfect). C_~
~
Then we have a natural functor
Homc(C , I) E T C ( R )
(natural topology on Home(C, I))
and this functor defines an e~ulvalence of categories Furthermore~. the con~u~ate category (w 3) ~
of
C
C_~ ~
TC(R).
is naturally
i
e~uivalent to
Dis(R).
Conversely, given any topolo~ically coherent and topologically coperfect complete ring
R,
then
TC(R) ~
is a locally noetherian cateEory,
and so in particular every projective of product of indecomposable prp~ectives. TC(R), e
TC(R) Further
is (uniquely) a direct R
is projective in
and every indecomposable projective is of the form
is a primitive idempotent, (i.e. e
orthogonal idempotents).
Re,
where
is not the sum of two non-zero,
Finally, if
R
is topologically coherent and
topologically cop erfect (but not necessarily complete) then
Dis(R)
the .conjugate of a locally noetherlan category, thus of the form
is
Dis(R),
where /R is also complete (this sort of completion will be studied below). PROOF:
Everything has been proved except the assertion concerning the
projective objects (R complete).
But in
TC(R) ~
every injective is an
essentially unique sum of indecomposables, and so ~y duality every projective in Further
R
TC(R)
is a unique direct product of indecomposable ones.
is projective in
TC(R)
as is easily seen.
be an indecomposeble projective ~ 0 of submodule of
P
such that
Then the map
P---> P/U
P/U
P.
Since
i) R
P
is indecomposable,
and let
is coherent in
is continuous,
and so the projective envelope of
TC(R),
P/U
Dis(R)
Finally, let U
be an open
and non-zero.
thus an eplmorphism in in
P - - > P/U
TC(R)
TC(R)
is a direct factor of
is the projective envelope
is then complete, topologically coherent and topologically co-
perfect, cf. w ~.
P
-
of
P/U.
But
P/U
is coherent in
230
-
Dis(R),
thus in particular a n finitely generated R-module, and so we have an epimorphism ~ R -->P/U--~0 1 in Mod(R), and this is a continuous map, thus an epimorphism in TC(R). n It follows that P is a direct factor of J_LR (n finite) in TC(R), 1 and since P is indecomposable, it is a direct factor of R (essential uniqueness of the decomposition of R), thus of the form a primitive idempotent.
That
e
Re,
where
e
is
can be arbitrary (primitive) is easily
seen. COROLLARY i.-
Any lineamly topologized topologically coherent and
topqlo~ically coperfect complete qing
R
ca n be represented as the
endomorphism ring with its natumal topology of a big injective in a %ocally noetherian category determined by
~,
whose equivalence class is uniquely
R.
END OF THE PROOF OF THEOREM 4 (w q):
Remark.-
! Apply Corollary I.
Since a big injectlve in a locally noetherian category is not
uniquely determined (a given indecomposable injective can occur an ambitramy number of times), we see that we can have R
being topologically isomorphic to
R'.
We will now study
closely and also determine the degree of choice of Recall that if ~
C~--~> C_~, without
~
and
I
more
R.
is an arbitrar-y Grothendieck categoz-y, ~
the spectral category of
R
P--> Spec(~)
an arbitrary injective object of
~,
then the kernel of the natural map (17)
Hom~(I, I)--> Homspec(~)(P(I) , P(I))
is exactly the Jacobson radical of the ring
Homc(I , I)
~his madical
w
will be denoted by
madj(Homc(I , I)~
[59], [19], [51~. FuPthemmore,
since (17) can he identified with the directed
lim
of the e Dimorphisms
- - >
(I
is injective)
Homc(I , I) --> Homc(V , I), V C I
of
I [51], (17) is an epimorphism.
Homc(I , I)/radj(Homc(I, I))
essential subobject
Thus the quotient ring
is naturally isomorphic to
-
Homspec(c)(P(I), P(I))
2 3 1
-
and thus it is a (von Neumann) reEular right
self-injective Pin E [51]. Now in particular, if
C
is locally noetherian then
discrete l')[27~ i.e. of the form
~
Mod(K i ) where
isomorphy classes of indecomposable injectives in i ~ S,
Ki
is the skew field
Then
I(C)
P(C)
of
C
C,
to get
P(C),
and where for The functor
i E S
occurs in
is
take the injective
Ki, {V i}
where the i ~
K. 1
P
and decompose it into a direc~ sum of indecomposables.
is a set of vector spaces over
dimension ovem
is
is the set of
Hom . . . . (P(i), P(i)). ~pec ~ )
easily made explicit on objects: envelope
S
Spec(C)
of
V. 1
I(C).
s
is the number of times the indecomposable
Thus in particulam
I(C)
is a big injective if
and only if
dimK.V i ~ 0, i 6 S . Let us say (cf. [25]) that I is a 1 sober injective if dimK.V i = i, i ~ S, i.e. if and only if every 1 indecomposable occums once and only once in I(C). From what we just have said it follows that the fact that
endomorphism ring of
I,
I
is sober can be easily seen on the
or more precisely on
for this Pin E is naturally isomorphic to
Homc(I , I)/radj(Homc(I , I))
I~EndK.(Vi). is
i
Since by the Theorem 4 of w W every lineaPly topologized topologically coherent and topologically coperfect, complete Pin E is the endomorphism ring of a big injective in a locally noetherian categor-], we get: COROLLARY 2.-
Let
A
be a linearly topolo~ized, to~gloKically coherent
and topologically coperfect complete rin~.
Then
A/radj(A)
Neumann regular right self-in~ective rin~ of the form
~-~EndK.(Vi), i ~S
where the skew fields
Ki
and the left K.-vectorspaces 1
is a yon
V. 1
i
are un~quel~
determined. Remark I.-
l)
A/radj(A)
More generally
is also left self-injective if and only if
Spec(C_) is discmete if
defined [25], and even more generally, if
C
has K~ull dimension C
is locally coirreducible
[423 (this is even a necessary and sufficient condition for discretenesS.
-
di-KV i <
2 3 2
-
V i [52].
l Re~rk 2.-
Corollary 2 is probably also true if we leave out the coherence
condition. DEFINITION 4.-
With the notations of Corollar~ 2, we say that
A
i__ss
sober if
dimK.V i = l, i ~ S. l Using this definition we get the important
COROLLARY 3.-
The n~p
A ~--> TC(A) ~
defines a one-one coz-~espondence
between the topologlcal isomorphy classes of linearly %opologized, topol0gically coherent, topologically coperfect and complete rin~s that ape sober, and the equivalence classes of locall~ noetherian categories I). If we say that two llneaply topologized topologically coperfect, topologically coherent, complete rings Morlta equivalent" if COROLLARY q.-
A
TC(A)--~> TC(B),
and
are "topologlcally
then we obtain:
Every linearly topolo~ized, topologically coherent,
topologlcall~ coperfect , complete rin~
B
is topolo~ically Morita
equivalent to a uniquely determined such a rin~ B
B
A
that is sober, an__~d
can be explicitly described as a topological n~trix rink over
A as
i_. [38]. [39]. The discrete rings of Corolla1~y ~ ape exactly the left coherent, right perfect rings (cf. R e ~ k
and example following Theorem 3 in w 3).
For these rings, the topological Morlta equivalence is the ordinary Morita equivalence ~],
[25] as is easily seen.
COROLLARY 5.-
A ~-> TC(A) ~
The map
Thus:
defines a one-one correspondence
b e.tween the Momita equivalence classes of left coherent, riEht perfect PinEs and the equivalence classes of locally noetherian categories that have a bi~ noetherian in~ective (i.e. every indecomposable injective is noetherian, and there are only a finite number of isomomphy classes of l)
This generalizes the result of Gabmiel ~4],
~5];which says that the
locally finite categories correspond to the pseudo-compact rin~s [25].
-
2 3 3
-
indecomposahle injectives). As we have remarked befome in w 3, example of a locally noetherian category Corollary 5 that is not locally finite.
ModIQ~)= C
TCI~ ~ I ~
is an
satisfying the conditions of
If however
C
is also a module
i
category, then
C
is locally finite [21].
Finally, let us observe that if ~ injective of ~, R = Homc(I , I), 9C
is locally noetherian,
I
a big
then the anti-equivalence
~--> HOmc(C , I ) ~ T C ( R ) m
tmansfor~ns injeetive objects into pmojective objects in
TC(R).
However,
these latem objects are not necessamily pmojective and not even flat in Mod(R)
in general.
The situation can he completely clamified if we use
the theory of Chase ~3]: THEOREM 7.(18)
With the notations and h2]potheses of Theorem 6, the functom ~ ~C
~
> Homc(C , I) 6 Mod(R)
transforms inject ive objects into pmo~ective (resp. flat) ones if and only if
R
is might cohement and left perfect (resp.
PROOF:
R
is might coherent).
Suppose first that (18) transforms injectives into projective
(mesp. flat) objects.
Then since
R = Homc(I, I),
we have that
~ - ~ R = H o m c ( ~ I , I) as a left R-module, and s i n c e ~ I is injective, K -- K K we have by hypothesis t h a t ~ R is projective (rasp. flat) as a left RK module for any K. Thus by Theorem 3.3. ~esp, the left-right symmetric of Theorem 2.1~ of [13] R
is might coherent and left perfect ~esp. might
coherent (cf. also Boumbaki [9], p. 63 Exercise 12)~. Conversely, suppose that be any injective in ~.
Since
R
is as in the last sentence, and let I
is a big injective,
J
J
is a direct
factor of a suitable s u m ~ I , thus Home(J, I) is as a left R-module a L direct factor o f ~ R , thus projective (mesp. flat) by the theorems of L Chase (-Boumbaki) just cited, and so the Theorem 7 is completely proved.
-
PROBLEM:
234
-
Is a left linearly topologlzed, left topologically coherent,
left topologically coperfect, complete, left perfect, rlght coherent ring R
discrete?
(If we omit the condition left perfect, then
necessarily discrete as is easily seen.)
is no!
If so, then the rings
Theorem 7 would be discrete in the projective case. is not necessarily amtinian on any side:
R
(~)
R
of
Such a discrete
R
is coherent and perfect
on both sides, but not aPtinian on any side. We now pass to a second application of Theorem 5 above. linearly topologized, and of
Mod(A)
T = Art(Dis(A))~Mod(A)
formed by the discrete artinian modules.
Let
A
be
the full subcategory Then
T
is evidently
an amtinian abelian categor~ that is closed under formation of kemnels and cokemnels (it is even closed undem the formation of subobjects and quotient objects in
Mod(A)), and by Theorem 5
MOdA~t(Dis(A))(A)
is the dual of
a locally noetherian categor,] in which dual the kernels, cokemnels, etc. ape the algebraic ones. 1)if
A
with its linear topology, itself belongs
to
A
is exactly what is called a stmictly linearly
MOdAPt(Dis(A))(A ) then
compact rin~ in 503, p. lll-ll2 [i.e.S.l.k. Ris E in [38], [39] and a Leptin ruing in ~4]Jand_ then the category
Lep(A)
MOdAmt(Dis(A))(A)
of Leptin modules over
A
can be identified with (el. Gabmiel [24], [25])
which is thus the dual of a locally noetherian category ~4],
[25~.
However in this case, it is not true in general that ever-] projective indecomposable object in
Lap(A)
is of the form
Ae,
wheme
e
is a
pmimitive idempotent (cf. examples in Gabriel [24J). COROLLARY 6 (of theomem 6).
If
A
iS a Le~tin rin~ (a strictly linearly
compact ming) then theme exists a lineamly topqlogized, topolo~ically cohement, topologically copemfect and complete ring potent
e
pamticulam
i_~n B, A
such that
A = abe,
B,
and an idem-
with its natural topology.
I__nn
is the endomorphism Pine (with its natural topology) of an
in~ective (not necessarily big) of a suitable locally noetherlan category. l)
The topology is the induced ... one.
-
Remark.-
235
-
It is probably not true that ever,] Leptin min E is topologically
coherent (if this were true, then we could of coumse choose e = 1
B = A,
in Comollary 6).
PROBLEM:
If the min E
fop a suitable
R
has a linear topology, so that
T CDis(R)
coperfeet and complete.
(T
as in Theorem 5),
Is the converse true?
then
R E Mod4R) R
is topologically
If not, characterize those
Pings that ame obtainable in this manner.
w 6.
TopoloKically coherently completed tensor products. This section will he used in w 7, notably fop an explicit description
of the Gabriel filtration of
TC(A) ~
when
A
is a linearly topologized,
topologically coherent, topologically copemfect and complete ring.
We
will introduce and study a notion of topologically coherently completed tensor product over
A,
~c ~A
denoted by
"
compact [this is for instance the case if
In the case when A
A
is pseudo-
is commutative - this follows
from the theorem 8.4 of Chase [1S]] then this tensor product coincides with the one introduced by Gabriel in ~ 4 3 i) and denoted by
~A
there,
and it is well-known that this last tensor product generalizes the usual completed tensor product for ~ d u l e s over noetherian local rings, used for instance in ~7] and in ~0].
As a ~tivating (and - as we will see below
in Theomem 8 and Remark 1 - an exhaustive) exa~le fop the introduction of B
~A'
let
~
and
D
be two locally noetherian categories,
the associated topological mines, and
covariant
left exact funetors from
C
Lex/ ~ -.. - (C,D) --
to
D
A
and
the category of
that commute with directed -.%
lim
.. ( , )
is defined in an analogous way I.
L e ~ lim(~,~) = Lex(N(~),~) >
w
~(C) = the categor~ of noetherian t_ objects of ~'I
and that
i)
We have that (el.
Cf. also ~6] for the commutative case.
2)
236
-
-
LeX/v lim(~,D) ~ ---> Rex/qlim(TC(A) , TC(B)). If
T
T(A)
is a functor belonging to this last category, then not only does belong to
TC(B),
but this
T(A)
also has a natural right A-module
structure, compatible with the left B-module structure: plication by by
T
k ~ A
into a map
T(A)
operation
T(.k)
of
M = T(A)
A
on
B-A-bimodule).
is a continuous map
as
A
T(.k) > T(A) in
.k,
'~ > A
TC(B).
right multi-
that is transformed
We will write this
and we will now prove that this right operation
is continuous (so that
M
is in fact a topological
Using the formula
m.~ - mo'~o : (m - mo)'(~ - ~o ) + mo'(~ - ~o ) + (m - mo )'~o we see that it is sufficient to prove the following three results: I)
The natural map
2)
For each
3)
H x A_._> M
mo~
H,
continuous at
0.
For each
~
O
~ A,
continuous at
is continuous at
the map
A-->
M
defined by
~-->
the map
M ---> H
defined by
m ~ --> m'~
To prove I) and 2), let
directed decreasing set of open left ideals of
Then
M/U
U
be an open B-submodule of is an artinian object of
creasing family of subobjects of
has a minimum element TC(B)
and
M/U
is
is
O
{C~L}
be the
that belong to
TC(A),
such that
H/U
is in
T.
and thus the directed deTC(B):
T(A)/U) M
Im(T(C~H)--> which implies that
in
> a . TaN = T e , e --
lim T(Ot e) = T(lim
A,
M = T(A) TC(B),
T a = Im(T(Ote) --> T(A) ~ >
in
mo "~
0.
We have just seen that 3) is true.
and let
(0,0).
But the directed
CK)e = 0,
are exact
and so
T(A) --> T(A)/U) = 0
H. Ot
C U,
and this proves both I) and 2) and even
eH
the (apparently i)) stronger result that the map equicontinuous
(cf. w 8).
M x A-->
H
is left
More precisely:
i) That the equicontinuity property is only apparently s t r o n g e r e ~ c a n proved in much the same way as the lemma 0.3.1 on p. 71 of [26] is proved (it is essential that
H g TC(B)
as a left B-module, cf. w 8).
be
237 -
THEOREM 8. -
Let
A
and
B
be as above.
Then the functor
RexN Iim(TC(A), TC(B)) ~ T ~--> T(A) 6 BTCT A < ,
~here
BTCT A
bimodules
is the category whose objects are those topological B-A-
M,
that belong to
TC(B)
are the continuous bimodule map~ M ~ BTCTA, M~AC
--,
as left B-modules, and whose morphisms
is an equivalence of categories.
the corresponding functor
TC(A)--> TC(B)
for the fo!lowin E reason:
For
N ~ TC(A),
Given
will be denoted by we have a natural
equivalence of functors: HOmTC(B)(M~CA N , V) e~> BiltOPB(M, N; V), where
BiltoPB(M , N| V)
define B-linear maps
is the set of continuous maps
M ~AN-->
I
~C
Remark i.-
We have seen in w 5 that
Now let
BiltoPB(M , N~ - ) 6 Lex(Coh(Dis(B)), Ab)
M ~ A N,
for a unique
M
be in
L [ - 6 Coh(Dis(B~.
This
L
defines
and one verifies that the Theorem 8 is true. Of course
BiltoPB(M , N| -) s Lex(Coh(Dis(B)), Ab) A
c
M ~A N
The Theorem 8 implies that There exists a "completion" AC
~C
M ~ A N ~-~> M ~ A N ~C
One verifies that Remark 2.-
BTCTA . Then
and so it is of the form
satisfies weaker conditions, and so we have a
such that
that
9 HOmTC(B)(L , -) s Lez(Coh(Dis(B)), Ab)
is anti-equivalence of categories.
HOmTc(B)(L, -)
M x N § V
V.
