Local Cohomology

Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, ZUrich

41 Robin Hartshorne Harvard University, Cambridge, Mass.

Local Cohomology A seminar given by A. Grothendieck Harvard University Fall, 1961

1967

Springer-Verlag. Berlin. Heidelberg. New York

All rights, especially that of translation into foreign languages, reserved. It is also forbidden to reproduce this book, either whole or in part, by photomechanical means (photostat, microfilm and/or microcard)or by other procedure without written permission from Springer Verlag. 9 by Springer-Verlag Berlin Heidelberg 1967 Library of Congress Catalog Card Number 67-29865. Printed in Germany. Title No.7361.

CONTENTS Page ii

Lntroduction w I.

Definition and Elementary Properties of the Local Cohomology Groups

1

w z.

Applications of Local Cohomology to P r e s c h e m e s

16

w 3.

Relation to Depth

36

w

Functors

48

w 5.

S o m e Applications

69

w 6.

Local Duality

81

Bibliography

on A - m o d u l e s

105

Introduction W h a t follows

is a set of lecture notes

seminar g i v e n b y A. G r o t h e n d i e c k in the fall of 1961. of local

preschemes.

and my d u a l i t y Furthermore, "Elements," worthwhile short, best

cohomology

This m a t e r i a l

and g e n e r a l i z e d

at H a r v a r d U n i v e r s i t y

The s u b j e c t m a t t e r

(or relative)

at H a r v a r d

it may appear chapter

III

introduction

[5].

treatment to it.

same as the first edition,

groups

I have

of a s u b j e c t

[16]

[17].

sections

available

of his

thought

again,

since

it a

is o f t e n the

The text is e s s e n t i a l l y except

on

in e x p a n d e d

of 1962

in 1963/64

However,

theory

of sheaves

seminar

in the later

to make these notes

elementary

is his

has since a p p e a r e d

form in his Paris seminar

for a

for minor

the

corrections

and an e x p a n d e d b i b l i o g r a p h y . The study of local c o h o m o l o g y origin paper

in the observation, FAC

varieties

[i0],

already

or c o m p l e t e

local rings.

and then p r o v e

implicit

that m a n y s t a t e m e n t s

can be r e f o r m u l a t e d

statements

groups has its in Serre's

about p r o j e c t i v e

in terms of g r a d e d rings,

This allows about

one to c o n j e c t u r e

local rings,

which

then

may be of use in o b t a i n i n g b e t t e r the finiteness on p r o j e c t i v e

theorems

of Serre

varieties b e c o m e

global

results.

for c o h e r e n t

statements

Thus

sheaves

that c e r t a i n

\

local c o h o m o l o g y modules duality theorem

are

"cofinite."

for p r o j e c t i v e

varieties becomes

t h e o r e m for local c o h o m o l o g y modules. was

developed

further b y G r o t h e n d i e c k

[16], w h e r e he studies relating

~I

X,

1 we define

of an abelian

w i t h respect

right derived sections

of

sheaf

1962 seminar

to

1

theorems,

and Pic of

2.8, w h i c h

as a direct

the local c o h o m o l o g y on a t o p o l o g i c a l closed

subset

in

Fy(F),

Y,

space as the the

Y."

2 we give the first c o n s e q u e n c e s

this theory w h e n applied

groups

in his

of the functor

"with support

In S e c t i o n

is Theorem

F

to a locally

functors F

This a p p r o a c h

sections.

In S e c t i o n groups

the

a duality

local and g l o b a l L e f s c h e t z

and Pic of a v a r i e t y

its h y p e r p l a n e

Similarly

to p r e s c h e m e s :

interprets

of

The m a i n result

the local c o h o m o l o g y

limit of Ext's.

In S e c t i o n 3 we give a s e l f - c o n t a i n e d

exposition

of the n o t i o n of depth and relate

(or h o m o l o g i c a l

codimension)

it to the local c o h o m o l o g y groups

(Theorem

3.8). S e c t i o n 4 contains functors

d e f i n e d on the c a t e g o r y

Noetherian functors

ring.

In particular,

and dualizing In S e c t i o n

cations

before

"general

nonsense"

on

of m o d u l e s

over a

we discuss

dualizing

modules.

5 we give some m i s c e l l a n e o u s

and some t e c h n i c a l

section.

some

(In the lectures,

results

useful

appli-

in the next

m o s t of this m a t e r i a l

came

that of S e c t i o n 4, and it is perhaps b e t t e r

read it first,

since it is c l o s e l y r e l a t e d

to

to S e c t i o n

3). S e c t i o n 6 contains are central

results

the duality

theorems w h i c h

of the local c o h o m o l o g y

theory.

R. H a r t s h o r n e Cambridge, July,

1967

Mass.

w 1.

Definition and Elementary Properties of the Local Cohomology Groups

For preliminaries

[4]

Godement

or Grothendieck

be valid for arbitrary abelian

groups Let

subspace

X

Z

[6].

topological

The results spaces

and let

F

X

space

X

we mean

be the subgroup

Fz(X,F)

sections of

whose

F

i m m e d i a t e l y that above.

Moreover,

support

Fz(X,F)

X

V C X

This is possible

V.

Z

of

be a locally closed

X.

since of

(We recall that a

B y an abelian sheaf on a topologi-

simply a sheaf of abelian

Choose an open subset

Let

sheaves

is locally closed if it is the

subset.

T h e category of all abelian sheaves on

i s closed i n

to

of t h i s s e c t i o n w i l l

be an abelian sheaf on

of a ~pological

space

we refer

and arbitrary

be a topological space, let

inter section of an open and a closed cal

algebra

over them.

o f X,

subspace

of h o m o l o g i c a l

groups

on

X.

will be denoted by

such that Z

F(V)

Z C V,

~(X). )

and Z

is locally closed in

X.

consisting of all those

is contained

is independent

in

Z.

One checks

of the subset

V

chosen

one sees also that the functor

z (x, F) from

~ (X)

to

(Ab)

of F with support i__nn Z.

is l e f t - e x a c t .

Note that if

We c a l l Z

r

Z

is closed,

(X,F) t h e s e c t i o n s it i s p r e c i s e l y

2

this; if

Z Let

of

X,

Fz(X,F) = F(Z).

is open in

X,

X,Z,F,V

be as above.

U

is any open subset

the natural restriction h o m o m o r p h i s m F(V) -* F ( V ~

induces

T h e n if

U)

a homomorphism

r z(X' F) -- rz~u(U, F[ u) Thus we may

consider the presheaf U~-r

One verifies

FZ~u(U,F

immediately

we will denote it by

~{X)

Definition. subspace,

derived

functors

Hp (X,F)

and

Let and of

r

coefficients

F

Remark: of

a

z

H PZ ( F ) , groups

to

X

F

tl~e c o h o m o l o g y

in

A s with

r Z,

is a sheaf,

and

one finds that the functor

Fz(F)

from

closed

.

that in fact this presheaf

r z(F).

F ~

is left-exact

Iu)

e., (x).

be a topological space, an abelian sheaf on

and

r Z,

X.

Z a locally T h e n the right

respectively, are denoted by

p = 0, I, ..., respectively, a n d are called (resp. , c o h o m o l o g y

a n d supports in

sheaves)

of X

with

Z.

Using the notion of a sheaf extended by zero outside

locally closed

subset

[4, II Z. 9],

we can give a different

interpretation of i n t e g e r s

of

on

r Z(F)

Z,

zero outside of

and

r z ( F ).

and let ~"Z,X

Z.