INDICATIONS OF THE PR00F OF THEOREM 8:
TC(B) 9 L
V ~ Coh(Dis(B)),
If
e A
(N
= eA
even if
M
in this case too. A
M 6 BTCTA
of
M
can also satisfy weaker conditions ...),
in the pseudo-compact case ~ ,
D = Mod(Z), B = End Z (Q/IQ/Z~), ~ ~
~.
then
ReXNlim(TC(A) , TC(B)) = Dis(A) ~ <
is also locally finite, then
where
is a certain pseudo-compact ris E introduced in ~ ] , ~ ,
AN
called the dual of
A.
Thus we have
so,~ indication that
AM
tensor product of
and of
B
Dis(A) ~
~--->TC(AN) [~'> PC(AK~,
and if ~
BTCTA--~> TC(AX),
and
and this ~ives
should be obtainable as a suitable topological A~
We have not tried to pursue this
further, nor have we tried to see whether there is a suitable lineamly
-
topologized Pin E
AM
238
-
in the general locally noetherian case (then
AM
is of coRPse not necessarily topologically coperfect and topologically coherent).
w 7.-
Expllcitstudy of the dual and the conjugate of the Gabriel
filtration of a locally .noetherian category. Let
~
be a locally noetherian category.
representation
C~>
w
TC(A) ~
Gabriel filtration of
~,
the case when
C
of Theorem 6 to describe explicitly the
and how
locally finite categories.
We will now use the
~
is built up by "extensions" of
We will have the most complete results in
is stable,
i.e. when the Gabriel filtPatlon is stable
under injective envelopes (cf. w 8 och w 9), but here we start first with some mesults in the genemal case. If ~
is an AB 5M-categomy with a family of cogenemators ~0]
(i.e. the dual of a ~othendieck categoz~/), then a subeategoPy
_F of
E
will be called a coclosed category if it is the dual of a closed subcategory [25] of category of
~
E_. ~
It is equivalent to say that
F
is a full sub-
that is stable undem the formation of subobjects, quotient
objects and
llm. In the same way we introduce the notion of a co< localizing subcategomy, the notion of pmoduct of two coclosed subcategomies
of
E
(cf. [2q, p. 395) etc.
If
[
is a coclosed subcategory of
ix
then the inclusion functor
~
E,
.N
, > E
has a left adjoint
is a colocalizing subcategory, then the quotient category
I EJ[
and if can be
.M
fommed and the natural functor
E--~EJ[
has a left adjoint
J,
that
is a full embedding, and so we have an "exact sequence" .x
o
>s
_z< ix
j, ,"
r.js -----~ o
j~
Now we will see that in case
= TC(A)
~.e.
E
is the dual of a
locally noethemlan categor~, then all these categories and functors can
-
2 3 9
be described in a very explicit manner.
-
Our results generalize and complete
those of Gabriel [25], p. ~00. THEOREM 9. -
Let
ideal0[
A
of
TC(A)
be as before.
that belongs to
The map that to each bilateral as a left A-module, associates
TC(A) i
the full subcate~omy
_F = TC(A/~)
M > TC(A)
(ix
is the natural em-
bedding), defines a one-one correspondence between these ideals ~ L the coclosed subcategories of w 6 we have
F
of
i.4i"(M) = A ~ A C
TC(A). M.
subcate~ories cortes p~ond~ng to 0~ 1 _~
If
and
With the notations above and F1
and
F2
a_nd ~ 2 ' and if
are two coclosed FI'_F2
is the product
subcategor~ [25J, then this coclosed subcategomy, cor~esponds to the bi_~lateral ideal TC(A)
C
~i
"O~t 2 '
and that contains ~ l
i.e. the smallest left ideal of "(~2
F
that is in
(this smallest ideal is bilateral and
it is also the image of the nultiplication map ~ l ~ C O ~ 2 ~artieular
A
-- 9 A).
In
is a eolocalizing subeate~ory if and only if ~ 2c
and in this case the in~redients of the exact sequence
0
9 TC(A/~) <
l
TC(A) <
TC(A)/TC(A/~) --> 0
> "
.)i
)'.
]
can be made explicit as follows: 9
.M
~
9
2) TC(A),
A
TC(A)/TC(A/oL)
can be fdentified with the full subeategomy of
formed by those
/% c~ 0%% ~ --> M 3)
A
M~TC(A)
is an isomorphism
such that the multiplication map (j,
is then the inclusion functor}~
The natural exact sequence
j,jX(M)-->
M - - > iMiM(M)--> 0
can he identified with the natural tensor product exact sequence
-
~
M ---> O ~ 9 M A
Finally,, if
A/OL
then
A = A/Of !IAe,
then
eAe
240
-
is onto). is the pro~ective envelope of
where
e
is an idem~otent of
A/Or A,
in
TC(A), -r %- c and if O ~ =
is with its natural topology a ~opologically cohement
(topologically coperfect is clear) complete ~ing, and (using 2)) TC(A)/TC(A/oI) functor
is naturally equivalent to
M~--> HomTC(A)(Ae, M).
Also
projective envelopes if and only if (gen. case). and then
TC(eAe),
by means of the
TC(A/oI)C_~TC(A) A/Oh
is stable under
is projective in
~ n this last case, we can choose
e
TC(A)
such that (~i = A(l-e),
(l-e)Ae = 0.]
The proof of this theorem is entirely based on the theory of w 6, .M
and the general functorial properties of
j,, ] .
(Compare also [48] .)
Now that we have Theorem 9 and w 5 - w 6, we can easily tmanslate the definitions and results of [2~, p. 382 ... into oum dual language: Let
COROLLARY I.-
O 1 of
A
A, O ~ T C ( A )
be as before. such that
there is a smallest one ~ o ' =~o'
so that
A/~o
TC(A/(~o)C__~TC(A)
envelope of and
--Fc ~i
is a left pseudo-compact ring~ and
A,~71~
in and
TC(A) ~iC~o
a dec~gasin~ filtration of A
such that
TC(A)
w
There is also a
that the image of
A/~I
i__nn
is pseudo-compact (this can be expressed with the projective
A/(> ~ =~i'
and this
is a pseudo-compact left A-module, -U--~c dh ~ is bilateral and G ~ =
is a qolocalizin~ subcate~ory.
smallest left ideal of TC(A)/TC(A/01 o)
A/Or
Then among the left ideals
A
too.)
This ideal O~ 1
. Contlnuin~ in this manner, we obtain by bilateral, left
=
AO6~oDOLID
is also bilateral
... O 0 % a D
...
TC
ideals
{~}
of
241
-
~f
-
~ has no predecessor, i.e. is a limit ordinal, then we put ~ a =
the biggest left ideal in
TC(A)
such t h a t ~ u C ~ 8
there is a unique smallest ordinal 0~
~ O, ~ < a".
TC(A) ~
This
H
~
such that
8 < u 1).~ Then
~H
= 0
and
coincides with the Kmull dimension of
and TC(A/OLo)~176
...~TC(A/oIs)~
coincides with the Gabriel filtration of
TC(A) ~
COROLLARY 2.-
TC(A) ~
The Gabriel filtration of
envelopes if and only if each O[~ = Ae~, A
,
where
...~TC(A) ~
is stable under in~ective em
is an idempotent of
[then eeA(l-e ~) = ~ .
Before we go over to a more detailed study of the stable case in the next two sections, we will first see how the conjugate categories fit into our picture. 'rHEOI~M 10.~
Let
~
be a l o c a l l y
topological ring and ~ =
Dis(A)
noetherian
category,
its conjugate.
A its
Then the map
associated
F~--> F
defines a one-one correspondence between the closed subcategories of and closed subcategories of ~_ ~nd O~b-@ Dis(A/o7 ~.
_FI ~ ~_ can be identified with
The localizing subcate~0ries cozTesp0nd and ~/[ =
= ~../.[~. In particula~ the Gabriel filtration J'v
C-oC ~ I C ' " ~ C - - u C
(19)
co_tic
...~C_
...c
gives rise to a filtration of _C = Dis(A):
c c
...c
,
which can be identified with Dis(A/O~o) C D i s ( A / O L I ) C
.. .~ Dis(A/6n )~ . . . ~
Dis(A). N
FumthePmore
C
is the smallest localizing subcategory of
"O
l)
By w 5 we have that 0]~ =
~ B<m
01.B
C
that
242
contains the simple and coherent objects
-
l)j N
~i
is the smallest
19calizlng subcategor7 of ~_ that contains those oh~ects whose images in ~/~_~ 9ye coherent and simple etc. ~i+I/~i
From the results above follows that
is locally finlte, thus of the form
PROBLEM i:
Can an
stable case)? PROBLEM 2:
AM
PC(AI") ~
be built up from these
A. x 1
(cf. w 6,Remark 3).
(at least in the
(Cf. w 6, Remark 3.)
Does every Grothendieck category
D
of Krull dimension zero
admit a filtration similar to (19) above?
w 8.
Generalized triangular ma.Trlx rinks with a linea~ topology, and
classification of stable extensions of locally noetherian categories. Let CJD
C
be a Grothendieck category,
D
a localizing subcategory,
the quotient category and .M 3
ix (2o)
o -->
D_
>c
~ cjs
the sequence of natural functors.
-->
o
Recall that here
ix
and
jx
have
!
right adjolnts iX(C)
by
D.
i"
= 0 < => C = i
and (D).
JM
that are full and faithful,
and
We will say that (20) is an extension of
_C/D _
Recall that Gabriel has proved that any locally noetherian category
can be obtained in a canonical way by means of successive extensions of locally finite ones [for more details see [25], Chap. IV (cf. also w 7)~ and that he has explicit results about the structure of the locally finite ones (some of which we have extended her~ to the locally noetherian case).
However, as far as we know, the problem to classify
the extensions (20) of locally finite categories or more generally of locally noetherian categories, has not been dealt with in the literature, and we will here give the rather complete results that we have obtained
1)
The smallest localizin E subcatego~y of objects of z e r o (w 3 ) .
~,
is of course
~
~
that contains all the simple
itself, since
~
is of KPull dimension
-
in the stable case.
2 4 3
-
[We say that (20) is a stable extension, or that
is a stable subcategoz~ of
~
object in ~,
D~
is still in
if the injectlve envelope in
= Qcoh(X),
of every
The stable case occurs frequently "in
practice" : It is for example well-known [25] that if prescheme,
~
X
is a noetherian
the categor,] of quasi~oherent sheaves over
then every localizing subcategory of
~
X,
is stable.
Thus consider now an "exact sequence" jH
i
(21) (D
0---> D
> C
9 E
a localizin E subcateEory of is locally noetherian (then
big injectlve of
~.
where Since
D
and and
~ E
the quotient categoz~), where are so too),
and let
Suppose that the sequence is stable.
a canonical decomposition of
(22)
~,
.9 0
I
I
be a
Then we have
([25], p. 375 cop. 2-3)
I = i (Zo)J_tj (II) I~ is a big injective in ~ and I1 is a big injeetive in ~. N. j 1M = 0, we get HOmc(i (Io) , j,(I1)) = 0, and so from (22) we
obtain a direct decomposition of abellan ~oups
(23)
A
=
(iH
Homc(I , I) = HOmD(Io,Io) tLHomc(JM(ll),i
and
Jx
(Io)) II HOmE(Ii,I1)
are fully faithful).
However, it is possible to rewrite (23) so that the Pin E structure of becomes apparent.
If we put
A ~ = HOmD(Io,Io) , A 1 = HOmE(If,If)
M = Homc(J,(Ii) , iM(Io)) , then
A~
and
A1
ape Pings and
natural way a left Ao-module and a ri_~Al-mOdule are compatible Now, if
(we will denote this by Ao, A1, AoMAI
M
and is in a
and these structures
M = AoMAI ) .
are arbitrary, we will denote by
(Ao Ao MA)~ the ring whose elements are tho Tri~les (ao,m,a I) 0 Ai q suggestively w~itten as (ao m )l ( a o C Ao, m ~ M~ a l ~ A l ) , 0 aI
Lmore where
A
-
244
addition is defined by componentwise addition, and where multiplication is defined by "matrix multiplication" !
(aO 0
m )(a~ m') = (aoa~ aI 0 a~ 0
aom'+m al) aleI
(This has a sense, since M = AoMAI.)
This kind of generalized Triangular matrix ring was first introduced by S.U. CHASE in ~l~ (cf. also [3~), and it is now easy to see that the assertion (23) can he made more precise by saying that we have a natural ring isomorphism
(24)
where
A = HOmc(I , I) "~> --
I HomD(Io, Io) -0
M
1
HomE(I l, II)
M = Homc(JM(I1) , ix(Io)).
Now in (24), to give the rings
A~ = HOmD(Io, Io)
and
(with their natural topology), is the same as to give
A1 = HomE(I I, II) ~
and E,
and thus
all information about The extension (21) should be contained in the Ao-Albimodule
M
(this module also has a linear topology ...). Thus The problem
arises To characterize those bimodules
M
that arise from stable exten-
sions. This problem will be solved completely in what follows, but first we will have to develop several pmeliminamies (some of them perhaps of independent interest) about generalized triangular matrix rings. Let
A = Ii ~
AoMAI~
be an arbitrary generalized triangular matrlx
AI/ ring.
We wish first To determine all the left ideals of
start with some examples of such ideals.
A~ AoMA11 where A 0 0 is a left ideal of left Ao-module
A,
A.
For that we
It is clear that
operates to the left Thl-ough matrix multiplication and this ideal can, in fact, be described as the
AoJ~M (M = AoMAI ) on which
follows that every left Ao-submodule
A
V~AoLIM
operates through
Ao.
It
defines a left ideal of
245
A,
that we will denote by A (0~
write AI,
(0V0)
M.~ C< 1 A : (0 v
[note that with this notation we can
M A IAM 0) = (0~ 0 ~ . On the other hand, if
then the left ideal of
0 (0
).
A
is a left ideal of is exactly
Finally any sum of these two types of left ideals of
0 0 ) + (0
M.C~ 1 o~ )
submodule of
Aoll M
restriction,
CK1 C A1
PROPOSITION 13.-
is evidently a left ideal of
such that and
of the form GtI
M. O~1 C W.
0
A1
(0 V~I),
where
V A1
A, which we will Ao/~ M
is an A o-
Except for this last
can be completely arbitramy: be an arbitrary genel-alized
The left ideals of
is a left ideal of
operates on
W = V + M. 0%1 C
W C Aoli M
Le.t A =
triangular matrix rin~.
where
(~i
0 0 (0 ~ )
generated by
write in the form l) (0 WO~l) , where
A
-
A
are exactly the subsets of
is an A o-submodule of
AoAL M,
such that
and where
M . O ~ C V,
and A
by left matrix multiplication. 1 Compute directly the left ideal of A, generated by a set of
PROOF:
(0 V )
generators. Remark.-
If Ot is a left ideal of
VO[C~ Ao/l M
and
O ~ C A1
A,
then the corresponding
are uniquely determined by O~.
Furthermore
((Ao//- M)/rot) A/or
can be identified
with i)
\
0
AI/~I /
, where
by left matrix multiplication which is well-deflned since
A
operates
M.O~ 1 C Vot.
This gives rise to an exact sequence of left A-modules
(25) where AI/O~ I.
0 A
> (Ao]-tM)/Vo~
operates through
A~
on
> AI/O~ 1
> 0
(Aolt M)/VoL , and through
A1
on
This sequence, which will be very useful below, does not split in
general, not even in the case l)
> A/C~
Vot = 0,
O~1 = 0 (i.e. O r = 0)~
We use here an extension of the notation introduced above.
- 246
COROLLARY.-
-
Wlth the notations above, Ot C A
%eft ideal, if and only jif O~1
and
is a finitely generated
Vo~/M. %
are finitely Kenerated as
left A l- and left Ao-modules Pespectivel Z. We now turn to the study of linear topologies on generalized triangular m a ~ i x rings 1).
Combining Proposition 3 of w 4 with Proposi-
tion 13, we obtain: PROPOSITION 14.A A = (0~ A1
M AI)
To give a (left) linear topology on the ring
is the same as to give a (left) lineam topology on the ring
and a linear topology on the left Ao-modul___.~e Ao/i M
such that the
maps:
(Aoli M) • (AolL M) ---> AoAJ_M , {(ao,m),(a~,m')}~-->
(i)
(aoa~,ao m')
and, (ii)
(Aoll M) x A 1 |
-> Ao/L M,
{(ao,m), al};
> (0,ma l)
.
are cont~nuoust and such that fumthe~nore the map (ii) is left equicontinuous in the sense that to each open Ao-SUbmodule is an open left ideal topoloKy on
A
O~ 1
of
A1
such that
V
of
Ao33_M
(Aoli M).0~IC V.
is then the product of the topologigs on
A1
there The
and on
AILM. o Those triangular matrix rings that ape of interest for us in connection with locally noetherian categories will have split linear topologies in the sense of the followin E definition. DEFINITION ~.-
We say that a left ideal
ideal if 0~ is of the forth ( left ideals,
U CM
i)
U o Oil) ' where
is a left Ao-submodule and
linear topolo~ on the ring A
O~ of
A
A A : (0~ O%iC A i
M AI)
is a split
(i = 0,1)
M.CEIC U.
ape
A left
is said to be a split linear topology if
has a fundamental system of open split left ideals9 Recall that in this paper, all topologies studied on rings ape supposed to be compatible with the ring structume.
-
Remark i.A A = (0~
247
To say that the topology on the linearly topologized ring
M AI)
is a split topology is the same as to say that the left A O-
linear topology on
Aoll M
such topologies on
A~
(Proposition 14) is the product topology of
and
M.
Thus P~oposition 14 implies that to give
a left linear split topology on the ring linear topologies on the rings left Ao-module
M
that the operation A
-
such that
A~ M
and
A
is the same as to give left
A1
and a linear topology on the
becomes a topological Ao-Al-bimodule such
M x A 1 ---> M
is left equieontinuous.
is then the product of the topologies on
in particular complete if and only if Remark 2.-
The discrete topology on
trivial examples are given by: A M THEOREM ii.- Let A = (0~ A1)
Ao, A 1
Ao, A 1 A
and
and M
The topology on M,
and so
system
{0%}
that every PROOF: that
A
is a split linear topology.
be a left linearly topologized ring that
o f split open left ideal neighboumhoods of A/Or
is coherent in
decomposition
(26) But if
C = TC(A) ~
A = Homc(I , I)
= HOmC(I/N , I), where -A/l(N) is coherent in 0M A(l-e) = (0 A I)
Then the
0
in
A
such
Dis(A). is a locally noetherian category,
corresponds in a natural way to a big injective
the topology on
Less
is a split linear topoloKy, and we even have a fundamental
We know by w 5 that A
is
are so.
is topologically coherent, topologically coperfect and complete. topology on
A
I
in
is defined by the left ideals
C,
that
I(N) =
N C I
is a noetherian subobjeet, and that every A 0 i0 Dis(A). Put e = (00) . Then Ae = (0~ 0 ),
and the decomposition I = I o l l I1
of
I
in
A = AelkA(1-e) C.
cozTesponds to a
Furthermore
Homc(I o, I I) = HOmTC(A)(A(I-e) , Ae). f
is an arbitramy left A-lineam map
f(1-e) = Ae,
thus
A(l-e) --> Ae,
f(l-e) = (l-e)f(l-e) = (1-e)%e = 0,
by direct matrix computation.
Thus
f = 0
since
then (l-e)Ae = 0
and a f o r t i o r i the right member
-
of (26) is O, so that Now let
C
248
-
Homc(Io, II) = O.
be the localizing subcategory of
C,
formed by those
C
--0
such that
HOmc(C , I I) = O.
I claim that _C,
tion of injective envelopes in C .
But if
C ~ C
--0
-oC
and that
is stable under the fommaI~
is a big injective in
were such that its injective envelope (in ~)
I(C)
--0
did not belong to factor
I
# 0
injective of
C,
then
I(C)
would have an indecomposable direct
that o c c u m s a s a direct factor of
_C and
I~ ~ -r ).
c
c
(Io/1 I 1
is a big
But the pullback diagram
" l(C)
3 Ca (
I1
C =~! a
gives mise to an object C @ 0 that belongs to C . We have a monoi a -o C the composition map morphism I a . a > I1 ' and since C ~ 6 -o 1 Ca ( > Ia ( ~ > Ii is zero and this gives a contradiction. Thus we have a stable exact sequence
0
> C_.o
'. '> C_
"I
> ~C/C
> 0
t.
where
I~
Now if
N
is a big injective of
-n)C, I = I o l L I1
is a noetherian s u b o b j e c t o f
I,
and
I1 ~-~> jMjM(II ).
then since both
I~
and
I1
are directed unions of noetherian subobjects, we get that there are noetherian subobjects thus
N o ~ I~
I(N)~I(No2INI).
left ideal of
A,
and
N 1 C I1
such that
N CNo/I
NI,
asd
It is clear that this last ideal is a split open
and that
A/I(NoJI N I)
is coherent in
Dis(A)
and so
the Theorem ii is completely proved, as well as the COROLLARY.-
With the notations and hypotheses of Theorem ii,
A~
and
A1
are topologically coherent, topologically coperfect and complete for their natural linear topologies. Remamk,assure that
We will later determine exactly those conditions on A
is topologically coherent.
M
that
-
249
-
We will say that a linearly topologized module
M
is topologically
artinian (r~sp. topologically noetherian, resp. topologically of finite length, resp. topologically coperfect) if {U}
of submodule neighbourhoods of
0,
M
has a fundamental system
such that every module
M/U
is
artinian (resp. noetherian, resp. of finite length, resp. coperfect) i). A M THEOREM 12.- Let A = (0~ AI) be a (left) line~ly topologized rin~. Then
A
is (left) topologically artlnian (resp. topologically noetherian,
resp. topologically of finite length) if and only if
A1
and
AoAIM
at?
so for their natural (left) linear topologies. PROOF:
Let O~ be a left open ideal of
where
Gt I
and
A.
Vot are open submodules of
Then ~ = A1
and
V (0 ~ ) ( M ' O ~ C Vc~): 1 A o ~ I M respectively.
Then by (25) above we have an exact sequence of A-modules
0 ---> (AoJIM)/Vot
(27)
where
A
operates through
> A/O~
A~
on
> A1/OC 1
(AolLM)/Vot
> 0
and through
A1
on
A1/0( 1 9 Since in an exact sequcnce (in any abelian category) 0 - - > CI---> C2--> C3--> 0 C2
is noetherian (resp. artinian~ resp. of finite length) if and only if
C1
and
A/Of
C3
are noetherlan (mesp .... ),
it follows easily from (27) that
is artinian (resp. noetherian~ resp. of finite length) as a left A-
module if and only if module and
AI/O~ 1
(AolL M)/V~t
is artinian (resp .... ) as a left A o-
is artinian (resp .... ) as a left Al-mOdule ~ and this
proves the Theorem 12. COROLLARY 1.-
A (0~
A split linearly %opologlzed rin~
M A1)
is(left)
topologically artini~an (r~sp. topologically noetherlan, resp. topologically of finite length) ~ if and only if
Ao, A 1
and
M
are so for their natural
1) These conditions are then verified for all open submodules
U.
-
250
-
(left) linear topologies. In the discrete case we obtain: A M COROLLARY 2.- A rings A = (0~ AI) is left artinian (resp. left noetherian) if and only if the rink and
Ao, A 1
are left artinian (resp. left noetherian)
is an artinian (resp. a finitely generated) left Ao-modtLl_____.~eel).
M
COROLLARY 3.-
Consider a stable extension of locally noetherian categories .M
0-->
s
9 c_
9 c_/~
9 0
Suppose that the endomorphism rink of a big injective I~/~
in
C/D
ID
the left
Example 1.-
and
iM(I~)ILj,(Ic_./~) in
Leptin rink if and only if for every noethePian subob~ect ~,
D
is a Leptin rink 2) (i.e. is topologically artinian).
the endomorphism rink of the bi~ in~ective
in
in
HomD(ID, ID)-module
Let
ToPs(Z)
HOmc(N , iM(ID))
N
C
Then is a
o_ff jx(I~/~)
is artinian.
he the category of torsion abelian groups, and
consider the exact sequence 0 Here
> Tors(~ZZ)
Tors(ZJ
~/Z and
~
[where P
and
> Mod(Z)
> 0
Mod(QQ) are locally finite, and have big injectives
respectively.
The endomorphism rings of these ape 7--[ Z ps p ~ P is the set of prime numbers, and ~p Z is the ring of p-adic
integers~ and Q~v respectively. rings.
> Mod(Q)~
Both these rings are of course Leptin
But the endomomphism Ping of Q IIQ/Z
in
Mod(Z)
ring, for ~Z is a noetherian subobject of ~Q' and
l)
is no__~taLeptin
Homz (ZZ, ~I~) = QIZ
is
According to Hopkins [16], a left artinian ring is left noetherian. If we combine this with Corollary i, we get that if artinian rings and module, then
2)
'
M
M
Ao, A1
are left
an Ao-Al-bimodule that is a left artinian Ao-
is finitely generated as a left Ao-mOdule!
If the endomorphism ring of one big injective in a locally noetherian category
[
is a Leptin ping, then the endomorphism ring of any
injective (big or not) in [
is a Leptin ring (cf. ~8~,[3~ ).
251
-
no_~tan artinian
Endz(Q/Z)-module , for it has the following strictly
decreasing chain of
Q/z= I I
-
--
p e
z
P "P|
Endz(Q/Z_)-submodules:
~,[I
z
p>__3--P|
~ 113 L~ I I z
p>_5 P ~ p>_? P|
p-C P Here
Z
-
pE
P
D II
z
D...
p>ll --P~
p~ P
p~
is the indecomposable injective of type
p
P in
Mod(Z),
cf.
[237. Example 2.p.
Let
~(p)
he the localized ring of
Z
at the prime number
This ring is noetherlan and of K~ull dimension i, and one vemifies
using Corollary 3, that here the endomorphism ring of a big injective i_~sa Leptin ring
A.
This ring
A
I
can not he topologically noetherian
too, for then it would be pseudo-compact, which is impossible, since the Krull dimension of
Mod(~(p))
is i.
Thus
Dis(A)
is a locally amtinian
(and also locally coherent) category that is not lopallygoetheria q.
The
theorem of Hopkins
[16_] cited above can be formulated as saying that if
R
Mod(R)
is a ring, then
locally noetherian.
locally artinian implies that
The example
Dis(A)
Mod(R)
is
shows that this theorem can not
be extended to Grothendieck categories, at least not in this formulation. A M THEOREM 13.- The (left)linearly topologized ring A = (0~ A1) i_~s topologically copemfect, if and only if Aoll M and Vo1 START OF THE PROOF: If 0~ = (0 01I) (M. O~1 C Vot) of
A,
A1
is an open left ideal
then the exact sequence (27) above shows that if
cally coperfect, then
AolIM
and
A1
are so too.
are so.
A
is topologi-
To prove the converse
we need the following generalization of the Corollar~ of Proposition 13: LEMMA l.-
With the notations above, the finitely generated left A-sub-
modules of
A ,u mo u o
0 (T0 + ~ ) '
where
A1/~l / T C (Aoli M)/V~t
A -submodule, and where O
~C
AI/~ 1
is an arbitrary, fiqitely Kenerated .is an arbitrary finitely generated
252
-
Al-SUbmodule
(M.~
M • AI/O~ 1
> (Ao/L M)/VG~ , which is well-deflned, since
PROOF:
is defined by the natural product mapping M.~lC
VO~).
Direct computation. By Lemma i, a decreasing sequence of
END OF THE PROOF OF THEOREM 13: finitely generated submodules of
A/O~
is necessarily of the fern:
"2
D
...
D
D
...
where (29)
Al/Ot I D ~ i ~ %
~ "'" ~ ~ n ~ ''' ~
is a decreasing sequence of finitely generated Al-SUbmodules and where the
Ti
are finitely generated Ao-sUbmodules of
(Aol/ M)/Vot
evem do not neoessamily form a decreasing sequence.
Since
A1
that howis supposed
to be topologically coperfect, the sequence (29)must become stationary from a certain index
n M on, and so in partlculam we get from (28) a
decreasing sequence
(30)
T M .M.)S N DT +~.,,'e N mT +M.'6 M _9... n n nM+l n nM+2 n
Now let
~f i ~ t [ n"+l ) i=l
f i = ti + nM+l nR ~i'
be an A -basis for o where 9~i
C M . ~ x, n
T
C (AolL M)VoL.
T'
and clearly
nN+l
T'
+ M.~,
nM+l
Then evez 7
nX+Z and
t i 6 T M" nM n
generate a (finitely generated!) submodule of by
D
= T
n
Now these
T M, which we will denote n + M.~,. Continuing in
nM+l
n
this manner, we see that the decreasing sequence (30) can be wmitten
(3l)
T n
.M.~
roT, n
nX+l
*M.Z.DT' n
.,.-4" nX+2
where
(32)
t iM n
(Ao/.[. M)/V0[ ~ Tnx 39 TnX+l/D T'n~+2~ ... m
~... n
-
253
-
is a decreasing sequence of finitely generated submodules of But
(Ao/LM)/Vo~
is coperfect
[AolL M
(AolIM)Vot-
is topologically coperfect3 .
It
follows that (32) must become stationary and thus also (31) is stationamy. Since (31) is just another way of writing (80), we finally see that (28) is stationary and so the Theorem 18 is proved. A M COROLLARY i.- If A = (0~ A1) is linearly topolo~ized with a split topology, then
A
is topologically copemfect if and only if
A~
and
A1
are so for their natural linear topologies. We have in fact that if object of
are
is topologically coperfect, then every
o
Dis(A o)
COROLLARY 2.A1
A
is coperfect. A M The ring (0~ A1)
is right perfect if and only if
A~
and
so.
PROBLEM:
Is it true that the topology on a linearly topologized, A M topologically coperfect, complete ring (0~ AI) is a split topology? We will not pass to the topologically coherent case, which by far is the most complicated one.
For our purposes (cf. Theorem ll) it is
sufficient to determine the necessary and sufficient conditions for A M A = (0~ AI) to be topologically coherent for a linear topology such that A
has a fundamental system
O,
such that every
A/O~
{Ol}
of split open ideal neighbourhoods of
is coherent in
Dis(A).
We will say that
A
is ss-topologically coherent if this is true [we do not know if this is the same as saying that the linear topology on
A
is split and that
A
topologically coherent (this could be called s-topological coherence~. In order to express our results in a convenient form we will first need some generalities about linearly topologized (hi-)modules and their tensor products. PROPOSITION 15.bimodule. that
M
Le_~t A ~
Suppose that
and A~
and
A1
be rin~s and let M
M
he an Ao-A I-
are left linearl 7 topolo~ized and
is a topological Ao-Al-bimodule when
A1
is ~iven the discrete
is
254
topology i). {fi};
Le__~ F
-
be a finitely ~enerated left
be a fixed finite set of ~eneratcrs for
submodules of
M $~A 1 F
A 1 -module~ and let F.
Consider the
A~ -
of the form t
f
f
t = Im (h_[u 1
V = VUI' "''' ~
where
U
a left
is an open A~
Ao-sUbmodule of
-linear topoloF~ on
of the finite set of ~enerators a topolosical left
A~
(F" of finite type),
( ) ~)Alfi --~i ~
t --->I_~M 1 M.
1 F,
M~A
{fi}t1 of
These submodules
:
V
that is independent s F.
Furthe,Pmome
-module and for ever,/ A 1 I~A?
M ~SAIF)
M ~AIF" ~
-linear ~ p M~AIF
define th9 choice ....
M ~ A 1 F is F" ~
F
is a continuous
A -lineammap. o PROOF: every
Let
{gi} s he another set of generators for F. I claim that 1 v~l' "''' gs contains a submodule of the form V Ufl' "''' ft
We have a factorization t z().f
t (33)
1
x
Al
> F
>
0
l s
1
( )'~i s
Al
0
1 Here
#
must be given by riEht multiplication with a matrix
elements in
I) This means that
>
M
~ ai~ij i=l
M
A1
j =i
is a left topological A-module, and that for any
the right multiplication map
in particular if for which
cf
AI:
{at} 1 '
a I E A1
(@lj)
M
9a 1
> M
is continuous [this is true
has some linear topology (not necessarily discrete)
is a topological right Al-mOdule3 .
255
-
By hypothesis there exists an open ~.~ijCU.
-
Ao--submodule
of
U
M
such that
Now if we combine this with (33) we set fl' "''' ft V~ ~_
El' "''' gs VU .
This shows that the linear topoloi~y on choice of the basis of
F.
M ~AIF
is independent of the
The other results of the Proposition
15
are
proved in the same way. DEFINITION 5.
-Let
finite type.
C
be a Grothendieck category that is locally of
We say that
p. 52) if fop every, map is so too.
If
A~
C"
C ~ f
A O -module! then
coherent if
M
O,
> C,
M
M
C"
is of finite type, K e r f M
a left linearly
is said to be topolozicall 7 psgudo-
has a fundamental system such that every
We say that
where
is a left lineamly topolo~ized rinF,
t0polo~ized
hoods of
is ~seudo-coherent (el. Bourbaki [9],
M/U
U of
A ~ -submodule nei~hboum-
is pseudorcohp.rent
[in
DiS(Ao) ].
i_~sstron~ly topolo~ical&y pseudo-cohgren ~ if in addition
there is an open submodule
W
the indus
M/W
topolo~ (then
of
M
that is topoloKically coherent fo~
is automatically pseudo-coherent, and it
follows that theme exists a fundamental system of such W:s). We can now formulate the main mesult of this section: THEOREM i~.
)
-The followin~ conditions of the left split linearly tops A~
~-Ing A = [ 0 \
M A1
l)
A
2)
(i) The rinks
ame equivalent:
is (left) ss-topolo~icall 7 coherent. A~
and
A1
are topolo~ically coheren T for rheim natural
left linear topolo~ies~ (ii)
For ever~ coherent
CE Dis(A1),
M~AIC
is Strongly topolo~ically
pseudo-cohere~ fop its natural topglop~3 , and fom every C"
@> C
i_~n Coh(DiS(Al)) ,
(I~
~ )-I(v) A1
monomorphigm
is tpDologically coherent
-
(induced topology) if
V CM
~AIC
256
-
is an open submodule that i9
topolo~isally cohe.r~nt (it even suffices to require this for one such V). Remamk i. -In the course of the proof of Theorem 14 we will see that (ii) can be replaced by the (apparently) weaker condition: (ii)" There is a fundamental system of
0
i_n.n AI,
a)
AI/OL 1
{OfI } of open left ideal neishboumh99d9
such that:
is coherent in
DiS(Al) ;
b) M ~ A 1 AI/G~ 1 is strongly topolgKically pseudo-coherent;
c)
FoP all finitely ~enerated
Al/Otl
FI, Fl~----> Al/Otl, M|
to~ olo~ically~p seudo- cohe rent, and
1 _ F1
~olo~icall~ of is topo
(IdM~AIi)'I(v)
finlte %3~e_for at least one open submodule
i~s
V C M~AIAI/Ot I
that is
topologically coherent. Remark 2. -In the discrete case Theorem i~ has a nice formulation in terms of Tot: COROLLARY 1. -The followin~ conditions on th9 arbitramy~enePalized triangular matrix rln~
A =
(:o
A1
~
l)
A
2)
(i) A 0
are equivalent:
is left coherent. and
A 1 am9 left coherent min~s;
(il) Fop every coherent pseudo-coherent for
C ~Mod(Al) , th9 lef-t Ao-module
i = 0
and co_herent for
A1 TotI (M, ~)
is
i > i.