Let

Z Z be the constant

be this same sheaf extended by

Then

F z ( X , F) = Horn ( Z Z , X ,

F)

and

Fz(F) = Horn

where

(mZ, X, F)

Horn d e n o t e s t h e s h e a f of g e r m s

of h o m o m o r p h i s m s .

T h e r e s t of t h i s s e c t i o n will be d e v o t e d to v a r i o u s of t h e s e c o h o m o l o g y g r o u p s a n d s h e a v e s with s u p p o r t s

properties

in a locally

closed subspace. proposition

a)

I . ].

Let

For any

r

Z

be l o c a l l y c l o s e d in

F s ~(X)

Z

X.

Then

,

(X,F) = H ~ I-

Fz(F) = Hz(F) b) of a b e l i a n

sheaves

If on

0 --" F ' --" F --" F " --" 0 X,

o - H z c x , r'~ -

and

is an exact sequence

then there are long exact sequences

.1,x

sheaf

~ z ~x,

r~

--

F,-

H z ~ x , r,,~ ~

4

0 -*H

where

the connecting c)

HP(x,F) L

If

for

Proof. H PZ

homomorphisms

F e ~(X)

: 0

z (F') --" H oZ (;) --" H oZ (;,,) h H Z (;,) _ Hi(;) ~) b e h a v e f u n c t o r i a l l y .

is an injective

p > O,

and

object of the category,

HP(F)

: 0

for

p > O.

T h e s e properties follow f r o m the fact that

are derived functors of

I" Z

and

r Z.

then

In fact,

and

H PZ

a,b,c

characterize the derived functors. Propo6ition

1.Z.

is the sheaf associated

H PZ (F)

u ~

Proof. H P'

With

(F)

Z , X, F

p = O,

This follows formally

= F z(F). the for

from the definitions. to the presheaf sheaves

above.

F

is injective, so is

H P Zc~j(U,FIU) = 0 p > 0.

functor s of

for p > 0.

F Iu

Then,

Moreover,

= the sheaf associated to the presheaf If

Set

is an exact functor,

f o r m a connected sequence of functors.

H~

p > O,

HE~. 7- U (u ' FIu)

equal to the sheaf associated

HP'(F)

for each

to the presheaf

s i n c e t h e o p e r a l J o m of t a k i n g a s s o c i a t e d the

as above,

for any

(U ~

FZr_~u(U, F IU)),

U; hence, all

Therefore, also,

HP'(F) = 0

But these properties characterize the right derived

Fz(F), hence

HP'(F) = HPz(F ).

for

(Excision Formula)

Proposition I. 3. closed in

X,

Then for any

and let

Proof.

~

H p (V,F Iv)

Proposition for any

are isomorphic

1.4.

be locally

and such that

F

there is a spectral

H P ( x , F)

of t h e f u n c t o r

F

-=~,=

for any ~

F ~,~

If

r

C(X)

functors from

(Spectral Sequence)

HZn (X,F)

where by

X

This follows from the fact that

F z ( V , F[ V)

X,

be open in

Z

Z _C V _C X.

F ~ C(X),

H PZ ( X , F )

in

V

Let

Z

z

(X, F) and to

(Ab).

is locally closed

sequence.

E zpq = HP(x, Hq(F))

F

we mean the right derived functors

r (F).

This proposition

will follow from the theorem

and lemmas

below. Theorem

Let

A.

~.. ,

~',

~.]' be abelian categories

with enough injectives, and suppose given two left-exact covariant functor s F:

C

Suppose furthermore i. e . ,

whenever

I

that

F

takes injectives into G-acyclic object s,

is an injective object of

~.- ,

RPG(F(1)) = 0

for

p > 0,

where

for each object

RPG

X

in

are the right-derived ~

the right derived functors

Rn(G

,

there is a spectral

of

F,G,

o F)(X)

~==

and

EPq=

F o r the p r o o f of t h i s t h e o r e m

G

o

I _C F

F

we refer

flasque, then

Then

RPG(RqF(X))

A n y sheaf can be e m b e d d e d

with

G.

F.

the reader of

I

t o [6]. abelian

sheaves

in a flasque sheaf, e.g. ,

the sheaf of discontinuous sections of itself. But if and

of

sequence relating

kin ~(X), t h e ca__q~29orZ A n y ~ n j e c t i v e Sheaf~s flasq-~e

Lernrna I. 5.

Proof.

functors

I

is injective,

is a direct s u r n m a n d os

F,

h e n c e flasque itself.

Lemma

Proof. Z

1.6.

If

F

is flasque, so is

Replacing

X

by an open subset

as a closed subset, w e m a y

w e m u s t s h o w that if

U

Fz(X,F) is surjective, So l e t

assume

that

Z

is any open subset of

--

(; ~ I ~ Z r ~ u ( U , F I U ) , Z r-~ U.

V

which contains

is closed in X,

X.

Then

the m a p

FZr. j(U,FIU)

where we suppose that

whose support is in

Fz(F).

i.e.,

F (~

Consider

is a given s is a section of

the zero

sheaf. F

s e c t i o n of

over F

U,

over

on

x,/

the open set their

X - Z.

domains

agrees

with

U t-~ (X - Z) = U -

a'

of

F

is

c;,

and whose

over

is flasque,

there

res~ziction

to

and

This

the open set

U ~

~" ~

(~

Lemma

1.7.

Proof.

Embed

to

F

F

(;"

is

the map

If

U).

X - Z

a section

(X - Z )

under

(Z ~

on the intersection Hence,

U <J (X - Z ) ,

restriction

exists

~

F

Clearly

above.

is flasque,

in a n i n j e c t i v e

to

U

Now since

F

over

all of

whose

~"

has

Hence

then

is a section

restriction

is zero.

of

~'

whose

there

of

X,

support

I'z(F)

I,

Z,

is flasque.

Hq(x, F) = 0

sheaf

in

for

and let

q > 0.

C

be

the cokerneh 0--,F--,I-~

Then C

I

is flasque

is also

flasque,

by lemma

HI(x,F) = 0,

HP(x,F) by induction

on

Proof which may

p,

--

Therefore,

1" (I) "-"- .I"(C)

and for

~

since

of Proposition

be written

1.5.

and we have an exact

0 - ~ 1" ( F )

Hence,

C-~0. Chapter

II, 3. 1 . Z ] ,

sequence

--~

0

p > I,

H p-l(x, C C

by [4,

) = 0

is also

1.4.

as a composite

flasque.

We have the functor functor

rz:

c(x) -- (AS),

r

Z

=r

9

F

Z

where

I~ z :

C(X) --" C(X)

In o r d e r t o a p p l y t h e t h e o r e m existence

of the s p e c t r a l

takes injective

quoted above,

sequence,

sheaves into

any injective is flasque,

r : C(X) --"(Ab).

and

we need only show that

F-acyclic

Fz(flasque

and thus deduce the

sheaves.

F

Z

But by the lemmas,

) = flasque,

and any flasque is

r -acyclic.

Lemma closed in on

X,

Z,

I . 8. and let

Let

Z

be locally closed in

Z" = Z - Z'

X,

let

Then for any abelian

Z'

be

sheaf

F

there is an exact sequence 0 -~ I"z,IF ) - - F z ( F ) -- FZ,,(F )

Moreover,

if

Proof.

F

in

Z',

X,

V

we can write a

Clearly we may assume

Fz,(F)

Then

is flasque,

that

0

on the r i g h t a l s o .

Z

is closed in

is the s e t of t h o s e g l o b a l s e c t i o n s

which clearly

contained in

is open,

Z" = Z r

and

V,

Then

I ' Z , , ( F ) = t h e s e t of e l e m e n t s

is in

Z"

rz(F). i.e.,

of

Letting

Z"

F

X.

with support

V-

is closed in

I Z' V.