Here we also have an (apparently) weaker formulation of (ii): (ii)~or. 1 is left a left
For all finitely ~ene.rated left ideals
A ~ -pseudo-coherent, and A 0 -module.
A1 Tot I (M, Al/~t I)
Ol1
of
A1, M ~AIAI/Otl
is of finite t~pe as
-
Remark S.
257
-
-There should also exist a formulation of the part (ii) of
condition 2) of Theorem 14 in terms of linearly topologized Tot:s, a formulation that reduces to (ii) in Corollary 1 in the discrete case. We hope
to retur~ to this later.
Remark ~.
-If
A1
is discrete, then Theorem 14 is simpler, and if
A1
is even a skew-field then we have: COROLLARY 2. -The followln~ conditions on the left split llnea~ly topologized ring i)
A
2)
Ao
A =
(o) 0
A1
(AI = skew-fleld) are equivalent.:
is left ss-topologically coherent. is left topologically coherent, and
as a left topological PROOF OF THEOREM 14:
A ~ -modul_.~e. Suppose that
A =
M
is 9tronKly pseudo-coherent
(:o,)
has a split linear
A1
~
topology, let (7[ =
o 011
be an open split left ideal of A
(MOtlC U)
and let us try to analyse the condition that
A/(A
is coherent in
Dis(A).
We have that
CAoo M~ 1
(34)
AI/OI 1
0
where
A
operates to the left by matrix multiplication
is well-defined since
M 0 1 1 C U~.
direct sum decomposition in
(ss)
where
A/01
A
(36)
A/0L
Dis(A):
A~
on the riEht factor of (35).
coherent in
" > M/U
It is clear that (34) gives rise to a
= Ao/Oto II ( 0
operates through
[M_ x AI/O~ 1
on
AI/OII M/U 1
Ao/Olo
and by left matrix multiplication
Thus:
Dis(A) <-->
l
AolOl o 0 MIU
is coherent in
Dis(A).
8) (0 AIlO~1"
is coherent in
Dis(A).
a)
-
It is clear that Gt=
(M Ok
0 (t
~)
1~
258 -
is equivalent to:
of
U)
A
For each open split left ideal
and for each left
A-module map
#
~i
arbitrary finite) :
t
Ker ~
t IA I ~
is of finite type.
MIU
But to give
1
r
~--> A~176
is the same as to give elements
9 t
{aoI}
of
Ao/~o
such that
~o,a'io : 0(i : i, ..., t)
in
Ao/~o:
The
1 , a-i
multiplication map
Ao
,,
,o,>
passes then to the quotient
Aol~ o
-i ca o
and
#
is then given by:
t < Ao/01c 1 [ i
AolO%o
>
ao/Ot o
M/5
t vi.-i
I
o
. ~
i=l
AI/C~ z
0
ao a o E
Ao/Oi o
aI =i
and so t z ( ).~i
(37)
1
Ker r --
, Ao/ o) I_I
Ao/
1
0
t
%
I_! AzI~ I 1 which thus has to be of finite type. Now let us turn to the condition is equivalent to: left
8)
of (36)~
For each open split left ideal ~ I
A-module map (t
arbitrary finite):
It is clear that of
A
8)
and for each
-
_I
00
1
.}. o.
is of finite type.
Kem u
(:,"
-
t):
O 1 o ,~i
AIIOI 1 u is the same as to give elements such that
0
These
= 0 (in
-
--'--'>
But to give
i=l
(i = i, .9
(38)
AIIO~ i/
259
A1101 I )
t
relations
M/U),
~U.a~ "
( 1 C:;I Ou~
U
o
=0
0~].
can be wmitten:
= 0
(in
M/U), o~i I .-i aI
= 0 (in
AZ/mZ)
(i = i, ..., 1:) the middle 9 is defined by restrictlon of the map
M x All01 1 § M/U]
and
(38) can be expPesse~by sayln E that the natumal multiplication maps -i ^.. "m ~ M/U Ao/O~ o
-i "M A~
(39)
-" MIU
~:> M/u
M
-i -aI 9 S/u
M/u
define quotient maps
-1
-i
A1/~A 1
9a 1
> AI/~ 1
J
(i = 1, ..., t) and
u is then given by
[we use the notations in the formulas to the
right of (39)] :
vl -i.h t fA
lj~0
o
l&
o
r~l~ Al/~l
~
a
o
0
m V 1.
al
o 1
~ >
--i
0
tZ vi al.a-i1 1
and so
+ m .al)1
260
-
-
t
(40) Ker ~' --
9
t
~1 Ker(~ tj AI/OI
0
~ i
.-i
r ( ) a1 i=l > AI/O~I )
]
which thus has to be of finite type. In order to analyse (37) and (q0), we now need the following Eeneral lemma, which Eeneralizes Lemma 1 above.
LEMMA 2. -Let A =
I Ao 0
M ) A1 he an arbitraz.I Keneralized trlan~ulammatrix
V
ring and
an arbitrar?/, left ideal o f
(MOIIC V)
,%,
t_he, finitely ~enerated
A-submodules of ~
(Acll M)/V
)
A.
The__~n
are exactly_ t h o
A -submodules of the for~:
ro * Im(M |
9 >
/_L(Ao~M)/v 1
e~
0
F1
t
llCAo,, 1
where
F1 C
t I IAI/~ 1 1
defined by PROOF:
is an arbltrary
Ac-submodule of finite type, and
is an arbitrarT. Al-.Submodule of finite t~l]~e [here
t t ~ ~ M x ~ [ A1/ol I _-t_"> ,I I (AomM)/V ( M . @ I C 1 1
9
is
~7 V)J,
Direct computation.
Now the Lemma 2 implles that to say that type is the same as to say that
Ker %
of (37) is of finite
-
261
-
t
z ( ).~i
t
Z~r([
1
o
I Aol~ o i
is of finite type as a left Th.us. ~)
[cf.(36)]
9 Aolmo)
Ao-module.
is equivalent tO:
The same Lemma 2 implies that
KeP u
Ao/Ot o
is coherent, in
DiS(Ao).
of (40) is of finite type if and
only if:
t A)
t F 1 = KeP(J_IAII~I 1
-i
Z ( ).a 1 1 , > AIIO~ I)
iS of finite type as a left
Al-mOdule.
+( )'a~
(AoI(~o.LL MI~)
KeP
B)
L1
MlVl is of finite
%
Im(M ~A}I
" 9 ~IIM/~)
type as a left Al-mOdule. The universal validity of A) is clearly equivalent to: cohe~nt in
A1/Oh
i__~
DiS(Al) , but the analysis of B) is much mope difficult:
t F1 = Im(j_iAl/O~l 1
t -i Z( ).a 1 1 ~ AIICtl).
Then the exact sequence t
o
9
9 IIAI/~ 1 ~ q
9
1 gives rise to an exact sequence t (41)
M
1
1
9 M|
A
Consider the commutative exact diagram the diagram) :
> o
1%
(Ro. R 1
and
f
are defined by
Put
262
-
0
-
0
(o,~f)(~1)
f > -I-L%t 8~o 1
T
(42)
E
0
>
t
R
; o
0
Heme
p
,~
1
I ,~j~~
,>
~ ,
1
T'
al/~ l
is defined by mes%Tiction of the map
defined by the onto map
M/U
1
., M|
RI
t
t
,,~,I.~%/.~0
T
>
T
M x AllOt I
,> s % 1 q
M x AIIOI1
" > MI~(M~I~)
" > M/U, ~
' and
~I
is
is the
restriction of ~ . If we compare (41) and (42) we see that the quotient of B) is the same as particulam that
Ro/(0,~I)(RI) . The analysis of A
ss-topologically coherent => A ~
for its natumal topology.
a)
above shows in
topologically coherent
Thus since we are intemested in provlng the
equivalence of i) and 2) in Theorem 14 we can and will suppose fmom now on that A~
is %opologlcally coherent for its natural topology, and it is then
pemmisslble %0 suppose that
Ao/61.o is cohement in
DiS(Ao).
Under this
t
hypothesis I I,Ao/ Oto ~ l
is coherent in
DIs(Ao)
of finite %~zpe as an Ao-mOdule if and only if ape of finite type.
and so Im f and
R l(00~ I)(~) o
is
Kem fief, (42)]
But it follows fmom (42) that t
(43)
Im f = Ker
t Ao/~o I/~ll
1 ~ Z( ).~z
M/U t
Im and that t t Z( ) . ~ Kem(//M/~l~ 1 Ker f = P(Rl)
M/U)
M/~
_ 1
~ MIU
=
263
=
Thus using the topology of Proposition 15 it follows from (43) that the univePsal validity of: (45)
f
finitely genePated is equivalent to:
FoP all finitely generated submodules
A1/~ A of
Im
M@A
1
FI~AI/~I,
the quotient m
by the natural (open) image of
F1
U
(u ~ u
~l)
is
To analyse (44), let us consider the commutative
pseudo-cohement.
exact
diagTam
t 0
=~
> R1
(46)
1
t
9 Ker(llMIU 1
I
M
| j_tAi i/c~i
9- i
- 9
~
z( ) az
t
0
,>
t
t
~ M/U) § I I MfO
Im(llM/~
~
0 wheme
M
|149
t
9- i
E( ) a 1 !
9 M/U)+ 0
0
%/ p is obtained by the natural faotorization of
p.
An application
of the snake le~na (191, Chap. i, w i, n ~ 4) to (46) gives an exact sequence
Ker p
*
~ > Ker p - - >
Coker
)
9 0
~t But
Coke~ (Pl)
Ker f
= Kerf
and
Ker
so that to say that
p = __J-~U/M(~%I , 1
is of finite type is equivalent to say that
t
t
(47)
',
1
is of finite type (the maps are the natural ones).
Note that in (47) 9
F1
is any finitely genemated subohjeet of
AI/O~,
t
~l } 1
is any finite set of genemators for
submodule of
M
such t h a t
FI,
and
~
is any open A o
~.-i aI = O(in M/U)(i=I, ..., t).
o
-
264
-
Consider now the exact sequence
| (48)
M ~AIFI
'
> M ~AIAI/cI.1
A1/ot1 il '
>M|
~
0o
From (48) it follows that with the notations of Proposition 15, (47) can
just be of
reforl,ulated by saying that the open submodule
M ~AIFI
is topologically of finite type.
we see in particular that
( I ~ Q A l i ) - l ( ~ U)
If we choose
F 1 = AI/~I,
is topologically of finite type.
Vu
Now from all that has been proved it follows easily that with the notations and hypotheses of Theorem 14 we have the implication l) => 2), where in 2) we have replaced (ii) by the weaker (ii)'.
ThaZ (i) + (ii)" =, i)
under the same hypotheses is also clear, and so we now have Theorem 14 with (ii) replaced by (ii)',
and this implies in particular that Corollary
1 [with the weakened (il)'Cor. l ~
and that Corollary 2 are valid.
Now the passage from (ii)" to (ii) [we suppose that (i) is satisfied] is an exercise in the use of topological
pseudo-coherence and exact
sequences, that we will only do here in the discrete case, where we will prove the more precise result requlred by Corollary i: M~A
1 A1/Ol I
A1 Tot I (M, AI/OI l)
is pseudo-coherent and that
type for all left ideals
Ol I C
of finite type,
AI
induction on the number of generators of coherent for all coherent
C:
if
C
We suppose that
C,
has
that
is of finite
Let us first prove by M ~AI
C
is pseudo-
~enerators, then we can
find an exact sequence 0 where
(311
generators.
(49)
"> C
n-i
> C-->
is of finite type, and where The exact sequence of
AI Tot I (M, AI/OI I)
Tot
AI/011 C.I
.~
0
is coherent and has
< n-i
gives:
> M~AICn_ I ~>
M ~AIC
> M ~AIAI/o~I
~ 0
-
and since the
Top I
265
-
in (49) is of finite type, its image in
M ~ A 1 Cn_ 1
is so too, thus this image is cohement, since by the induction hypothesis M ~ A 1 Cn_ 1
is pseudo-coherent.~
But then the quotient of
with this image is pseudo-coherent,
0
' 9 PC 1
and so we get from (49) an extension
9 M ~A 1 C
9 M ~ A 1 Al/Ot I - >
where the extreme terms are pseudo-coherent. too. C,
Now that we know that
M ~ A 1 Cn_ 1
M ~A 1 C
0
Thus the middle term is so
is pseudo-coherent
fop all coherent
the exact sequence A1 > Tom I (M, A1/O[ I)
0
gives that type of
A1 ToP1 (M, AI/~ I)
M ~AlO~l,
coherent module
9 M~AIOL 1
~ M qgAl A I / G t l - - 2
is coherent, fop it is a subobject of finite
which is pseudo-coherent,
(A1
9 M
since the ideal O71
is a
is coherent).
From this we get, essentially as above A1 by induction on the number of generators of C, first that ToP 1 (M, C) is of finite type fop all coherent
C.
A1 A1 Top 2 (M, AI/OII) ~-~-> Top I (M, ~ i ) can 8o one step further and show that coherent
COROLLARY 3. - If A (0~
is of finite type, and using this we A1 Tom I (M, C)
is coherent fop all
C, and now one passes easily to the higher Tom:s and so the
CoPollary 1 is completely proved.
then
Thus it follows in particular that
M AI)
M
~
As an immediate consequence we have:
Al-flat and the ~ing
is left coherent if and only if
A1
A~
is left n oethePian,
is left coherent.
If we now confront the results of this w with those of w 5, then we obtain in pamtiaulaP:
0
-
THEOREM 150 -Le__~t C O
and
.
.
.
coperfect,
-
C1 be locally noetheri~ categories,
the associated linearly toDolog~ized .
266
.
?
~
A~
and
A1
Topologically Coherent! topologlcally ,
,
complete, sober rin~s.~j~ The map that to each left linearly
topolo izedzzeecoco letele~e~oto,ological Al-bimodule
M
Ewith
M x A1
>
M
left
equicontinuous~ ;hat satisfies the condition (ii) of Theomem 14, associates the extension : 0
A TC(0~
9 TC(Ao)~
H o AI)
9 TC(A1)~
0
defines 9 one-one correspondence between th 9 topological - isom0r~Phy classes of these bimodulea
M,
0
and the equivalence classes of stable axtensions
>--oC
9
1
> 0
of locally noetherian categpries. Remark i. -Theme ape also othe_____y.rstableextensions, where the middle caregory is not locally noetherian. Remayk 2. -More details about Theorem 15 in the discrete case will be given in w 9 - w I0. w 9.
Change of Krull dimension in stable extenslon of locally ngetherian
categories. For simplicity we will only study the discrete case in this section, Thus let
A~
and
A1
be left coherent, right perfect rings, and
M a
blmodule as in the Corollary 1 of Theorem 14 and consider the corresponding stable exact sequence (50)
0
9 TC~) ~
A . 9 TO(0~
M o A1 )
> TC(AI)~
> 0o
Of course all categories in (50) have finite Kr~ll dimension, and according to a general result of Gabriel [25], we have the inequality:
(51) m~(dimTC(Ao)~ , dimTC(A1)~ where
A A = (00
of
the condition that the middle dimension in (42) attains a g lven value
M
M AI).
dimTC(A)~ ~ dimTC(Ao )~ § dimTC(Al)~ + I,
Thus the pmoblem arises to formulate explicitly in terms
between the bounds of (51). dimTC(A1)~ = 0~ THEOREM 16. - L e t
Here we will only study the case when
i.e. the case when A~
A1
is left artinlan.
be a left cohement, might perfect rin[., A 1
a m
-
left artinian tins, and of eor.l ~ n ~ - ~
let
M A
be as above.
dimTC(A) ~ = dimTC(Al)~
cf. 2))
If
M
l)
is not finitely ~enerated
dimTC(A) ~ = dimTC(Al )~ + i,
Further
TC(Ao)~
step of the Gabriel filtration of -in
-
anAo-Al-bimodule satisfying the conditions
as a left Ao-mOdule__._.___ 2), then
C ~ 0
2 6 7
Mod(AI) , M ~ A 1 C
TC(A) ~
TC(A) ~
otherwise
is exactly the last
if and only if for all cqherent
is not coherent (i.e. not of finite type,
i__nn Mod(Ao).
The proof of this theorem is not difficult, and one can even prove that it is sufficient to suppose
Example:
If
K
is a field, then
C
cyclic in the last condition.
K |IK')~
dim TC
f~o 0
= i,
and using
K
variations of this example, one can construct for any integer coherent, right perfect ring
A,
such that
that the (finite) Gabriel filtration of envelopes.
Such rings
A
dim TC(A) ~ = n,
TC(A) ~
n,
a left
and such
is stable under injectlve
will be called stable (it is easy to formulate
the stability condition purely ring-theoretically, using w 7)and they will be studied in detail in the next section.
w i0.
Application i:
The structure of ~ight perfect~ left coherept~
stable rings. To obtain what the title promises, we now only have to put together suitable pieces of w 5 and w1678-9. THEOREM 17.A
Let
A
This gives:
be a left coherent, light perfect, stable ring.
can be built up in one and onl[ one way as follows:
There exists a finite strin~ of left artinian rin~s I and an Ao-Al-bimodule M~ such that Tot9 (M~
z)
Then
Note that by Hopkins [16] all left ideals of and thus all finitely generated modules in
2) Since to: M
M
is left pseudo-coherent,
is not left coherent.
M
A1
AO = Ao,AI,...,A n, is Ao-pSeudo-coherent, are finitely generated,
Mod(A l)
are coherent.
not of finite type is equivalent
268
-
but not coherent for A
Put T
Coh r 0
and
-
i = 0,
and coherent for
A 1 = (0~
A1).
There exists an A1-A2-bimodule
M1
such that
A2 . 1 or i (M , Coh) i_ss ..... es Finally we arrive at an An_l-An-bimodule
condition and such that Mo,...,~-I
(and
A~ = A ~
above, and construct
A
perfect and stable and COROLLARY.-
If
A
Mn-i A n ).
A = (An-I
and conversely , if we give
A
i Z i.
Mo
AI,... ,An_ 1 ) Al:s
Mn-I
satisfying the Tor-
Fumthermore
A~
1
,An
are uniquely determined by
a__nd Ml:s
and A
satisfyin S the conditions
as indicated, then
A
is left coherent, Eight
dimTC(A) ~ = n.
is a left coherent, right perfec!, stable rin~, then
is seml-primary, i.e.
radj(A)
is nilpotent, and
A/radj(A)
is a
semi-simple am%inian ring. This corollary is proved by induction, using the fact that radjAt_ 1 radj(A t) = ( 0 Remark i.-
We doubt that
M t-I ),
1 < t < n
(An = A).
radjA t
radj(A)
is nilpotent if we leave out the
stability condition in the corollary. Remark 2.-
Theorem 17 above implies that
Mod(A)
can be built up
canonically as an iterated comma category of module categories over amtinean rings. Mod(Ai+l)
Ti
In fact, tensoring with the Mod(A i)
Mod(A i)
defines functors
and it is vemy easy to see, that with the notations
of [37] we have natural equivalences 0 ~ i~n-l.
MI
(Mod (A i) , Ti) -~> Mod(Ai+l),
Fumthermore, one can verify that in this case, not only is
a localizing subcategory of
Mod(Ai+l),
but it is also a tri-
localizing one, (i.e. the quotient functor has a left adjoint, which has a left adjoinS.
We hope to return to a systematic study of this.
For
the bilocalizing case, see [48]. The tetralocallzing subcategomies etc. are not so interesting since they are direct factors.
-
w ii.
"Application" 2:
A
Mod(.R)
A
is said to be a (left) quasi-Frobenius
is left artinian and left self-injective (then
right quasi-Fmobenius). ring, then
-
~uasi-Frobenius cateKories.
Recall [16] that a ring ring if
269
R
In [21] it was proved that if
R
A
is also
is an arbitrary
is quasi-Frobenius if and only if every injective in
is projective.
[The "dual" of this is also true, cf. [20]~
Thus it is natural to introduce the following DEFINITION 6.-
Le__~t ~
b_e a Gmothendieck category.
a qua si-Frobenius category, if every injective in Remark i.-
We say that ~
C
is
is also projective.
It is unnatural to use the "dual" result [20] as a basis for
a definition, since there are non-zero Grothendieck categories (even satisfying AB 6) having no projective objects except
0
[49].
It was proved in [53] that a quasi-Frobcnius category decompositions of injectives. AB 6,
then
~
C
has bounde4
satisfies the axiom
is locally noethePian, and the injeetive envelope of
every noethcPian object in
~
it follows from [21q, that if R
If further~nome
~
is quasi-Frobenius.
is noethemian [53]. ~
As we have said above,
is also a module categor,/ Mod(R), then
In the general case we do not even know if
locally noethePian + quasi-FTobenius = > ~
C
locally finite, but we will
now reformulate this in a purely Ping-theoretical manner, and then prove that it is true in the special case when THEOREM 18.-
~
has a big noethePian injective:
The followin~ conditions on a linearly topolo~ized, topolo-
gically c~herent, topologically coperfect andcomplete mink
A
art
of
A,
equivalent: (i)
TC(A) ~
(ii)
A
is a quasi-Frobenius category~
is an in~ective object in
TC(A);
(iii) FoP every topologically coherent closed left ideal ~ every continuous left A-linear map ~ - - > by an element of
A
is the might multiplication
Aj and ever[ Ae (e primitive) is artinian in TC(A).
-
2 7 0
-
We do not know if the last part of (iii) is a consequence of
Remark.-
the first part. PROOF:
Follows from w 5 and from what has been said above.
PROBLEM:
Let
A
satisfy the equivalent conditions of Theorem 18.
true that
A
is pseudo-compact?
true that
A
has a fundamental system of neighbourhoods of
by bilateral ideals C~ a ring [thus
A "~>
same as
such that every
lim A/grLa.~? <
More precisely, if
A/Ol e
A
Is it
is sober, is it 0,
formed
is a quasi-Fmobenius
In this case, is the dual ring
AM
the
9
A~
Suppose now That
A
is discrete.
Then the ideals of condition
(iii) of Theorem 18 are exactly the finitely generated (f.g.) left ideals, and (ill) can be expressed by saying that
A
is
f.g. left self-injective.
Now, both J.-E. BjSrk [7] and B. StenstrSm (unpublished) have proved independently that if
A
is a right perfect, left coherent and left
self-injective ring, then
A
is a quasi-Frobenius ring, and the proof
of Stenstr~m has the merit that it extends immediately to the f.g. selfinjective case: THEOREM 19.-
A right perfect, left coherent, left f.g. self-injective
ring is a quasi-Fr~enius rin~ (i.e. the problem above has a positive solution in The discrete case). PROOF (Stenst-~Sm): Let
A
I(S) = {a 6 A I aS = 0}
and
LEMMA i:
If
rl(I) = I PROOF:
A
he a ring with unit,
S
a subset of
r(S) = {a 6 A ~ Sa = 0}.
A.
We need:
is left f.~. self-injective and left coherent, then
for all finit@ly generated right ideals
I of
A.
This is essentially a slight reformulation of what is proved in
[16], p. 399-~00. LEMMA 2.-
If
A
right ideal, then PROOF:
Put
is left coherent, and l(I)
I
is a finitely ~enemaTsd
is a finitely Kenemated left ideal.
This follows from Chase [13], Theorem 2.2.
-
LEMMA 3.- If PROOF:
It
A
271
-
is as in Theomem 19, then
A
is mlght noethemian.
suffices to show that any ascending chain ~In~
genemated might ideals is stationary.
of finitely
By Lemma 2, ~l(In)~ is a descending
chain of finitely genemated left ideals, and so it is stationamy, since A
is left copemfect.
Now Lemma 1 completes the pmoof.
To pmove Theomem
19, it is now sufficient to memamk, that a small modification of the pmoof of theomem 1 in ~0] gives that a might noethemian, left f.g. selfinjective Ping is a quasi-Fmobenius Ping.
w 12.
Final memamks. The biggest gap in the theor,j developed above is evidently the lack
of a theory of genemal (non-stable) extensions of locally noethemian categomies.
To fill this gap, we would have to generalize the theory of
genemalized triangulam matmix PinEs of w 8 to a theory of maiTix Pings of the forum A
M
A = (N~ where
M
and
N
A1) ,
ape not only Ao-A I- and Ai-Ao-bimodules respectively,
but wheme theme is also given an Ao-A ~ -bimodule map an Ai-Al-blmodule map
N~)A M o
~ > A1
M ~ A 1N
> Ao
and
such that the diagmams
IdM~)~
IdNC_~~
. lA1 l~91dM
I~
Ao ~ A oM - - 2
M
1
AI~AI N
are commutative, and where multiplication in multiplication (using ~
~91dM
and
#).
A
II > N
is defined by ma%Tix
This seems to be a formidable algebraic
problem, and we might try to make an attack on this at a later occasion. It also remains to make a detailed study of the associated topological
-
272
-
rings in the examples 1)-7) of the introduction.
The case 2) seems to be
particularly rewarding, since any locallzing subcategory of
Qcoh(X)
is
stable, thus in particular the Gabriel filtration is stable under injective envelopes, and the bimodules that describe the cor~-espondinE extensions should be of interest. etc .... .]
[We should make a duality theory for
Qcoh(X)
It should be remarked, that in the examples 5) and 7), parts
of the associated topological rings have already been studied explicitly with nice applications in [I/] I), [29] and [2;]. Finally, so~e stl,ucTume ~heorems for some classes of strictly linearly compact rings, can be found in Ill, [2], [18] and [32].
1)
Brumer [11] u s e s a more r e s t T i c t l v e t h a n t h e u s u a l one - he r e q u i r e s o f open b i l a t e r a l
definition
of pseudo-colpact rings
the existence of a fundamental system
ideal neighbourhoods of
0 ...
-
273
-
BIBLIOGRAPHY
[1]
I.
KR. AMDAL
-
F. RINGDAL, Categories unis~rielles, Comptes rendus,
s~rie A, 267, 1968, p. 85 - 87 and p. 247 - 249.
[2]
B. BALLET, Struetume des anneaux strictement lin~airement compacts qommutatifs, Comptes rendus, s~rie A, 266, 1968, p. ill3 - ii16.
[a]
H. BASS, Finistic dimension and a homoloKical generalization of semi-primar~ PinKs, Trans. Amer. Math. Soc., 95, 1960, p. 466 - 488.
[4]
H. BASS, In~ective dimension in noetherian minks, Trans. Amer. Math. Soc., 102, 1962, p. 18 - 29.
Is]
H. BASS, Lectumes on Topics in Algebraic K-theory, Tara Institute of Fundamental Research, Bombay, 1967. J.-E. BJ~RK, RinKs satisfying a minimum condition on principal ideals (to appeam in J. reine ang. Math.).
[71
J.-E. BJ~RK, On perfect minks, QF-rinKs and rings of endomorphisms of injective modules (to appear).
[8]
N. BOURBAKI, Topologie K~n~rale, Chap. i-2, 3e 6d., Hermann, Paris, 1961.
[9]
N. BOURBAKI, A1g~bre commutative, Chap. 1-2, Her~ann, Pamis, 1961.
[10]
N. BOURBAKI, Alg~bre commutative, Chap. 3-q, Hermann, Paris, 1961.
[11]
A.
BRUMER, Pseudo-compact algebras, profinlte groups and class
formations, J. Algebra, 4, 1966, p. 442 - W70. H. CARTAN - S. EILENBERG, Homologlcal Algebra, Princeton University P~ess, Princeton, 1956. S. U. CHASE, Direct prqduets of modules, Trans. Amer. Math. Soc., 97, 1960, p. 457 - 473. [14]
S.U. CHASE, A generalization of the rink of %-~iangular matFices, Nagoya Math. J., 18, 1961, p. 13 - 25.
[lS]
J. M. COHEN, Coherent Eraded rings and the non-existence of spaces of finite stable homotopy type (to appear in Comm. Math. Helv.).
-
[16]
-
C. W. CURTIS - I. REINER, Representation T h e o ~ o f and AssoclativeAlgebras,
[17]
274
Finite G~oups
Wiley, New York, 1962.
P. DELIGNE, Cohrmologie ~ support propre et constpuctlon du !
foneteur f', Appendix ~D [34]. [18]
J. DIEUDONNE, Topics in Local Al~ebra, Notre Dame Math. Lectures, n
[19]
O
I0, 1967.
C. FAITH - Y. UTUMI, quas!-inlective modules and their endomorphism Pings, Arch. der Math., 15, 1964, p. 166 - 174.
[20]
C. FAITH, Rings with rscendin ~ condition on annihilators, Nagoya Math. J., 27, 1966, p. 179 - 191.
[21]
C. FAITH - E. A. WALER, Direct-sum representations of injective modules, J. Algebra, 5, 1967, p. 203 - 221.
[22]
P. FREYD, Functor Categories an_d ths
Application to Relative
Hrmology (mimeographed notes), Columbia Univemsity, 1962. [23]
P. GABRIEL, Objets injeetifs dans les categories ab~liennes, S~minaire Dubreil-Pisot, 12, 1958 - 1959, Expos~ 17.
[2~]
P. GABRIEL, SuP les categories ab~liennes loealement noeth~riennes et leuPs applications aux_alg~bmes ~tudi~es pap Dieudonn~, S~mlnaime J.-P. Serre, Coll~ge de France, 1959 - 1960.
[25]
P. GABRIEL, Des cat~omies ab~liennes, Bull. Soc. Math. Ft., 90, 1962, p. 323 - 4q8.
[26]
P. GABRIEL, C~oupes formels, S~minaire Demazure-Grothendieck, I.H.E.S., 1963 - 196~, Fasc. 2 b, Expos~ VII B (cf. Math. Rev. 35, 1968,~4222).
[27]
P. GABRIEL - U. OBERST, Spektralkategomien und regul~me Ringe im yon Neumannschen Sinn, Math. Z., 92, 1966, p. 389 - 395.
[28]
P. GABRIEL - R. RENTSCHLER, SuP la dimension des anneaux et ensembles qrdonn~s, Comptes rendus, s~mie A, 265, 1967, p. 712 715.
-
[29]
275
-
v v E. S. GOLOD - I. R. SAFAREVIC, 0 ba~ne pole d klassov, Izv. Akad. Nauk SSSR, set. mat., 28, 1964, p. 261 - 272. Engllsh translation in: American Mathematical Society TPanslations~ Set. 2, 48, 1965, p. 91 - 102.
[30]
A. GROTHENDIECK, SuP guelques points d'alK~bre homoloKique, Tohoku Math. J., 9, 1957, p. 119 - 221.
[31]
A. GROTHENDIECK, Technique de descente et th~or~mes d'existence en K~om~trie alg~brique, II: Le th~or~me d'existence en th~orie for~nelle des modules, S~minalre Boumbaki, 12, 1959 - 1960, Expos~ 195. (Has also been published by Benjamin, New York, 1966.).
~2]
A. GROTHENDIECK (and J. DIEUDON~), El~ments de g~om~tTie alK~bri~ue, Publ. Math. I.H.E.S., n ~ 20, 196~.
[33]
M. HARADA, On a special
type
of hereditary abellan categories,
Osaka J. Math., 4, 1967, p. 243 - 256.
[3.]
R. HARTSHORNE, Residues and Duality, Springer, Berlin, 1966.
[3s]
I. KAPLANSKY, Dual modules over a valuation ring, Proc. Amer. Math. Soc., 4, 1953, p. 213 - 219.
[36]
I. KAPLANSKY, Infinite Abelian Groups, University of Michigan Press, Ann A~bor, 1954.
[37]
F. W. LAWVERE, The category of categories as a foundation for mathematlcs, Proceedings of the Conference on Categorical AiEebra, La Jolla 1965, Springer, Berlin, 1966, p. 1 - 20.
[38]
H. LEPTIN, Linear kompakte Moduln und Ringe, I., Math. Z., 62, 1955, p. 241 - 267.
[39]
H. LEPTIN, Linear kompakte Moduln und Ringe, II., Math. Z., 66, 1956 - 1957, p. 289 - 327.
[.0]
E. MATLIS, Injective modules ov@r noetherian rings, Pacific J. Math., 8, 1958, p. 511 - 528.
[.1]
B. MITCHELL, Theoz7 of Categories, Academic Press, New York, 1965.
[421
C. N~STASESCU - N. POPESCU, qgelques observations sur les topos ab~liens, Rev. Roum. Math. Pumes et Appl., 12, 1967, p. 553 - 563.
-
2 7 6
-
Y. NOUAZE, Cat6~ories loealement de.type fini e t categories localement noeth~riennes, Comptes rendus, 257, 1963, p. 823 - 82,.
[.q
F. OORT, Commutative Group Schemes, Springer, Berlin, 1966.
[.5]
Z. PAPP, On algebraically closed modules, Puhl. Math. Debrecen, 6, 1959, p. 811 - 327.
[.6]
J.-E. ROOS, Sur les foncteurs d~riv~s de rendus, 252, 1961, p. 3702 - 370,.
["7]
lim. <
Applications. Comptes
J.-E. ROOS, Bidualit~ et structure des foneteups d~riv6s de dans la cat~gorle des modules sup un anneau r~gulier, rendus, 25,, 1962,
[.8]
p. 1556 - 1558
lim
Comptes
and p. 1720 - 1722.
J.-E. ROOS, Caraet~risation des categories qul song quotients de cat6gories de modules par des sous-ca!~ories biloealisantes, Comptes rendus, 261, 1965, p. ,95, - ,957.
[,9]
J.-E. ROOS, S.ur les foncteums d~riv6s des produits infinis dans les cat6gorles de Grothendieck. Exemples ' et contre-examples, Comptes rendus, 263, 1966, p. 895 - 898.
[50]
J.-E. ROOS, Sup la condition AB 6 et ses variantes dans les categories ab~liennes, Comptes rendus, s6rie A, 26,, 1967, p. 991 99,.
[51]
J.-E. ROOS, Locally distributive spectral categories and strongly regular mines, Reports of the Midwest Category Seminar, Springer, Berlin, 1967, p. 156 - 181.
[52]
J.-E. ROOS, Sup l'anneau maximal de fractions des A ~ - a l g ~ b r e s e t des anneaux de Baer, Comptes rendus, s6rie A, 266, 1968, p. 120 123.
Is3]
J.-E. ROOS, SuP la d6compositio 9 born~e des objets injectlfs days les ca t~Kories de Grothendleck , Comptes rendus, s~rle A, 268, 1968, p. 449 - ~52.
[5.]