I~(V, F 1 V) w h o s e s u p p o r t

N o w it is clear that the natural restriction m a p

in

r : F(X,F)--F(V, FIV) induces a m a p ~Z((;) = 0 in

Z'.

4Z : Fz(X'F)

is to say

Hence, Now

if

F

is fla sque,

~

3 (~' s r ( X , F )

Z', a'

m u s t have

Proposition I. 9. F

is zero on

o"

" Furtherfore,

Z", i.e.,

a

to say

ha s support

our sequence is exact.

Z,,(X,F),

let

-" F Z " ( X ' F )

be any abelian

o -

support

Let

i s surj ective.

Hence,

restricting to

(;.

in

Z'.

Z, Z', Z"

sheaf on

rz,(X, ~ -

X.

Hence,

if

But since ~Z

is surjective.

be as in the l e m m a ,

Then there

are

exact

and

sequences,

rZ(X,F) -- r z,,r F) -- HI,r

F) -- HIcx

and 0 -~ F z , ( F ) -~ F z(F)

Corollary i. 9. abelian sheaf on

X,

If there

Z

--" r Z,,(F) -* Hz,(F)--- H I ( F ) --~

is closed are

exact

in

X,

and

F

is any

sequences

0 --- F z ( X ,F) --" F(X,F) -~ r(x - Z,F) ---H Z1 (X, F) --" --

H I(X,F) - * H I(X - Z,F) ---

and 0 -- F z(F) --~F-~j.(F I X -

Z) - - H Z1 (F) -~ 0

for

p > 0

, F)-~

10

where direct

j : X - Z ~

image

is the natural

X

injection,

and

j$

is the

functor.

Proofs.

Take an injective

resolution

~

= (Ii)

of

F:

0 -~ F --- I0 -~ I1 - - . . .

Then the

I.

are all flasque

by lemma

1.5,

and so by lemma

1.8

1

we have an exact sequence

of complexes

o--rz,( ~ )-- rz(~) )-- rz,,(~))--o and similarly

for the sheaf functors

give rise to the exact sequences For the corollary, and write

Z

for

Z'.

r

F

z"

written

z'

rx(F

9 The se complexes

above.

we take the particular Note that

rZf

case where

) = F,

FX-Z

by the definition of the direct

image

HP(F) = 0

HPx_z(F)= R ~ ( F Ix- Z).

for

p > 0, and

Proposition

1.10.

the following conditions

(i)

Let

F

functor,

and

r X is exact,

be an abelian

sheaf on

( F ) = j . ( F [ X- Z) so

X.

Then

are equivalent:

F is flasque.

(ii) F o r a l l l o c a l l y c l o s e d s u b s p a c e s all

Z = X,

Y

i > O,

i.

H~(X,x F) = 0

and

i

Hy(F) =

0

of

X,

and for

II

(iii)

For all closed subspaces

Y

of

1

X,

Hy(X,F)

= O.

[

Proof. closed.

Let

subset.

Then

(i) => V C X

(ii). Suppose

F

flasque, a n d

be an open set containing

F IV

is flasq~e,

Y

Y _C X

locally

as a closed

so by the excision formula

(proposition

I

we may assume corollary for

that

V = X,

of P r o p o s i t i o n

i > 0,

since

F

i.e.,

Y is closed in

1.9, and noting that is flasque,

X.

Now using the

Hi(X, F) = Hi(x - Y, F) = 0 i Hy(X, F) = 0

we find that

for

and

i>Z,

I'(X,F) --" r ( x - Y , F ) -- H I ( X , F ) --- 0

B y t h e d e f i n i t i o n of f l a s q u e n e s S , U _C X,

F IU

Passing

to associated

Hy(F) = 0

also

(X, F)

= O.

i ,FlU ) = 0 Hyr_~u(U

is also flasque, so

sheaves

H y1

(Proposition

For any open f o r i > O.

1. Z), w e f i n d t h a t

f o r i > O.

(ii) => (iii) i s t r i v i a l . (iii) => (i) f o l l o w s f r o m t h e d e f i n i t i o n of f l a s q u e n e s s exact sequence,

for any closed F(X,F)

Proposition V

= X

-

integer.

Y,

Y,

-~ F ( X - Y,F)

1.11.

Let

l e t F be a n a b e l i a n

and the

1 --Hy(X,F)

Y be a c l o s e d s u b s e t of s h e a f on

X,

and let

Then the following conditions are equivalent:

n

X,

let

be an

1.3),

[2

(i)

For all

(ii)

i H y ( F ) = 0.

i<_ n ,

For all open subsets

HI(U r-~ V, F )

is injective for

Proof.

(i) => (ii).

X,

the map

~ . : H i ( u , F ) -~ I

and an isomorphism

i = 0,

for all

i < n.

S i n c e t h e c o n d i t i o n (i) i s of a l o c a l n a t u r e ,

it will be sufficient to prove must

of

U

(ii)

in the case where

U = X.

So w e

show that the map oz. : H i ( x , F ) -~ H i ( x - Y, F ) I

is injective for exact sequence to show that is a spectral

i = 0,

and an isomorphism

of the Corollary

to Proposition

H~y(X, F ) = 0 f o r

i < n.

Since by hypothesis

1.9,

Using the

it will be sufficient

But by Proposition

1. 4 t h e r e

the

H yq (F) = 0

H yi ( X , F ) = 0

U,

and for H yi

associated

sheaves,

remaining

case is where

Hq(F))

Pq = H P ( x , < = E~.

for

(ii) => (i). O u r h y p o t h e s i s for any open

i < n.

sequence

i X Hy(,F)

the abutment

for

i = 0

(F) = i=n>

0

for

q<

it follows that

n,

i < n. clearly insures and for 0.

i < n. i = 0

H iy ~ u ( U , F I U ) = 0

that

Hence, and

Then for each

passing to

i < n. U,

The only we have

13

H n- 1 (U, F) ~ H n- 1 (U :'~ V, F) --" H yn ~ u ( U ,

This exact sequence as we pass from sequence

commutes

one

U

of associated

presheaf

U ~

with the restriction

to another,

sheaves.

Hn(u, F)

But the sheaf associated

is zero,

since

let

Y

abelian sheaf on

X.

theorem

also for

i Hy(X, F),

F

X

(F)

--~

So w e h a v e

0

be a Zariski space of d i m e n s i o n

T h e n for all

S e e [6,w

analogous

and

Let

be a locally closed subset of

Proof.

n > 0'

to the

~ y (F) = O.

Proposition I. 12. n,

homomorphisms

hence gives rise to an exact

H n- I(F)-~ R n" 1,(F[ V) --" H yn

which shows

F) --" Hn(u, F)

flasque

is proved

and let

F

[4, II, Th. 4.15.Z], Hi(x, F).

once one observes

so

Passing to associated sheaves gives

i Hy(F) =0.

where the

Hy(X,F)

= 0

(Proposition 1. I0)) and that t h e f u n c t o r

again a Zariski space of d i m n,

and

The same proof works

that

with direct limits and finite direct sums.

be any

i H y ( X , F) = 0,

i > n,

and for

X,

for I"

i Hyt..aU (U, F[ U) = 0 Hi(F) Y

= 0

for

i >

C ornn%ut

Y

A n y open subset

i > 0

UCXis for n.

i > n.

e s

14

Remark. cohomology

agrees

paracompact groups

For s u i t a b l y n i c e t o p o l o g i c a l with

singular

and locally contractible)*

defined above in terms

precise,

let

(respectively

Y

spaces

the constant

cohomology

one may interpret

of known relative

be c l o s e d i n

X

sheaf

and let G

in w h i c h s h e a f

on

the cohomology

cohomology.

G X).

(e. g. ,

To be

be an abelian

group

Then

H y ( X , G ) = Hi(x, X-Y;G)

where

the expression

of X relative to X - Y , f u n c t o r i a l isomorphism, E x a m p l e . Let subset, so that each

y s Y,

the pair

X

X

with coefficients in G. This is a canonical, c o m p a t i b l e w i t h the u s u a l e x a c t sequences. be a topological space, and

looks locally like

We c o n s i d e r in

such as above,

is homeomorphic the cohomology

Y,

Y

a closed

Y X ~Rd..That is, such that for V

there is a neighborhood

(Y r-~ V , V )

have support

singular cohomology

the right is

to

sheaf

and we can calculate

of

y in

(Y ~ V X

X,

such that

{0} , Y ~ V

X

Hy(X,~).

This sheaf will

it locally.

Over a

hRd).

V

it looks like

(One can calculate this by ordinary topological m e t h o d s via the r e m a r k above: pair

The pair (]Rd, S d-l),

(JR d, IR d- {0} ) is of the s a m e h o m o s

type as the

so w e have l

*see [14, e x p o s ~ X I I I p.3] and [15, e x p o s e XX, p. i] . C o m p a r e also [4,II5.10.i] w h i c h says that these also a g r e e w i t h C e c h cohomology.

15

i } (~l Z ) -~ H i (m d z)-9 ..-" Hi-l(IRd; Z ) - ~ H i-I (Sd-l" Z ) - " H{0 ,

We k n o w i {d-

Hi(]Rd;z)

1,

Z

= 0

for

i > 0,

H dy(X,

Z)

Y.

Y

X. T h e s p e c t r a l s e q u e n c e of

Call this sheaf

H i (X, Z ) ~ 0 Y

']ry, x,

only when

~ry,x

s e q u e n c e of

i s c o n c e n t r a t e d on

Z

t h e s h e a f of t w i s t e d i n t e g e r s o f Proposition 1.4 degenerates,

i = d,

a n d we f i n d

i Hy(X, m) m Hi-d(x, Ty,x) = Hi-d(Y,

since

9

if

i s l o c a l l y i s o m o r p h i c to

along

since

Hi(s d-l; Z) = 0

9

otherwise.)

Hence the sheaf

in

and

,

Y.

~"y,x )

S u b s t i t u t i n g in t h e e x a c t

P r o p o s i t i o n 1 . 9 , we h a v e

... -* Hi{x,

~..) -* Hi{x

- y,

7..) -* Hi+l-d(y,Ty,x

) -~ Hi+I(x,

Z)

-~ ....

.

w Z.

Application of L o c a l C o h o m o l o g y to P r e s c h e m e s.

In this section we will apply the notions of local cohomology, as developed X

i n S e c t i o n 1, t o t h e c a s e w h e r e

is a prescheme.

We w i l l f i r s t

showing how to compute Koszul complexes. preschernes, in terms

to theorems

on projective

U and

and let

V

V

are

into

be true if sheaf of

[10,

results

of

groups

and sheaves

are very closely

w 69] r e l a t i n g

cohomology

of sheaves

space to "Ext" groups.

~X'

and

These

by means

valid on quite general

the local cohomology

of Serre

Let

Proposition Z. I. sheaf

groups

Then we will give a theorem

which interprets

space

s t u d y t h e c a s e of a n a f f i n e s c h e m e ,

the local cohomology

of the "Ext" functors.

related

our topological

X

Y

open.

X

V

be a prescherne, U

-

Assume

that the natural

is locally Noetherian.

for all

By Propositionl.lwe

Proof.

.

.

--~ Hu(F)n .

) If

.

n "-" H y ( F ) .

.

n>

F

subset,

where

injection maps

immersions.

then the sheaves

~x-m~

with structure

be a locally closed

are quasi-compact

~X-mOdules,

sheave s of

=

X

of

U

(This will always

is a quasi-coherent

Hy(F)

are quasi-coherent

0.

have an exact sequence n+l --- H V (F) --~

of sheaves

17

on any p r e s c h e m e SinceAthe kernel and cokernel of a h o m o m o r p h i s m sheaves

are quasi-coherent,

coherent

sheaf by another

and since an extension is quasi-coheren~

n

s h o w that the sheaves

Hu(F )

n

and

Hv(F )

j : U --"X

is the natural

injection,

R n j , ~ F [ U).

Since

is a quasi-coherent

an immersion

is separated,

following theorem,

Theorem morphism X.

B.

sheaves

on

be a closed X,

A-module

is open in

U

X,

n

then the sheaf

Hu(F )

sheaf on

our proposition

direct

on

f : X -- Y

and is just

U,

will result

We

from

and since the

F

images

Rnf~(F)

s u b s e t of

H yi ( X , F ) .

be a quasi-coherent sheaf on of

F

are all quasi-

Y.

Z. 2.

and for any

be a q u a s i - c o m p a c t separated

and let

See [5, I . i . 6 . 3

Proposition Y

Let

of p r e s c h e m e s ,

Proof.

are quasi-coherent.

which we suppose known.

Then the higher

coherent

of one quasi-

it will be sufficient to

have already seen (Corollary I. 9) that if

F IU

of quasi-coherent

Let X.

i > O,

a n d III

X = Spec A

1.4.10 ]

be an affine s c h e m e , and let

T h e n for any quasi-coherent sheaf Hy(F)

F

is the sheaf associated to the

M o r e o v e r , there is an exact sequence

0 -~ H y0 (X, F) -~ H 0 (X, F) -~ H 0(X-Y, F) -- H I(X, F) -~ 0 * S i n c e the q u e s t i o n is local, it is s u f f i c i e n t to c o n s i d e r an a f f i n e scheme. T h e n the f i r s t s t a t e m e n t f o l l o w s f r o m [5, I. 1.3.9] a n d [5, I . i . 4 . 1 ] , a n d the s e c o n d s t a t e m e n t is [5, III. 1.4.17] .

18

and there

are isomorphisms

Hi(X- Y ,F) -% Hyi+l(X, F) Proof.

For the first

i>0

statement,

we apply the spectral

sequence

(Proposition I. 4) n

H y (x, F) < = E pq = HP(x,

By the previous

proposition,

quasi-coherent. p > 0.

Therefore,

So o u r s p e c t r a l

H yq (F))

q Hy(F)

we know that the sheaves since

sequence

X

is affine,

degenerates

E pq = 0

are for

and we have

n Hy(X ,F) --~H0(X, ~y(F))

or,

in other words,

n

Hy(F)

is the sheaf associated

to the A-module

Hy(X, F). n

The second of Corollary for

1.9,

statement since for

follows directly X

affine,

from the exact

the groups

sequence

Hi(x, F) = 0

i > 0. We n o w p r o c e e d

an affine scheme. expressed

study of the local cohomology

Since the cohomology

in terms

we will restrict

to a detailed

of the groups

n

sheaves

Hy(X, F)

our attention to the latter.

n

Hy(F)

of

are well

by the above proposition,

19

We f i r s t r e c a l l

the definition of the Koszul

[11, C h . IV] o r [5, C h . I I I , w 1. 1]). L e t and let

f

be an element

(homological)

complex

of

A.