J.-E. ROOS, Sum la structure des categories ab~llennes loealement nqeth~riennes, Comptes rendus, s~rie A, 266, 1968, p. 701 - 704.
[55]
S6minalre H. CARTAN, ii, 1958 - 1959, Expos6 15. (Has also been published by Benjamin, New York, 1967.)
-
is6]
277 -
J.-P. SERRE, GrouDes pr~alg6bmiques, Publ. Math. I.H.E.S. n ~ 7, 1960.
j.-P. SERRE, Al~b,,e LocaZe.
SultipZlciZ6s
(r6dig6
par P. GabrieZ),
SprlnEe~ , Berlin 1965. J.-P. SERRE, Cohom91ogle Gal01sienne., Springer, Berlin, 1965. Y. UTUMI, On a theorem on modular lattices, Fr~c. Jap. Acad., 35, 1959 s p. 16 - 21.
[60]
J.-L. VERDIER~ Topologies et faisceaux~ Fasc. 1 of S6minaiz~ ArtinG~othendieck, I.H.E.S., 1963 - 196~.
- 278 -
KAN EXTENSIONS,
COTRIPL~AND
ANDR~
(CO)HOMOLOGY
by
Friedrich Ulmer I
INTRODUCTION
The aim of this note is to p o i n t out that the "non-abelian"
derived
and B a r r - B e c k
[2] are the abelian
Kan e x t e n s i o n
and to sketch how the p r o p e r t i e s
H
functors
can be o b t a i n e d w i t h i n
and g e n e r a l i z a t i o n s
results
and insights.
derived
functors of
framework.
of the A,
turns out to be the s t a n d a r d for c o m p u t i n g
of p r o j e c t i v e s
limit ourselves
(cf.
the left [20] this
We m o s t l y
and leave
it to the reader to state the dual theorems
to d e a l i n g w i t h h o m o l o g y
sure that this works
try to avoid exactness Grothendieck's
of
of k n o w n facts as w e l l as to new
algebra
by means
To make
and
In many
volume).
cohomology.
[i]
An instance of the latter is the
in h o m o l o g i c a l
functors
of A n d r ~
leads to s i m p l i f i c a t i o n s
m e t h o d of acyclic models w h i c h procedure
H,
derived
the a b e l i a n
cases the a b e l i a n v i e w p o i n t proofs
A, and
conditions
axiom AB5),
for
in practice,
on the range c a t e g o r y
we (i.e.
[12]).
i Part of this w o r k was s u p p o r t e d by the F o r s c h u n g s i n s t i t u t ffr M a t h e m a t i k der E.T.H. and the D e u t s c h e F o r s c h u n g s gemeinschaft.
-
However
a few results
without
it.
to be very useful
representable
functor
is a composite
of
[A,-]
: A
the S-fold
Ej
given by M 6 M
sum of
A.
category
: [M,A] ---> [~,~]
"restriction" small,
then
unless
~
triple
in
and
Ej(t)
extensions illustrated
~).
But if
> C
(right)
(e.g.
t = A|
and
on M
A|
If
~
is a subcategory
and
A
of the M
is not
t : M--->
Ej(t)
relationship
the higher
If
if there
then
S
Kan extension z
s.J.
representable
Ej ;i.e.
to a set
be a functor
for every
properties
A
[M,-]
as the left adjoint
The close
for
A generalized
where
assigns
The
[21]
the left adjoint
by the fact that the functors
vanish
2 If of
sums.
and generalized
of
G
with
AO
need not exist
= A|
objects
triple
J : M
is defined
has special
acyclic Ej
Let
A|
Rj : [~,A] ---> [ M , ~ ] , s ~ - > Ej(t)
functors
to a category
> ~,
and
Recall that
M
false
of Kan
in this context.
: M---> S
> S.
be an abelian
representable
from a category
A|
is a hom-functor
and are probably
is based on the notions
[13] and generalized
which prove
-
depend on AB5)
Our approach extensions
2 7 9
is a coexists
between
functors
Kan
is also
A|
are
left derived
functors
is a category with a coof G-projectives,
J is full and faithful, then Ej(t) is an e x t e n s i o n t : M > A. This explains the terminology.
-
then c o m p o s i t e analogous
functors
t.G
280
-
: M
to the one of g e n e r a l i z e d
The axioms of Andr~ essentially
A
> M
representable
and B a r r - B e c k
ties of the functors information
about
t.G
A
and B a r r - B e c k in seminars
of 1965-66 results
and
and
A|
appear
presented
contain
at the F o r s c h u n g s i n s t i t u t
of B. Eckmann),
in the Math.
this note
(or rather
functors
derived
E. Dubuc
func-
some of the
found i n d e p e n d e n t l y
Zeitschrift,
For the notation,
proper-
during the w i n t e r
by several
(in a thesis u n d e r the [9], U. Oberst
The paper of U. Oberst,
[23])
H
a lot of
In the m e a n t i m e
A m o n g them are M. B a c h m a n n
(unpublished).
and
first o b s e r v e d w h e n
their n o n - a b e l i a n
and summer of 1967. 3
supervision D. Swan
A
H
of this summary 4 were
authors.
functors.
This and the above m e n t i o n e d
A good deal of the m a t e r i a l was
tors
for
assert that they are the left d e r i v e d
of the Kan extension.
Andr~
have p r o p e r t i e s
[18], and
which
is to
led me to revise part of
to include
some of his results.
terminology
and the p r e l i m i n a r i e s
3 Some of the m a t e r i a l herein was first o b s e r v e d d u r i n g the w i n t e r of 1967-68 after I had received an early v e r s i o n of [2]. The second half of [2] Ch. X was also d e v e l o p e d d u r i n g this period and illustrates the mutual influence of the m a t e r i a l p r e s e n t e d there and the c o r r e s p o n d i n g m a t e r i a l here and in [23]. 4 Details
are to appear in another L e c t u r e Notes v o l u m e
[23].
-
about g e n e r a l i z e d
2 8 1
representable
we refer the reader to
[20]
-
functors
(1)-(12)
and Kan extensions
which
is c o n t a i n e d
in
this Lecture Notes volume. I am indebted many s t i m u l a t i n g
discussions
not have its present Let category every
natural idtM
and let
> C
t 6 [M,A]
be the i n c l u s i o n
X 6 M
and object tM |
under the Yoneda
= 9 (tM) f f by f : M
of a small sub-
M 6 M
[M,-] ~
isomorphism
t
is a m o r p h i s m
tM |
which,
restricted
> X,
is
: tM ---> tX.
tf
{idtM}M6 M determines ~(t)
: 9
(tS |
there is a
[M,-],
t] ~
to
[tM,tM].
[M,X] =
on a s u m m a n d
a natural
For
w h i c h corresponds
[tM |
> tX
(i)
the paper w o u l d
be an abelian c a t e g o r y w i t h sums.
transformation
Its value at
ties
A
without which
Barr for
form.
J : M
functor
to Jon Beck and M i c h a e l
tM
indexed
The family of identitransformation
[M,-])
---> t
M
w h i c h by the above the r e p r e s e n t a b l e tive to the class which
functors ~
are o b j e c t w i s e
follows [20]
is an objectwise A |
split exact
(12) we showed that if
A
does.
standard h o m o l o g i c a l
[M,-]
are p r o j e c t i v e s
of short exact sequences
that there are enough
projectives
split epimorphism.
[M,A]
~
(cf.
[20]
from
rela-
[M,~]
(4)), it
in
also enough
Thus we obtain
algebra
(3),
-projectives has
in
Since
[M,A].
In
(absolute)
[20]
(ii) by
-
(2)
of a s m a l l with
Theorem.
sub-category
sums.
Then
and its r e l a t i v e A
Let
has e i t h e r
2 8 2
J
of
C
-
> C
9 M
and
A
be an a b e l i a n
Ej
: [M,A]
the Kan e x t e n s i o n left d e r i v e d
enough
functors ~-L
projectives
derived
exist.
functors
coincide by
for
latter n i o.
A,(,-)
tors A
~ - L , Ej
(3)
P
The only
For every
(t)
: ....
to = t
and the p r o p e r t y sequence gives
of f u n c t o r s
of
plexes
(t')
in
[~,A].
it m a k e s
5 i.e. 6 P,(t)
> EjP
iums
The
~-L,Ej
are e x a c t
denotes
(t)
of
Ej
the e p i m o r p h i s m
~(to) ,
~(t)
9 (toM| M6M
etc. 6
> t
if in
resolution.
> t"
> O
~(t)> t
Using
[20]
(9)
a short exact
> 0
in
[M,A]
sequence --~ EjP
(t")
> O
long e x a c t h o m o l o g y into an a b s o l u t e
in
L.Ej(-)
is t h a t the func-
functors
projective
0 ---> t'
also
and ~ - L n E j
~ , one can s h o w t h a t
rise to a s h o r t e x a c t
0 ---> EjP
with
AB4)
If
homology.
--->
t I = ker
L.Ej
we denote
derived
9 (tlM| M6M
and
[~,A]
Grothendieck's
Ln EJ
to p r o v e
t 6 [M,A]
rise to a r e l a t i v e
9
where
thing
are the a b s o l u t e
is AB4).
(i) gives
it the A n d r ~
category
exist.
functors
In the f o l l o w i n g
and call
Proof.
case the
>
Ej
or s a t i s f i e s
a x i o m AB4), 5 t h e n the a b s o l u t e In the
be the i n c l u s i o n
of c h a i n
sequence
exact
com-
associated
connected
A.
the n o n - a u g m e n t e d
complex,
i.e. w i t h o u t
t.
- 283 -
sequence
of functors.
Since
~-LnE J
vanishes
for
n > o
on sums
9 (tM | [M,-]), it follows, by s t a n d a r d homoloM gical algebra, that ~ - L , E j is left universal. In other words,
the functors
functors
(4)
LnE J
~-LnE J
are the
of the Kan e x t e n s i o n
A c o m p a r i s o n w i t h Andre's
H(,-)
:
[M_,A_]
[C,A_]
in
[i] p.
agrees with the Kan e x t e n s i o n
it follows Since
Since both
Hn(,-)
: [M,A]
sums of r e p r e s e n t a b l e
15
Ej
Ho(,-)
from the exactness 9
of
functors
is valid. because
This h o w e v e r functor
3-5, shows
is not so.
t : M
and defines
H
> A n
(-,t)
H,
Ej
a complex
in Ho(,-)
are right exact,
for
H
( ,-)
n > o
on
H,(,-)
are the left
_~ A , ( , - )
= L,Ej(-)
is "by chance"
in an e n t i r e l y
Recall
that,
on sums of r e p r e s e n t a b l e
It may seem at first that this
Andr~ c o n s t r u c t s
[~,A].
it follows by s t a n d a r d homo-
derived
Hence
9
theory
vanishes
algebra that the functors Ej.
: [M,A]
(3) that they coincide.
[~,A]
functors,
left d e r i v e d
[i], the functor
and
logical
of
Ej
homology
view of his second a x i o m on page
functors. 7
(absolute)
d i f f e r e n t way.
that he associates
with every
of functors
: C
to be the n-th h o m o l o g y
C
(t) of
C
(t)
> A (cf.
[I] p. 3).
7 In the n o t a t i o n of Andre, C should be r e p l a c e d by N. Note that in view of [20] (~) a "foncteur ~ l ~ m e n t a i r e ~ of Andr~ is the Kan e x t e n s i o n of a sum of r e p r e s e n t a b l e functors M----> A.
-
It is not difficult on
M
of
Cn(t) .J
an objectwise construction
Namely,
choose
an
[1] prop.
Ej(S,.J) of
A,(,-)
of
1.5 because Ej-acyclic
is valid. 9 which,
the Kan extension
is
Thus Andre's
procedure
in homo-
functors
of
of
t; apply
Ej
is true for his computational the restriction resolution
of
of the complex S.J
and
generalize
(2),
results
of Andr~
(3) and the nice b e h a v i o u r
on representable
functors.
Theorem (a)
For every Ap(J-,t)
Ej.
We now list some of the properties
in part, of
Cn(t)
C,(t) .J
t.
the left derived
The same
They are consequences
(5)
Moreover,
resolution
of
and that the Kan
Ej-acyclic 8 resolution
is an = S,
Cn(t ).
to compute
and take homology.
M
functors
turns out to be the standard
algebra
on
is
split exact
logical
device
-
to show that the restriction
is a sum of representable
extension
S
284
functor : M ---> ~
t
the composite > ~
8 A functor is called Ej-acyclic vanishes on it for n > o.
if
is zero for
p
>
O.
[s
LnE J : [M,A]
9 In [20] we show that this computational closely related with acyclic models.
method
is
[1]. of
-
(b)
Assume
that
an o b j e c t (M,C) p
(c)
>
A
in
C
Ap(-,t)
sketch
Then
Ap(C,t)
Assume moreover
that for every
functor
: C ---~ S
[JM,-]
C
be
vanishes
for
index categories.
A,(-,t)
> A
: C
As f o r
M
it follows
vanishes
this
it
direct
from
is
a s s u m i n g that
of
limits
such direct
assumption
are finitely
(a) and
on a r b i t r a r y
applications,
this w i t h o u t
of
the hom-
Then
also preserves
(In most examples
sums,
M 6 M
preserves
over d i r e c t e d
: _C ---> _A
establish
and let
o~
has finite
Mv 6 M.
category
such that the comma category
fied if the objects
M
-
is an AB5)
is directed. I0
limits.
If
285
sums
great A
is satis-
generated.)
(c) that @ M~,
where
importance
is AB5).
to
We will
later how this can be done.
The p r o p e r t i e s sequences
of
(3),
that in an AB5) categories
(a) -
(2), footnote
category
direct
(c) are s t r a i g h t f o r w a r d 15),
[20]
con-
(9) and the fact
limits over d i r e c t e d
index
are exact.
i0 A category of objects morphisms morphisms
D is called d i r e c t e d if for every pair D,D' i~ D there is a D" 6 D t o g e t h e r with D ---> D", D' ~ D" and--if for a pair of y,l: D0---~ D 1 there is a m o r p h i s m u: D 1 ---> D 2
such that uy = ~l. (M,C) are m o r p h i s m s
Recall that objects of the category M > C, where M 6 M.
286
-
(6)
A change
sequence small J'
(cf.
[I] prop.
full sub-category
: M' --~ C
A'
and
A"
J = J'.J", composite Ej,
and
it follows of
>
of
containing
~
: M
> M'
Ej
2.4.1). objects
A
is AB4)
A
into E j , - a c y c l i c
representable The
A
and by
Since is the
homological a spectral
algebra
sequence
(-,t) P+q
has enough that
objects.
projectives
(cf.
[13]
Ej,, takes E j - a c y c l i c
But this is obvious
the Kan e x t e n s i o n
from
of a r e p r e s e n t a b l e
The same holds
for p r o j e c t i v e
functors. II "Hochschild-Serre"
[i] p. 33 can be obtained Likewise, to a u n i v e r s a l
Denote by
with
t : M --~ A
is again representable.
be a
the inclusions
[M',~]
One only has to verify
(9), because
functor
or
M.
Thus by standard
functor
M'
: [M,A] ---> [~,A]
A' (-,A" (-,t) ) " ~ P q
provided
let
Andr4 homologies.
that
[~,~].
is for every
rise to a spectral
For this,
Ej,, : [M,~] --~
(7)
[20]
J"
gives
8.1).
the associated
: [M',A]
there
of models
-
sequence
of Andr~
in the same way.
a composed
coefficient
spectral
coefficient
spectral
functor
gives
rise
sequence.
ii The a s s u m p t i o n that M' is small can be replaced by the following. The Kan e ~ t e n s i o n Ej, : [M',A] --~ [~,~] and its left derived functors L,Ej, exist and LnE J, vanishes on sums of representable functors for n > o.
-
(8)
Theorem.
Let
t : M
functors,
where
and
A'
A
have enough projectives. functors
L,F
F
: A --~ A'
are either AB4)
categories
Assume
and
be or
that the left derived
F
has a right adjoint.
Then
sequence
L F.A
P
for
-
> A
exist and that
there is a spectral
provided
2 8 7
(-,t)
;
A
q
the values
(-,F-t) P+q
of
t
are F - a c y c l i c
(i.e.
LqF (tM) = 0
q > o ).
(9)
Corollary.
~(-,t)
: ~
A,(C,F.t) product
> ~
If for every
vanishes
on an object
~-- L,F(Ao(C,t)).
formula,
preserving. the property
provided
For,
let
%(C
p > o
This gives Ao(-,t)
C = @ C
,t) = 0
for
the functor C 6 ~,
then
rise to an infinite
: ~
> A
is sum-
be an arbitrary p > o.
co-
sum with
Then the canonical
map =.
(i0)
@ A,(C~,F.t)
is an isomorphism
because
L,F(O Ao(C~,F.t))
> A,(@ C~,F.t)
9 L,F(Ao(C~,F.t))
~ L,F(Ao( ~ C~,F.t))
holds. 12
12 This applies to the categories of groups and semi-groups and yields infinite coproduct formulas for homology and cohomology of groups and semi-groups w i t h o u t conditions. For it can be shown that they coincide with A, and A* if A' = Ab.Gr., A = C-modules, t = Diff and F is tensoring With ~ ~-~mmi~g into some C-module (cf. [2] Ch. 1 and Ch. X).
- 288 -
(ii)
Cgrol!ary.