A

complex

be a commutative

Then we denote by

of A-modules

(see

K(f)

defined as follows:

ring,

the

The module s

are

Kl(f) --~ K0(f) ~ A

;

K.(f)~ = 0

for

i> 1

and t h e m a p d : Kl(.f)--" K0(f ) is multiplication

If f and if

M

by

= (fl'

f. ....

f

n

)

is a finite family

is an A-module,

of elements

then we denote by

K,(f_;M)

of

A,

the

homological complex K{fl)

where

M

|

"

is considered

We d e n o t e b y

K ~_;M)

" ~A

K{fn) |

as a complex

A M

concentrated

the cohomological

in degree

zero.

complex

HOmA(K,(f_; A), M)

The homology

and cohomology

of t h e s e

complexes

will be denoted

Xc

by

H , ( f_; M) If

then there

m

and and

H (f_; M ) , r e s p e c t i v e l y . m'

is a natural

are two positive

integers

map of the Koszul complexes

such that

m' > m,

ZO

K(fm' ) --. K(fm ) defined as follows: in degree

In degree

z e r o i t i s t h e i d e n t i t y m a p of

one it is multiplication

by

s

elements

also maps follows

of

A

with respect

of t h e K o s z u l c o m p l e x e s

to a module

of t h e h o m o l o g y a n d c o h o m o l o g y (letting

_ f m = (f~l . . . . .

and

One extends this

definition in the obvious way to give maps n

A,

M,

of

and therefore

of t h e s e c o m p l e x e s ,

as

fIn))n :

K,(_rm;M) H,(fm';M)-

H,(fm;M)

and K * ( f m ' ; M ) ~ K (fro;M) H (fr n ; M ) ' - H

(frn;M)

Thus in the first case we have inverse homology complexes

A,

and cohomology

~-. 3.

Let

X; l e t f. = ( f l '

and let the variety

be an A-module. functor s

of c o m p l e x e s

groups; in the second case we have direct

Theorem spectrum

systems

(i > 0)

systems

and of

groups.

A 9 ..,

be a c o m m u t a t i v e fn )

ring with prime

b e a f i n i t e f a m i l y of e l e m e n t s

of t h e i d e a l t h e y g e n e r a t e

T h e n t h e r e i.s a n i s o m o r p h i s m

be

Y C X;

let

of c o h o m o l o g i c a l

of M

21

lim

i -- H y ( X ,

Hi(fro;M)

~I)

m

The complete notes,

p r o o f of t h i s t h e o r e m

and besides,

i s b e y o n d t h e s c o p e of t h e s e

is fairly well known.

with outlining the proof,

We w i l l c o n t e n t o u r s e l v e s

and refer to the literature

for details.

[5, Ch. n I ] The first

step is to interpret

cohomology in terms i Hf (M)

let us define

t h e d i r e c t l i m i t of K o s z u l

of C e c h c o h o m o l o g y . lirn

to be

For simplicity

Hi(f_m;M).

If

~

of n o t a t i o n ,

is a collection

in

of o p e n s e t s o f

by

~i( ~_

X,

,F)

the

ith

t o t h e f a m i l y of o p e n s e t s Finally,

if

of e l e m e n t s sets

X = Spec A, of

A,

F

and if

i s a s h e a f on

Cech cohomology ~,~

,

and if

, n,

_f = ( f l '

where

U.

1

the variety

we will denote

g r o u p of

and with coefficients "'''

t h e n w e w i l l d e n o t e by

(U.), i = 1 . . . .

X,

X

with respect

in the sheaf

fn )

is a finite family

~f

t h e f a m i l y of o p e n

is the complement

in

F.

X of

1

of t h e i d e a l g e n e r a t e d

by

f.. 1

Proposition

C.

[5, Ch. III, 1 . 2 . 3 ]

and with the notations above,

With the hypotheses

there is an exact sequence

0 --~H (M) --~ M -~ ~ O ( ~ f ,

~I) - - H f 1 (M)

-.

0

of T h e o r e m 2 . 3 ,

22

where

~

is t h e n a t u r a l

HI( ~ , f , Moreover,

restriction

~[)

~

map and there are isomorphisms

H +I(M)

the exact sequence and isomorphisms

T h e p r o o f of t h i s p r o p o s i t i o n

that t h e c o m p l e x coefficients ~f

(M)

for

in

~*(

~f,

l~I)

of t h e c o h o m o l o g i c a l

with the dimensions

are functorial

is elementary.

of

M.

of

~f

with

to t h e d i r e c t l i m i t

Koszul complexes

s h i f t e d by o n e .

in

One shows explicitly

Cech cochains

is canonically isomorphic

~I

i > 0

K (f --

m

;M)

,

but

That is to say, there is a

commutative diagram

o--

(M)- ~ f ( M ) -

)~

~( (M)-

~~

0-- cO(~f,~)

~~

--

~( ~ . f , ~ ) -

Now since taking direct limits is an exact functor, i H f (M)

a s t h e c o h o m o l o g y of t h e c o m p l e x

the exact sequence and isomorphisms proof is functorial isomorphisms

in

M,

D.

Let

by o p e n a f f i n e s u b s c h e m e s , X.

~f

*

one can compute

(M),

and one finds

of t h e p r o p o s i t i o n .

so t h e r e s u l t i n g

The entire

exact sequence and

are also.

Theorem

on

...

X

be a s c h e m e , and let

Then there is an isomorphism

F

let

~

b e a c o v e r of

he a quasi-coherent of c o h o m o l o g i c a l

X

sheaf

functors

(i> O)

Z3 ~i( where

Hi(x, F )

F-functor

F) ~ Hi(x, F)

,~,

denotes the

from the category

ith

right derived functor of the

of a l l a b e l i ~ n s h e a v e s

on

X.

! For the proof see This theorem in the previous open affines.

[5, Ch. III, P ~ o p .

since

~,~.f

Z. 3.

have both been characterized of t h e m a p

Hi'I(x

- Y, l~i)

each

i.

for

I"(X-

Hence they are isomorphic

That the isomorphism

commutes

i n t h e p r o o f of P r o p o s i t i o n

i Hf (M)

of t h e f u n c t o r

0 Hf(M}

thing to say if

M

Definition. groups. an

for

of c o c h a i n c o m p l e x e s

A

is a Noetherian

ring,

are the right derived functors of

A-modules.

A-module,

Let {Mm)m> 0

be a n i n v e r s e

Mmw --- M m

functors for

with connecting homomorphisms

is an injective

such that the map

is the zero map.

i > 0

in the category

We s a y it i s e s s e n t i a l l y

m' > m

i Hf (M)

and

C.

O u r n e x t g o a l w i l l b e t o s h o w if then the functors

Z. Z, w e c a n c o m p l e t e

Y, 1~I) f o r i = 0, 1, a n d a s

can be seen by going back to the isomorphism described

by

i n t h e s a m e w a y , n a m e l y , as t h e k e r n e l

a : M-"

i > 2.

X - Y

i Hy{X,I~I)

For the functors

H i ( x - Y, l~I)

1~) by

i s a c o v e r of

N o w w i t h t h e a i d of P r o p o s i t i o n

the proof of Theorem

and cokernel

Hi( ~[r

allows us to replace

proposition,

1 . 4 . 1].

then

z e r o if f o r e a c h

It i s t h e s a m e

i Hf (M) = 0

system

for

of a b e l i a n

m_> 0,

there exists

i > 0.

24

Remark. zero,

1)

If t h e i n v e r s e

then it follows that its inverse

converse

system limit

(M

)

rn

liin ~-

M

is

essentially

is zero.

In

The

is false. z)

If we have an exact sequence

of i n v e r s e

0 - - (M~n) - - ( M i n ) - - ( M ' ~ )

then the middle ones are

one is essentially

essentially

zero.

-- 0

systems ,

zero if and only if the two outside

We l e a v e t h e p r o o f a s a n e a s y e x e r c i s e

for the reader.

Lemma f = (fl' i > 0

9..,

2.4.

fn )

Let

A

be a commutative

be a finite family

be an integer 9

of elements

Then the following statements

(i) H fi (M) - 0

for all injective

A-modules

fin (H i ( f ; A ) ) i n > l i s a n e s s e n t i a l l y

(ii)

Proof.

of

ring, A, are

let and let equivalent:

M.

zero inverse

system 9

By definition,

i

Hf (M)=

l

Hi(fin M ) In

If

M

is injective,

with passage

then

to homology,

Hi(fin;M)

If

(Hi~-

in

;A))in>l

Horn(

, M)

is exact.

Hence it commutes

and we have

~ H O i n A ( H i ( f i n ; A ) , M)

is essentially

zero,

then for each

m

there

is

25

an

m'

> m

s u c h that

Hi~_m;M) --~ Hi(fin';M)

is zero,

i H f (M)

and hence the direct limit Conversely,

we can imbed

suppose

H.(fm;A)

{i) i s t r u e .

of t h e s e m o d u l e s

Given an integer

in an injective module

M.

is zero.

m > 0,

Let

cr s HOmA(Hi(fm;A),M) -~ Hi(fm;M) be the i m b e d d i n g m a p .

i H f (M)

If t h e d i r e c t l i m i t

is zero,

then

m 9

there must be an is zero;

m' > m

in other words,

H.(f is zero. arrow

Since

m

cr

!

such that the image of such that the composed

;A) --~

H.(f

In

in

h~

m

(f_

w

;M)

map

M

;A) ~

is a monomorphism,

it follows that the first

is zero.

Lemma

Z.5.

Let

A

be a Noetherian

be a f i n i t e f a m i l y of e l e m e n t s

of

finite type.

system

Then the inverse

(H.(fin; N)) 1

is essentially

zero for

m>l i > 0.

A,

and let

ring, N

of A - m o d u l e s

let

_f = ( f l '

be a n A - m o d u l e

"''' of

fn )

Z6

Proof.

We proc&ed by induction

Then the only value of that

H l(fun;N)

elements

is multiplication

Noetherian

rn

by

sequence, N

and

multiplication

by

rn

n.

First

is

i = 1.

of

N

which must

of finite type.

consisting

Nmw --~ N r n , N

m

since

In other words,

there

N

rn

the map from

.

r n ~ _> r n , of

N

A

is

system

(Nrn)

i.e. , the zero

is essentially

Now let proved

,

n

modules

N

of

of finite type

N,

let

~ 2,

< n

of finite type.

Therefore,

N m , -~ N

sequence

Given

~ = (fl'

[ll,

if is

rn

Hence the inductive

Ch.

and suppose

elements

"'''

of

_f = ( f l . . . . . f

1 )" n

is an exact

m 0

zero.

be an integer

for all sequences

map.

form

is an

m 0 f

n = 1.

of those

for

be stationary,

all of the modules

m ' = r n + m 0,

suppose

One sees immediately

Now the subrnodules

annihilates

.is given,

N

frn. The map

~1 !-rn

by

and rn 0 f

such that

to consider

is the submodule

annihilated

an increasing

i

on

A,

the lemma and for all

fn ) s A,

Then for each

and a module i > 0

there

~

IV],

0 -~ H0(~nn ;Hi" ( ~ r n ; N ) ) --~ H"(fm;N)l-- - - H1 (i~na;H'n1-1 ~ r n ; N ) } -~ 0

To show that the middle sufficient, systems

by the remark are

essentially

inverse

system

2 above, zero.

is essentially

zero,

it will be

to show that the two outside

inverse

27

On t h e l e f t , w e f a c t o r t h e i n v e r s e (where

system map as follows

m ! > m): ~l~n t

HO(

m I

-..<

;H.(g

;N))

rrl

HO(nf~ ;Hi(g

;N))

H 0 (if~;Hi (gin;N)) By the induction hypothesis,

the inverse

system

essentially

zero for

i > 0.

Therefore,

when

than

the map

~

m,

the left-hand inverse We p e r f o r m

w i l l be z e r o ,

(Hi(~m;N)) m m'

is enough greater

and hence also

system is essentially a similar factorization

is

~ o ~.

Therefore,

zero. on t h e r i g h t - h a n d

inverse

system.

Hl(nf~

'

(m'

;H.:_I

"N))

I HI (n~n;Hi- I(~m;N))

H I(nf~ ,H.:_l(~m;N))

f

For given

i,m,N,

the A-module

H. (~m;N) 1-1

i s of f i n i t e t y p e .

Therefore,

applying the induction hypothe sis to this module,

and to

28

the single element

f s A,

we find that for

m'

enough large r than

n

the map

fl

system

is zero.

Hence,

fl o ~

on the right is also essentially

Proposition 2.. 6. f = (fl' .... f )

--

be an injective

Then

By Lemmas

2.4 and 2.5.

2.7.

Let

spectrum

X,

and let

sheaf

on

X

Proof. any closed

is surjective.

Q , E . D.

be a Noetherian ring,

A-module.

Corollary

l~i

A

zero.

A

M

be an injective

To show that

for

be a Noetherian

is a flasque

Y C X,

Hie (M) = 0

A, i

let a n d let

>

M

0.

ring with prime

A-module.

Then the

sheaf. ~/

is flasque,

we must

show that for

the m a p

M

r(x

- Y,

Let

fl .....

i~i)

fn

generate

t h e i d e a l of

X - Y

is the union of the open sets of the family

above,

and from

F (X-Y, ~I)

and the inverse

be a finite family of elements of

n

Proof.

Let

is zero,

the definition of Cech cohomology

is i s o m o r p h i c

to

~o

(~[f,~I).

~,~

Y.

Then

defined

it follows that

So b y P r o p o s i t i o n

C

quoted above, there is an exact sequence

M K

But

1

Hf (M) = 0 is surjective.

F(X-

for

1

Y,YW--Hf(M)

-" 0

IVi injective by the previous proposition,

SO

m,

Z9

Remarks.

Conversely,

a s we s e e by P r o p o s i t i o n

the corollary implies the proposition

C above.

Using results

of i n j e c t i v e m o d u l e s o v e r N o e t h e r i a n r i n g s d i r e c t p r o o f of t h e c o r o l l a r y a s f o l l o w s . ring

A

on t h e s t r u c t u r e

[3], [9]* one c a n g i v e a

An i n j e c t i v e o v e r a N o e t h e ~ i a n

i s a d i r e c t s u m of i n j e c t i v e h u l l s

I

of r e s i d u e c l a s s f i e l d s P

k(p)

of p r i m e ideals

p

of A.

Since a direct s u m of flasque

sheaves is flasque, it suffices to s h o w that

I-

is flasque for each

P p. A

But

I P

is a direct limit of Artin m o d u l e s over the local ring

. Hence

[

P of

is a constant sheaf P

Spec A, and zero outside.

I P

on the closed subset

V(p)

Since a constant sheaf on an irreducible

Noetherian space is flasque, the contention follows. N o w w e c o m e to the m a i n t h e o r e m of this section, relating the local c o h o m o l o g y groups on a p r e s c h e m e

Let and let

X

F

be a p r e s c h e m e ,

be a q u a s i - c o h e r e n t

be a q u a s i - c o h e r e n t let

~n =

for each

let

~X / n

Y

be a c l o s e d s u b s p a c e o f

s h e a f of

~X-mOdules.

s h e a f of i d e a l s d e f i n i n g

~n.

Then

~'n

to a direct limit of Exts.