Then every gives
finite
: C ---> A' --
satisfied. if
i
sub-sum
9 C~i i
Proof EC
: [C,A]
> A
evaluation tions
on
of
F
and
t
>
for
the c o n d i t i o n
in
(5c)
is
,F.t) --~ 9 A . ( C v , F . t )
(C~i,t) 9 C~
(8) and E6
of f u n c t o r s
: ~
~').
is v a l i d
for e v e r y
*
of
and
(but not
A,(-,t)
formula
A,(e C
--~ 9 A
is AB5) for
coproduct
In o t h e r w o r d s , A , ( e C~i,t)
A
formula
provided
8
i
finite
that
coproduct
rise to an i n f i n i t e
A,(-,F.t)
holds
Assume
.13
(ii).
(Sketch)
: [C,A']
--~ A'
associated
imply
with
By we d e n o t e
C 6 C .
the
The a s s u m p -
t h a t the d i a g r a m
_A
/E'(F.-) [M,A] J >
is c o m m u t a t i v e .
The d e r i v e d
can be i d e n t i f i e d
with
spectral
arises
sequence
E c'.EJ'(F.-)
into
formula
A
of
for
[C,A'] '
Ec.E J
(-,t)
: C
~" u
functors
t ~
A
) A'
of
(C,F.t)
E 6 . E ~ ( f . - ) :[M,A] ---~ A' .
As a b o v e
f r o m the d e c o m p o s i t i o n F.
and >
A
The
infinite
in
(6) the
of coproduct
can be e s t a b l i s h e d
by m e a n s
(5c).
13 This a p p l i e s to all f i n i t e c o p r o d u c t t h e o r e m s e s t a b l i s h e d in B a r r - B e c k [2] Ch. 7 and A n d r ~ [i] w i t h _A, _A' , t and F as in f o o t n o t e 12.
- 289 -
Thus (8).
it also holds
for the E 2 - t e r m of the spectral
One can show that the d i r e c t
sum d e c o m p o s i t i o n
E 2 - t e r m is c o m p a t i b l e w i t h the d i f f e r e n t i a l s iated f i l t r a t i o n one obtains
A.(-,F.t)
of the spectral
an infinite
An abelian
resolutions
and the assocIn this way,
formula
interpretation
and n e i g h b o r h o o d s
paper of U. O b e r s t version
of the
for
A'
:
(12)
sequence.
coproduct
sequence
[18].
is c o n t a i n e d
remained
this we b r i e f l y
review
non-abelian
in a f o r t h c o m i n g
We include here a s o m e w h a t
of ~his i n t e r p r e t a t i o n
problem which
of Andre's
improved
and use it to solve a central
open in B a r r - B e c k
[2] Ch. X.
the tensor p r o d u c t
|
For
between
func-
tors w h i c h was
i n v e s t i g a t e d by D. B u c h s b a u m
[5], J. F i s h e r
[10], P. Freyd
[ll], D. Kan
[17]
C. Watts product
[24], N. Y o n e d a
[13], U. O b e r s t
[25], and the author.
The tensor
is a b i f u n c t o r
(13)
8_ : [M_opP,Ab.Gr.]
x [M,A]
d e f i n e d by the f o l l o w i n g u n i v e r s a l s 6 [s_~
t 6 [M,~]
and
(14)
[s | t,A] ~
in
s,t
and
A.
> A
property. A 6 ~
isomorphism
natural
[18],
[s,[t-,A]]
For every
there
is an
-
It can be c o n s t r u c t e d A-modules,
namely = A
where
A
choose
a presentation
X @ Y
to b e the c o - k e r n e l
where
Y~ = Y = Yu
of c o n t r a v a r i a n t Z |
[-,M]
Y
"
---> 9 A
=
free)
X
---> 0
X
and d e f i n e
9 Y~---> 9u Y u ,
is p l a y e d b y the f a m i l y
functors where
M 6 M
and
Z
denotes
(13)
follows
8 t = tM
The u n i v e r s a l
property
(2) in the
of
following
bifunctors
(L s |
--~ L
that the d e r i v e d and
sense
s @
(8 t)(s),
and the v a l u e s groups.
--~ [Z 8
of
Under
Tor
(s,t)
makes
Tot
(s,t)
can be c o m p u t e d
s
these
and has
way: [-,M],[t-,A] ]
(15) and the c l a s s i c a l
> ~
abelian
A
[tM,A] --~ [T,[tM,A]]
: [M~
AB5))
of
[-,M])
[20]
One can s h o w by m e a n s
(resp.
~
module
of the i n d u c e d m a p
The role
lemma
[-,M] | t,A]
property
(e A v) | Y = 9 Yv,
9 A
as above.
f r o m the Y o n e d a
t
2)
3) for an a r b i t r a r y
(Z |
about balanced
A 8 Y = Y;
between
Thus we d e f i n e
(15)
[w 8
product
= Y;
---~ A b . G r . ,
the integers. 14
and c o n t i n u e
l)
representable
: M~
-
like the t e n s o r
stepwise:
and
2 9 0
argument
functors
: [M,A]
> A
provided
A
are free conditions
its u s u a l
by p r o j e c t i v e
of h a v e the
is AB4)
(resp.
torsion
the n o t i o n
properties,
e.g.
or f l a t r e s o l u t i o n s
14 N o t e t h a t the f u n c t o r s Z @ [-,M], M 6 M are p r o j e c t i v e and f o r m a g e n e r a t i n g f a m i l y in [Mopp,~b.Gr.]. This f o l l o w s e a s i l y f r o m the Y o n e d a l e m m a [ 2 ~ ~ ) and (5).
-
in either variable. representable
291
-
We remark w i t h o u t
functor
A |
[M,-]
: M
proof that every >
A
is flat.
should be noted that for this and the following one cannot
replace
the condition
AB4)
It
(until
(22))
by the assumption
that
has enough projectives.
(16)
U. Oberst
exact sequences objectwise of
s @
~(L,s and
@ t
@)(t)
@
abelian
He shows
C 6 C
: M~
to ~
@ t)(s)
Thus the notion
[J-,C]
He establishes
an isomorphism
(17)
A,(C,t)
t : M---~
abelian
of
is a relative Andre's
projective
result that
complex
C
(t)
C
: C
A
p. 3) or a n o n - a b e l i a n special
(C,t)
variable.
of which
~
C
[J-,C],
of
Z |
of
at
C C
With
are free
t)
out that a non[i] p. 17
[J-,C].
Thus
either by the
(cf.
(4) and
[1]
turns out to be a
fact t h a t ~ T o r , ( - , - )
projective
s
is the inclusion.)
can be computed
evaluated
on
a functor
and points
resolution
by a relative
are
functors
sense.
in the sense of Andr4
case of the well known
be computed
|
resolution
> A
makes
J : M
~
which
any conditions
the values
~_~Tor,(Z
for every functor
of short
have the property
is a s s o c i a t e d
(Recall that
resolution
[M,A]
~-Tor,(s,t)
there
~
that the derived
without
---> Ab.Gr.
groups.
the class and
relative
~ ~(L,
every object
considers
[M_~
split exact.
and
t.
in
[18]
resolution
can
of either
- 292
U. Oberst also observes that a n e i g h b o r h o o d of
C
(cf.
of
Z | [J-,C].
("voisinage")
[1] p. 38) gives rise to a relative projective Therefore,
it is obvious that
A,(C,t)
can
also be computed by means of neighborhoods. (18)
The notion of relative
difficult to handle in practice. sequences
(7) and
(8)
~-Tor,(-,-)
is somewhat
For instance,
the spectral
and the coproduct formulas
(i0) and
(ii) cannot be obtained with it because of the m i s b e h a v i o u r of the Kan extension on relative projective Moreover, also
Andre's computational m e t h o d
[i] prop.
t
Call a natural transformation
a ~-natural
transformation
9 imp(M)
> O
and
The above difficulties [M,A]
>
[~,A]
and
not preserve ~ - n a t u r a l
if for every
O ---> imp(M)
M 6 M
---> t'M
9 t'
the morphisms are split.
arise because the Kan extensions [M~
--~
transformations.
(17) etc. of the relative ~ - T o r
Oberst
| [J-,C],t)
~ : t
[c~
do
Our notion of
absolute Tor, does not have this disadvantage.
similarly
The proper-
can be e s t a b l i s h e d
for the absolute Tor, using the techniques
[18].
of
U.
We now sketch a different way to obtain these.
The fundamental Kan extension
(cf.
in question need not be relative
projective.
ties
1.8
(4)) cannot be e x p l a i n e d by means of ~ - T o r , ~
because the resolution of
tM
resolutions.
relationship between
0
and the
Ej : [M,A] ---~ [~,~] is given by the equation
-
(19)
(Z |
where
t
and
C
respectively.
[J-,C])
293
|
t--~ E
are arbitrary
To see this,
-
(t)(C)
J
objects [J-,C]
let
[M,A]
of
= lim
and
[-,M ]
C be
>
the canonical direct
representation
limit of c o n t r a v a r i a n t
index category
for this
[J-,C]
of
representation
(M,C).
and it follows
(15) and Kan's
~Z |
[J-,C])
8 t = lim t M --
the homology
(20) Thus
(3) of
A A,(C,t)
tions of
Z |
= E
(t)(C).
t,
Since
[J-,C],
A
(-,t)
is
(t).J
t)
either by p r o j e c t i v e
(e.g. n o n - a b e l i a n
C
[-,M~]
.
(Z |
can be computed
resolutions
= lim Z | --->
c o n s t r u c t i o n 15 that
neighborhoods 16) or flat resolutions or Andre's
[J-,C]
with
E P (t), where P, (t) is the J 9 it follows from (19) that
(C,t) --~ Tor
[J-,C]
is isomorphic
Z |
as a
Note that the
J
of the complex
flat resolution
Hence
__>
> S
hom-functors.
the comma category from
: M~
and
of
resolu-
resolutions t
S ,
(e.g. cf.
and P, (t)
(4)).
15 Kan [13] 14 constructed Ej : [M,A] > [C,A] objectwise. He showed that Ej(t) (C) is the direct limit of (M,C) ---> A, (S > C)~ tM. 16 Further examples are provided by the simplicial resolutions of Barr-Beck [2] Ch. 5, the projective simplicial resolutions of type (X,o) of Dold-Puppe [8] and the pseudosimplicial resolutions of T i e r n e y - V o g e l [19]. A corollary \ of this is that the Andr~ (co)homology coincides with the theories d e v e l o p e d by Barr-Beck [2], D o l d - P u p p e [8], and T i e r n e y - V o g e l [19] w h e n C and M are defined appropriately. Note that there ar~ more pr--ojective resolutions of Z | [J-,C] than the ones described so far (for instance, the resolutions used in the proof of (21) below).
-
2 9 4
-
The above m e t h o d s prove v e r y u s e f u l in e s t a b l i s h ing the t h e o r e m b e l o w w h i c h is b a s i c for m a n y
(21)
Theorem.
a category However,
C
Let
M
be a full small s u b - c a t e g o r y of
w h i c h has sums.
if a s u m
@ Mi 6 C
M
n e e d not h a v e finite sums.
is a l r e a d y
a s s u m e d t h a t e v e r y s u b s u m of
@ Mi
ing t o
M,
t : M
> A
M ---> 9 M~
where
M
6 M.
p > o
v a n i s h e s on a r b i t r a r y
Proof. resolution
M,
M 6 M
it is M.
Moreover,
and
e M~
6 C
factors t h r o u g h a s u b s u m b e l o n g Let
a sum-preserving
Then for
in
is also in
assume t h a t for e v e r y p a i r of o b j e c t s every morphism
applications.
A
be an AB4)
functor, i7
the f u n c t o r 9 M~,
sums
c a t e g o r y and
%(-,t)
where
M
: C
9
A
6 M. I8
The idea is to c o n s t r u c t a p r o j e c t i v e
N, (@ M~,M~)
of
w |
[j-, ~ M~]
: M_Opp
; Ab.Gr.
w h i c h remains e x a c t w h e n t e n s o r e d w i t h | t : [M~
> A
.
W i t h e v e r y p a i r of o b j e c t s and e v e r y full small s u b - c a t e g o r y
17 The m e a n i n g is that e x i s t in M.
t
M
C 6 C of
Mc
and
M 6 M
C ,
has to p r e s e r v e the sums w h i c h
i8 I am i n d e b t e d to M i c h e l A n d r 4 for an i m p r o v e m e n t of an e a r l i e r v e r s i o n of the theorem. He also f o u n d a d i f f e r e n t p r o o f b a s e d on the m e t h o d s he d e v e l o p e d in [i], a s s u m i n g that M has finite sums and that e v e r y m o r p h i s m M > 9 M factors t h r o u g h a finite subsum.
-
Andr~
associated
augmentation gives ~
a s.s.
M
(M,C)
denotes
full
complex
M~
which
defined
for e v e r y
Z |
n
to
M.
of a s u m
ascending
arrows
Hi
where > Mi-i
morphisms.
M~
contraction
Sn(M)
6 M,
on
of t h o s e
resolution
of
Z
| M,(-,e
~n(M,C). M
be the of
is
(~ Mv,~)
M~)
[-,~]).
where
subsums
N
and c o n s i s t s
More
of
6 M_
o _< i _< n
for
Mo
9 M
> 9 M
complex
> Ab.Gr.
precisely,
there
is o b j e c t w i s e
a sum
the c a n o n i c a l
(e M ,M)
N
is a s u m m a n d
and the d o u b l e
denote
: N n(s S ,~) (M) ~ For
It
,
and let
--> 9 M v
and
as follows.
38).
of s u b s u m s
The a u g m e n t e d
of f u n c t o r s
defined
Mi
M~
(Z |
an
Ab.Gr.
group
The
S
chain
Mn_ 1 [-,~n],
to
consisting
M,
[i] p.
M~
where
to be a s u b c o m p l e x
in d i m e n s i o n
Mn
of
belong
from
with
of f u n c t o r s
the free a b e l i a n
C = (9 M~,
sub-category
together
(cf.
[JM,C],
| M C-,C) ---> ~, | [J-,C]
Let
> ~ |
[J-,e M ]
s p l i t exact.
Nn+ I(s S ,~) (S)
~ M
and a s u b s u m
The is
9 M
i ~i S~ 6 M. 9 M -----> ~ M the c a n o n i c a l m o r p h i s m , i ~i ~ T h e n t h e r e is f : M > 8 M~ be a m o r p h i s m , M~ 6 M.
denote Let
by
a unique Mf
decomposition
~ 9 M
tains
the
assigns > Mf
M
> Mf=>
is the s m a l l e s t
"image"
The m o r p h i s m
M
;
-
(M,C)
set
rise to an a u g m e n t e d
Z | M--n(M,C)
2 9 5
of
S_l(M)
f. : Z |
to a b a s e e l e m e n t in the c o m p o n e n t
9 M
subsum
Obviously [JM, ~ M f : M Z |
]
such
that
which
con-
is an o b j e c t > 9
> ~ M
[M~M f]
M
of Mf
f
of
(~ | its
indexed
of
~.
[M,Mo])
factorization by M f
> ~ M
+
296 Likewise,
Sn(M)
: Nn( ~ M ~ , ~ ) ( M )
to a b a s e e l e m e n t indexed
by
M
M
Mf
> Mn
subsums check
s
,M)(M)
natural
in
t i o n of
Z |
.)
Z |
....> e~ M~.
Since
[JM, S M
Hence
is
].
~ |
[M,M n]
indexed Mf
and
by ~i
are
it is n o t d i f f i c u l t
to
of (However
N , ( ~ M~,~)
s
(M)
is n o t
is a p r o j e c t i v e
and the p - t h h o m o l o g y By
Ap(
assigns
factorization
[M,M f]
o < i < n,
[J-, 9 Mu]
M ,M) | t
its
is a c o n t r a c t i o n
> Z |
M
> 9 M ~
o
where
(M)
of the s u m m a n d
component
"'" ~ o ,
> N n + l ( ~ My, ~) (M)
> Mn
... M
in the
9 M
that
N,(~ M
>
~"
of
f : M
n
> Mf
-
(15) the
resolu-
of the c o m p l e x
latter
consists
of a s u m
tMn >
in d i m e n s i o n homology
except
a certain present
n.
s.s.
- -
Mn
hn
-"'" M ~ by
t% > %-1
M
~ > 9 M
~
~> %
~ ~I~i >
complex
I originally
M. A n d r ~
is r e d u n d a n t
n
has t r i v i a l assumed
the Kan c o n d i t i o n
a contraction
h,
which
then pointed
because
is a s u b s u m
maps
indexed
"'" M O
as follows.
chain
9 M i ~i
the s u m m a n d
of
tM~i
9 M ~
.
The morphism of the com-
by > 9 M~
out
the a b o v e
is d e f i n e d
of an a s c e n d i n g
that
(which is
t h e c a n o n i c a l morphism.
> 9 t/qn+ 1 =
zero,
I know).
condition
Mui
: 9 t/qn
ponent %
has
t h a t this
in d i m e n s i o n
t h a t the o b j e c t >
Denote
To p r o v e
in all e x a m p l e s
subcomplex
o
set s a t i s f i e s
to me that this
Recall
M
identically
on the
-
2 9 7
-
m
component M~i
of
> %
to c h e c k A
tM
(e M p u
>
that
,t)
>
n-i
h
= 0
9 tMn+ 1
i n d e x e d by
"'"
M
> 9 M ~ v "
o
is a c o n t r a c t i o n . for
This
It is r o u t i n e
shows
that
p > o. Q.E.D.
(22)
Corollary.
consisting
L,E
Ej,
be the f u l l s u b - c a t e g o r y
: [M',A] ~
= A' ,
j,
representable sequence
M'
of sums of o b j e c t s
Kan e x t e n s i o n functors
Let
in
M.