Y,

X2

Let

and for each

is a sheaf concentrated

on

n>l, Y,

and

there is a natural injection Hom~

( ~n' F) -" r y ( X , F) -X

For a homomorphism

of

of t h e u n i t s e c t i o n of

~

in

~n n

into

F

is determined

, w h i c h m u s t be a s e c t i o n o f

by the i m a g e F

with support

Y.

* S e e a l s o [17, II w 7] for a c o m p l e t e d e s c r i p t i o n of the s t r u c t u r e of i n j e c t i v e ~X- m o d u l e s on a l o c a l l y n o e t h e r i a n p r e s c h e m e X. U s i n g t h e s e r e s u l t s a l l o w s one to a v o i d the p a i n f u l (2.3) - (2.6). It a l s o g i v e s a m o r e d i r e c t p r o o f of P r o p o s i t i o n 2.1 for X l o c a l l y noetherian.

30

Letting

F

range over the category of 0 K - m O d u l e s ,

consider on the one hand the derived functors

we

Ext,. ( ~n, F)

of

2k

H~

(~n' F), and on the other hand the cohomological functor

Hy1 (X, F), which is a p r i o r i Having a m a p

property i n

into

a derived functor in this category'*

of these functors for

by t h e u n i v e r s a l

As

not

varies,

n

i

H y ( X , F),

i = 0,

w e deduce one for

i> 0

of d e r i v e d f u n c t o r s . i

,

-~

these

Hy(X,

Ext's

form a direct

system mapping

so there are homomorphisms

n

"X

n

(*)

-~ H y ( X , F)

Performing these h c m o m o r p h i s m s

locally, and passing to associated

sheaves, w e have h o m o m o r p h i s m s

of sheaves

lirn n

Theorem coherent, X

xtiffx((~'n'F)

Z. 8.

If

X

(~)

is local ly Noetherian, and F quasi-

then the h o m o m o r p h i s m s

is Noetherian,

i

-- H y ( F )

(~) are isomorphisms.

then the h o r n o m o r p h i s m s

If furthermore

(*) are also isomorphisms.

The proof will be deferred until after s o m e auxialiary propositions and l e m m a s. *In fact, it is a d e r i v e d functor, b e c a u s e b y lemma 2.9 below, any i n j e c t i v e ~ X - m O d u l e is flasque, and b y P r o p o s i t i o n i.i0, f l a s q u e s h e a v e s are F y - g c y c l i c , so may b e u s e d to c a l c u l a t e the c o h o m o l o g y g r o u p s H~(X,F).

31

Lemma sheaves of category

[4, II, 7.3.2] Z. 9.A Let X be a p r e s c h e m e , and suppose that

(~X-mOdules,

of

~'X-mOdules.

Proof.

We m u s t

Then

let

U

and

G

be

G is injective in the

HOm~x(F,G

s h o w , if

F

) is aflasque sheaf.

is open in

X,

then the natural

re striction

Hom~.

(F,G) --- Horn

(FIU,GIU)

-x

is surjective. to

U

is

FU

Fu

F] U,

homomorphi of

Let

a n d w h i c h i s z e r o o u t s i d e of

sm of

into

denote the unique sheaf whose restriction

F[ U

G.

Since

such homomorphism

into G [ I t

U.

Then a

is the same as a homomorphism

F U _C F, a n d s i n c e

G

extends to a homomorphism

is injective, of

F

into

[4,II 7.3.3] Proposition 2. 10.ALet X be a prescheme, and let G

be sheaves of

~'__-modules. -X n

Ext-

Proof.

(F,G)

F

EXt~x(F,G))

Hom/~ (F,G) "X

functor

H~

G.

and

Then there is a spectral sequence

<= E q = HP(x,

One can represent

any

= I"(Hom0,x(F, G))

as

a composite

32

By Lemma injectives

Z. 9 into

apply Theorem sequence

Homl~, ( F , G ) $ c o n s i d e r e d a s a f u n c t o r i n G , takes -X F-acyclic objects (Lemma 1.7). Therefore, we may A quoted above to deduce the existence

of t h e s p e c t r a l

of d e r i v e d f u n c t o r s .

[4, ch. II,w 4.12] Proposition 2. II. ALet X be a Noetherian topological space, and let

(F.)

be a direct system of abelian sheaves on

X.

T h e n for

I

each

p > 0,

HP(x, lira F.) = lira HP(x,F.) --~

Proof. F. a n d of 1

F,

Let

1

"-~

1

F = lira F..

T o c a l c u l a t e the c o h o m o l o g y of the

take injective resolutions

of the

F.,

1

such a way that they form a direct Then

C(F)

limits

of i n j e c t i v e s .

space,

C(F. )

is a resolution

of

system,

F

r

"-~

functor commutes

HP(x, Fi) =

C(F) = lira C ( F . ) . 1

by sheaves which are direct

Now since the direct limit functor is exact,

lirn

and let

Using the quasi-compacity

one f i n d s that the

in

1

of a N e 6 t h e r i a n with direct limits.

we have

HP(F (C(F)))

It will therefore be sufficient to s h o w that any direct limit of injective sheaves is flasque, since then w e m a y use the c o m p l e x calculate the c o h o m o l o g y of

F.

But by L e m m a

C(F)

to

I. 5, any injective

sheaf is flasque, and an easy argurnent using qua si- compacity shows

33

that a direct limit of flasque sheaves on a Noetherian space is flasque. Example.

The previous

proposition

Noetherian

hypothesis.

For example,

of l x ~ t i v e i n t e g e r s

is false without the

with the discrete

let

X

be the space

topology.

Let

F.

be the

1

sheaf whose

stalk is

into

by killing the

Fi+ 1

alone,

Then

Z

for

lim F. = 0, --~

n>

ith

i,

and

type.

Z. lZ.

X.

Let

Then there

F.

else

but

Let M,N

are

A

a tilde denotes

/ o

z neE

be a Noetherian

be two A-modules

canonical

functorial

i i E x t A ( M , N ) N ---- E x t , , X

where

Map

1

neE

spectrum

n < i.

stalk, a n d leaving e v e r y t h i n g

nm F(F i)=-[]- Z / G Lemma

0 for

ring with prime and let

M

be of finite

isomorphisms

N)

taking t h e q u a s i - c o h e r e n t

s h e a f on

X

as.sociated

to the given A-module. Proof. that

For

H o r n ( M , N)

presentation ~f

i = 0, commutes

the isomorphism

follows from

with localization

when

[5, I, 1 . 3 . 12 (ii)].

As

M

varies

M

the fact

is of finite

in the category

i N o f m o d u l e s of f i n i t e t y p e o v e r A , ExtA(M,N) is the ith (with respect to t h ~ f i r s t variable) right derived functorAof H o m A ( M , N) . Hence, there are canonical A

homomorphisms of sheaves

of

(of functors ~X-mOdules)

in

M,

from

t f

to the category

34

EXt~(M,N)~-

Extl~

(M, N)

(i)

X Since

is Noetherian,

A

every

M e ~ f has a p r o j e c t i v e --~%

r e s o l u t i o n b y finitely g e n e r a t e d show

(i) is an isomorphism,

side is zero w h e n

M = Ar

free A-modules.

we need only show the r i g h t - h a n d for then they will be derived functors

i Since Ext ~ _ X Commutes with finite direct sums, to show that Extl~ ~y-module

Hence to

(A, N) = 0

for

i > 0.

it is sufficient

In fact,

for any

F,

Hom~/ ( ~X, ~X i the Ext ~

Extl~ (A, F) = 0, since A = ~X, and vx (Recall that F) = F is an exact functor in F. are defined as derived functors

in the category

2 %

of

~X-mOdules,

with respect to the second variable.)