[~,A]
exist and that
functors
(7) c o l l a p s e s
for
n > o.
Assume a n d its
LnEj, Then by
and one o b t a i n s
of
t h a t the left derived
vanishes
on
(21) the s p e c t r a l
an i s o m o r p h i s m
A, (-,t) --~ > A,' (-,A"o (-,t)) where
t : M
) A
is a f u n c t o r
: M'
> A
is its K a n e x t e n s i o n
A"(-,t) O
~
: ~
) A
with a certain (cf.
(33)
realized the
(21) a n d on
M'. 19
~
The v a l u e A'
as in
of
(22)
lies
in the f a c t t h a t
can be i d e n t i f i e d w i t h cotriple
(34)).
C
(the m o d e l
induced
associated cotriple
In this w a y e v e r y A n d r ~ h o m o l o g y
as a c o t r i p l e
latter carries
in
the h o m o l o g y
homology
and all information
o v e r to the f o r m e r
c a n be about
and vice versa.
19 If A is AB5), t h e n the t h e o r e m is a l s o t r u e if M' an a r b i t r a r y full s u b - c a t e g o r y of ~, s u c h t h a t fur every M' 6 M' the a s s o c i a t e d c o m m a c a t e g o r y (M,M') directed. T~is f o l l o w s e a s i l y f r o m (5b) a n d (7).
is is
-
2 9 8
-
It has b e c o m e a p p a r e n t in s e v e r a l places s m a l l n e s s of
M
be removed.
that the
is an u n p l e a s a n t r e s t r i c t i o n w h i c h s h o u l d
A n d r 4 did this by r e q u i r i n g t h a t e v e r y
has a n e i g h b o r h o o d in
M.
C 6
A n o t h e r w a y of e x p r e s s i n g the
same c o n d i t i o n is to a s s u m e t h a t e v e r y f u n c t o r Z |
[J-,C]
: M~
) Ab.Gr.
admits a p r o j e c t i v e
resolution.
It is then c l e a r f r o m the a b o v e t h a t t h e r e is an e x a c t c o n n e c t e d s e q u e n c e of f u n c t o r s properties n > o.
A
Since
o
(,-) M
= E
J
A,(,-)
and
A
n
: [M,A]
(-,A |
[-,M]) = O
is not small, not every f u n c t o r
n e e d be a q u o t i e n t of a sum of r e p r e s e n t a b l e
L,Ej = A,.
ples this is h o w e v e r the case, e.g.
if
G-projectives
C.
of a c o t r i p l e
G
in
M
[~,~] w i t h the for t : M
functors
c a n n o t a u t o m a t i c a l l y c o n c l u d e that
(23)
,)
~
and one
In m a n y e x a m -
c o n s i s t s of the
So far w e h a v e o n l y d e a l t w i t h A n d r 4 h o m o l o g y w i t h
r e s p e c t to the i n c l u s i o n
J : M
homology associated with a cotriple
> C G
and not w i t h the in
C
d e f i n i t i o n of a c o t r i p l e we refer to B a r r - B e c k
(for the [2] intro.).
One r e a s o n for this is t h a t the c o r r e s p o n d i n g m o d e l c a t e g o r y is not small.
A n o t h e r is that the p r e s e n c e of a c o t r i p l e
is a m o r e s p e c i a l s i t u a t i o n in w h i c h t h e o r e m s o f t e n h o l d under weaker conditions
and proofs are easier.
The a d d i t i o n a l
i n f o r m a t i o n is due to the simple b e h a v i o u r of the Kan e x t e n sion on f u n c t o r s of the f o r m
M-I~
M
the r e s t r i c t i o n of the c o t r i p l e on
M.
our a p p r o a c h w o r k s
~
~,
where
G
We now o u t l i n e h o w
for c o t r i p l e h o m o l o g y .
is
-
(24)
Let
G
be a c o t r i p l e
any f u l l s u b - c a t e g o r y G-projectives
(X)
: GX
of
~,
and include
t h a t an o b j e c t
2 9 9
X 6 C
> X
every
admits
With every
efficient involves
>
t.
the v a l u e s
H,(-,t) G
called
t
M
are
C 6 C.
where
(Recall
if
c : G
: ~
) id
GC
C
are
>
Barr-Beck
functors cotriple
Their construction
of
is a l s o w e l l
where
t
derived
A , also
functor
of w h i c h
The objects
functor
the c o t r i p l e
: ~
and denote by
GC,
a section,
called
H,(-,t) G
C
is c a l l e d G - p r o j e c t i v e
of the c o t r i p l e .
[2] a s s o c i a t e
in
the o b j e c t s
is the c o - u n i t free.)
-
homology of
H,(-,t) G
on the f r e e o b j e c t s
defined when
t
with co-
of
only
C.
Thus
is o n l y d e f i n e d
on
M.
(25)
Theorem.
The Kan e x t e n s i o n
exists.
It a s s i g n s
cotriple
derived
Ej(t.G) AB4)
= t-G
functor
H
is valid.
t : M
(-,t)
o
G
: [M,~]
~
: C --
(Note t h a t
A
the
> A. --
~
)
[~,~]
zeroth
In p a r t i c u l a r
n e e d n o t be AB3)
or
for this.)
Proof. extension,
According
we have
restriction We
to a f u n c t o r
Ej
map
to s h o w t h a t
[Ho(-,t)G,S]
limit ourselves
and leave
to g i v i n g
it to the r e a d e r
e a c h other.
A natural
rise to a d i a g r a m
to the d e f i n i t i o n for e v e r y
~
of a K a n
S : C
[t,S.J]
A
the
is a b i j e c t i o n .
a m a p in the o p p o s i t e
to c h e c k
>
direction
t h a t t h e y are i n v e r s e
transformation
~ : t
>
S.J
to
gives
-
tE (GC)
tG2C
s~ (GC} i
.
-
i
for every
The top row is by construction
a coequalizer. SC
functors. t : M tG
----> t
: [~'~]
>
> A
is an
[~,A]
in [2] Ch. I
imply that
is an exact connected sequence of
(objectwise split)
'-)G ~
we obtain
algebra the following
>
[~,~]
is valid.
exist and
Moreover ~-L,Ej(-)
~ H,(
'-)G
denotes the class of short exact sequences
which are objectwise
w i t h o u t proof that for of representable
epimorphism,
The left derived functors of the Kan
Ej : [M,~]
holds, where [M,A]
established
H (-,t.G) = 0 for every functor n and since the canonical natural transformation
L.Ej(-) ~ H,(
in
transformation
Since
Theorem.
extension
In
> S.
by standard h o m o l o g i c a l (26)
Thus there is a unique m o r p h i s m
a natural
The properties '-)G
of
w h i c h makes the d i a g r a m commutative.
this way one obtains Ho (-,t) G
SC
-
C 6 C.
~
s~ (c)
SGC
i
S G E (C)
H,(
Ho (C,t) G
, (GC)
SG2C
Ho(C,t) G
-
>> tGC
, (G2C)
Ho(C,t) G
3 0 0
split exact.
n > o , Hn(
functors.
'-)G
We remark
vanishes
on sums
-
(27)
C o r 0 1 1 a r ~.
in
C
jectives
-
The cotriple
the G-projectives. 20 G'
3 0 1
homology
In particular,
only on
two cotriples
give rise to the same homology coincide.
depends
G
and
if their pro-
One can show that the converse
is also
true.
(28)
It is obvious
Barr-Beck
[2] Ch.
acyclicity
III for
criterion
resolution
of
tG,
in h o m o l o g i c a l tG,
on
M
: C
is an Ej-acyclic
(29)
It also
(-,t)
(for
follows
[2] by means
resolution G,
and Andr~ homology
see
As in
(-,t)
(4) the
of the s~ procedure
the restriction of
t
[2] Chap.
of
and because I).
(26) that cotriple A
property
the standard
is b e c a u s e
from
G
the models deduce
holds
the u n i v e r s a l
is actually
This
of
are the usual
of functors.
in
) A
algebra.
= tG,
*
sequence
H,(-,t) G
Ej(tG,.J)
H
H,(-,-) G
for e s t a b l i s h i n g
of an exact connected constructions
from the above that the axioms
coincide,
homology provided
*
for the latter
are
M.
this from the first half of
can only be obtained set theoretical
One might be tempted
to
(26), but apparently
it
from the second half.
difficulty.
For details
The reason
we refer to
is a
[23].
20 In many cases the cotriple homology depends only on the finitely generated G-projectives. An object X 6 C is called finitely generated if the h o m - f u n c t o r [X,-] : C ~ S preserves filtered unions of subobjects.--
-
F r o m this previously
it is obvious
established
homology.
fications
which
result
(3O)
The a s s u m p t i o n when
following: t : M
M
In adjoint.
(8)-(11)
It suffices
sub-category
(31)
M
derived and
G : C
directed
instead
M
is as in functors
[M,A]
~
(24), and
: [M_~ is defined [M~
Tor,(Z
|
~
M
direct F
that
limits. need not have a right
F
is right exact. 11 we obtain
a spectral
for a
sequence
> Ap+q(-,t)
~,
in
and the functor
and
A
denote [~,A]
the left >
[M,A]
product , ~ , sxt ~
(14) but may not exist However
[J-,C],
of these properties.
respectively.
• [M,A] as in
some useful modi-
is seldom p r e s e n t
of the Kan extensions [~,A]
carry over
can be replaced by the
the functor
of
The tensor
hold.
(5c), which
Hp(-,Aq(-,t))G
where
s s
in
(26) , (7), and footnote
From small
We list below
is not small,
preserVe
that the properties
from direct proofs
The cotriple
> A
-
for the Andr~ homology
to the cotriple
examples
3 0 2
(19) and
s @ t for every
(32)
t) ~ H,(C,t) G ~ - T o r , ( Z
| [J-,C], t)
-
The first i s o m o r p h i s m
shows
by either a p r o j e c t i v e
3 0 3
-
that
H,(C,t) G
resolution
of
Z |
can be c o m p u t e d [J-,C]
or an
E -acyclic r e s o l u t i o n of t. The former is a g e n e r a l i z a t i o n J of the main result in [2] 5.1, b e c a u s e a G - r e s o l u t i o n is a projective
resolution
the acyclic model
(33) there
of
Z |
argument
[J-,C];
in
[4]
(cf. also
W i t h every small s u b - c a t e g o r y is a s s o c i a t e d
cotriple G : ~
(cf. > ~
a cotriple
[2] Ch. X). assigns
G,
w h i c h belongs
The co-unit
M.
r e s t r i c t e d on a s u m m a n d (26) and A,(-,-)
(25)).
of a category
c a l l e d the model
to an object f : df
to
M
[20]
induced
C 6 ~
> C,
df
is
the sum
the domain
~(C)
: 9 df
f : df
9 df, f df of ) C
) C.
The theorems
(22) enable us to compare the Andr~ h o m o l o g y of the inclusion
the model
M ~
induced cotriple
of objects
M
6 M
G
~ in
w i t h the h o m o l o g y C.
Since every
is G-projective,
Theorem.
satisfies
the c o n d i t i o n s
preserving
functor
A
Assume
we obtain the
t : M
(-,t)
that the i n c l u s i o n
in
(21).
~
A
is an isomorphism,
A
> C sum-
the c a n o n i c a l map
(-,t)) o
provided
M
Then for every
-- > H , ( - , A
*
is an AB4)
of
sum
following:
(34 )
C
Recall that its functor part
indexed by m o r p h i s m s
9 M
the latter g e n e r a l i z e s
G
category.
-
If Droduces,
t
likewise
A*(-,t)
A
-
is c o n t r a v a r i a n t
we obtain
provided
304
H * (-, A O (-, t) )
that Andr4
be r e a l i z e d under rather weak conditions even of a m o d e l
in B a r r - B e c k
formulas),
(homology sequence
9.1,
9.2
(Mayer Vietoris)
Moreover,
(co)homology as
[2] 7.1,
7.2
(coproduct
over to Andr4
and
[2]
(co)homology.
tends
to classify
extensions
(cf.
illustrate
the use of this r e a l i z a t i o n with some examples.
(35)
carries
of
In this way,
of a map)
cohomology
can
(co)homology
also apply to Andr4
the fact that cotriple [3])
G
induced cotriple.
the c o n s i d e r a t i o n s [2] 8.1
~n~o
category.
The t h e o r e m asserts
a cotriple,
sums
for c o h o m o l o g y
<-
is an AB4)
and takes
cohomology.
We
Examples
a)
Let
C
be a c a t e g o r y
rank
(C) = u
that
if
[14]).
M
free algebras
~
of
(34) and
sense
> ~,
where
on less than
of u n i v e r -
(cf. L a w v e r e
(27) one can show that
of the free c o t r i p l e (co)homology
Recall
[15].
is a c a t e g o r y
in the c l a s s i c a l
the A n d r e
clusion
, then
By means
(co)homology with
in the sense of L i n t o n
= M0
sal algebras
of algebras w i t h
in
~
coincides
associated with
the objects ~
of
generators.
~
inare
-
b)
Let
C = Ab.Gr.
of finitely
305
-
and let
generated
M
abelian
be the s u b - c a t e g o r y
groups.
one can show that the first Andr~ AI(c,[-,Y]) extensions
is i s o m o r p h i c of
C 6 Ab.Gr.
in the sense of Harrison. is a category
of
Using
(34),
cohomology
group
to the group of pure w i t h kernel
Y 6 Ab.Gr.
The same holds
A-modules,
where
A
if
C
is a ring
w i t h unit.
c)
Let
C = A-algebras
category Let
C
of finitely
and let
generated
be a A-algebra
and
Y
M
be the sub-
tensor algebras. be a A-bimodule.
1 Then E
A
(C,[-,Y])
> C
classifies
w i t h kernel
A-module e x t e n s i o n Harrison
The cases of s i m i l a r examples,
Y
is pure
singular e x t e n s i o n s
such that the u n d e r l y i n g in the sense of
(cf. b)). 21
(b) and which
of pure group e x t e n s i o n s
(c) set the tone for a long list indicate
that Harrison's
can be c o n s i d e r a b l y
2 1 1 am indebted to M. Barr for c o r r e c t i n g made in this example.
theory
generalized.
an error
I had
-
3 0 6
-
BIBLIOGRAPHY
[1]
Andrd, M., M~thode simpliciale en alg~bre homologique
et alg~bre commutative, Mathematics,
[2]
#32), Springer, 1967.
Barr, M. and J. Beck, structions",
(Lecture notes in
"Homology and standard con-
(to appear in Lecture Notes in
Mathematics).
[3]
Beck, J., "Triples, algebras and cohomology", Disserta-
tion, Columbia, [4]
(1964-67).
Barr, M. and Beck, J., "Acyclic models and triples", in;
Conference on Categorical Algebra, pp. 336-343, Springer,
[5]
(1966).
Buchsbaum, D., "Homology and universal functors", in; (Lecture Notes in Mathematics,
[6]
Dold, A., "Zur Homotopietheorie der Kettenkomplexe",
Math. Annalen, 140; 278-298, [7]
#61), Springer, 1968.
(1960).
Dold, A., S. MacLane, U. Oberst, "Projective classes and acyclio models", in; A. Dold, Heidelberg and Eckmann
Seminar,
(eds.), Reports of the Midwest Category (Lecture Notes in Mathematics,
78-91, Springer, 1967.
#47);
-
[8]
3 0 7
-
Dold, A. and D. Puppe, "Homologie nicht-additiver Funktoren", Ann. Inst. Fourier,
[9]
Dubuc, E., "Adjoint triangles", Mathematics,
[i0]
#61), Springer,
ill; 291- (1961).
in; (Lecture notes in 1968.
Fisher, J., "The tensor product of functors, satellites and derived functors", J. of Algebra,
[ii]
to appear.
Freyd, P., Abelian Categories, Harper and Row: New York, 1964.
[12]
Grothendieck, A., "Sur quelques points d'alg~bre homologique",
[13]
Kan, D., "Adjoint Functors" 8_~7; 295-329,
[14]
TDhoku, Math.
J., ~; 119-221,
Tran8
Amer. Math
(1957). Soc ,
(1958).
Lawvere, F., "Functorial semantics and algebraic theories", Proc. Nat. Acad. Sci., 50; 869-872,
[15]
Linton, F., "Some aspects of equational categories" Conference on Categorical Algebra,
Springer, [16]
(1966).
MacLane, S., Homology, Springer,
1963.
pp. 84-94,
(1963).
8
in;
-
[17]
308-
Oberst, U., "Basiserweiterung in der Homologie kleiner Kategorien", Math. Zeitschrift, i00; 36-58,
[18]
(1967).
Oberst, U., "Homology of categories and exactness of direct limits", to appear.
[19]
Tierney, M., and W. Vogel,
"Simplicial resolutions and
derived functors", Mim. Notes, E.T.H., [20]
(1968).
Ulmer, F., "Acyclic models and Kan extensions",
this
volume. [21]
9 "Representable functors with values in arbitrary categories", J. of Algebra, ~; 96-129, (1968).
[22]
"Properties of Kan extensions" E.T.H.,
[23]
Mim. Notes,
(1966).
. "On Andr~ and cotriple
(co)homology and their
relationship to classical homological algebra", (to appear in Lecture Notes in Mathematics). [24]
Watts, C. E., "A homology theory for small categories", in; Conference on Categorical Algebra, pp. 331-335, Springer,
[25]
(1966)9
Yoneda, N., "On Ext and exact sequences", J. Fac. Sci. Tokyo, Sec. I, 8; 507-576,
(1961).
Offsetdruck: Julius Beltz, Weinheim/Bergstr.