Proof of Theorem 2.8 question is b c a l ,

In the first assertion,

so we may assume that

X

the

is the spectrum

N

of a Noetherian ring

A.

Let

~

= I

and

F = N.

Then we must

show that the h o m o m o r p h i s m s " ~ ~ i ~ lim Extm9 (A/In, N)- Hy(N) n ~X -are isomorphisms. By p r o p o s i t i o n 2.2, the r i g h t - h a n d

is equal to

i X ,N)N. Hy(

Moreover

using Lemma

2.12 and

noting that direct limits commute w i t h the o p e r a t i o n I, 1.3.9

side

N

[5,

(iii)], we reduce to showing that the h o m o m o r p h i s m s

n

35

are i s o m o r p h i s m s .

This is a h o m o m o r p h i s m

f r o m the category of A - m o d u l e s

of cohomological functors

into itself. For

i= 0

the functors

are isomorphic, their cornrnon value being the set of elements of annihilated by s o m e p o w e r of vanish:

I. For

i> 0

N

and N injective, both

The one on the left since the Ext's are derived functors; the

one on the right by Corollary 2.7 and Proposition I. 10.

This charac-

terizes both functors as derived functors, so they are isomorphic.

;) and -X as the abutments of spectral sequences, using propositions

;or the second assertion, we express Ext

I I

H i (X, F) Y

i. 4 and Z. I0.

N o w one observes that a direct limit of spectral se-

quences is a spectral sequence, and that w e have functorial h o m o morphisms

of spectral sequences, which give the f~)llowing commutative

diagram:

E~q=

lirnn HP(x' E X t ~ x ( ~ n ' F ) ) - ~

lirn Ext n T o show that the h o m o m o r p h i s m

(~n,F)--" H y ( X , F )

of the

E pq

But this follows f r o m the i s o m o r p h i s m

(~)established above, and f r o m Proposition 2. II, since a s s u m e d Noetherian.

($)

of the abutments is an i s o m o r p h i s m ,

it will be sufficient to s h o w that the h o m o m o r p h i s m s t e r m s are i s o m o r p h i s m s 9

HP(x'H---qY(F)) = E~pq

X

is

w 3.

In t h i s s e c t i o n , codimension

w e r e c a l l t h e n o t i o n of d e p t h o r h o m o l o g i c a l

introduced

give a theorem

Relation to Depth

by Auslander

(Theorem

and Buchsbaum

[1].

Then we

B. 8) r e l a t i n g t h e n o t i o n o f d e p t h t o t h e l o c a l

cohomology groups. First

let us recall

a commutative the variety I.

V(I) If

ring,

of

and I

I,

prime

If

A,

If

A

we denote by i d e a l s of

is

V(I)

A

containing

is a c l o s e d s u b s e t of Spec A. N

is an A-module,

Ass

N

ideals

isomorphic

i s a n i d e a l of

w h i c h is t h e s e t of a l l p r i m e

w h i c h i s the s e t of p r i m e denote by

some notations and definitions.

A

ideals

?

A

such that

N

A

such that

primes

N1

of N:

~ 0.

They

contains a submodule

We

are those N1

A/~ is a commutative

a s e q u e n c e of e l e m e n t s if for each

of

t h e s e t of a s s o c i a t e d of

to

w e d e n o t e b y S u p p N t h e s u p p o r t o f N,

i = 1....

fl'

ring, and

9..,

, n,

f.

f

n

of

M A

an A-module,

then

is said to be M-regular

is not a zero-divisor

in the module

1

/(fl . . . . . divisor in

f"1 - 1 )M " (In particular, this m e a n s

is not a zero-

M. )

Lemma ideal,

fl

and let

S. 1. M

Let

A

be

be a n A - m o d u l e

conditions are equivalent:

a Noetherian

ring,

of f i n i t e t y p e .

let

I

be an

Then the following

37

(i) H o m ( N , M )

= 0

support is contained in

for all A - m o d u l e s

whose support is equal to No associated

{iii) ~ f s I

Noetherian

3.Z.

r i n g A.

whose

for s o m e

A-module

of finite type N

V(I). prime

such that

Sublemma

N

V(1).

(ibis) Horn(N, M ) = 0

(ii)

of finite type

Let

of M c o n t a i n s

f

N,M

is

I.

h/i-regular.

be modules

of f i n i t e t y p e o v e r a

Then

A s s (Hom(N, M)) = Supp N r-~ A s s M Proof.

Using well-known facts about associated p r i m e s of

m o d u l e s of finite type over Noetherian rings, w e have Ass( H o m ( N , M)) _C Supp (Horn(N, M)) _C Supp N Moreover,

if we express Ar--,N

N

a s a q u o t i e n t of a f r e e m o d u l e

--, 0

then we have

0 -~ H o m ( N , M ) -~ H o m ( A r ,M ) = M r from which follows

Ass(Hom(N,M))

C Ass M r = Ass M

Thus we have an inclusion

A s s ( H o m ( N , M)) _.CSupp N r-~ A s s

M

38

Conversely,

s S u p p N r - ~ A s s M. M 1 _~ A / ~

s Supp N ,

In particular

all of Spec

A/~

has positive

rank over the integral

and therefore injection

N'

the support (since

isomorphic

N/ N must be

of

r

N/$N

N is of finite type),

domain

to an ideal of

has a submodule

M

z Ass M,

which is torsion-free

N' --" M 1,

composition

~

Then since

Since

quotient module

i d e a l of A s u c h t h a t

be a prime

let

A/~

H e n c e it h a s a

a n d of r a n k o n e o v e r

A/~

Thus there

s Hem

and we can define

(N,M)

is an to be the

of t h e f o l l o w i n g h o m o m o r p h i s m s

B y our construction, the s u b m o d u l e of Horn(N, M ) generated by is i s o m o r p h i c to

A/9.

Proof of L e m r n a

,

so $ s A s s

(Horn (N,M)).

Q.E.D.

3. I.

(i) => (ibis). Trivial. (ibis) => (ii). If Horn(N, M ) = 0, and Supp N = V(I), then by the sublemrna,

V~I) r-~ A s s M = A s s

associated p r i m e of M

contains

(Horn (N, M)) = ~,

SO

no

I.

(ii) :> (i). If Supp N C V(1), then again by the s u b l e m m a , Ass(Horn(N, M)) _C V(I) r-~ A s s M =

by hypothesis,

so H o m

( N , M ) = 0.

39

(iii).

.....

~

(which are finite in number

since

(ii) < = > of

M

to say none of the

Let

"~ i

~l

contains

I

be t h e a s s o c i a t e d

r M

primes

is of finite type).

Then

is the same as to say their union

r

~-J ~ i= 1

does not contain

i

an element element of t h e

f ~ I f s A

~ i'

is

M-regular

let

3.3. M

i s t h e s a m e a s t o s a y t h a t it i s i n n o n e

Let

A

be a Noetherian

be an A - m o d u l e

let

of f i n i t e t y p e , a n d l e t

= 0

(ibis)

be an

I

n

ExtA(N,M ) = 0

for s o m e

elements

fl

. . . . .

of finite type

A-module

fn s I

N

of f i n i t e t y p e

i < n.

forming an M-regular

sequence.

Proof.

(i) => ( i b i s ) .

(ibis) => (ii).

H o m { N , M ) = 0, This

Trivial.

We p r o c e e d

there is nothing to prove.

M-regular.

be

i < n.

Supp N = V(I), and for all integers 3

N

for all A - m o d u l e s

such that Supp N C V(I), a n d for all integers

(ii)

ring,

Then the following conditions are equivalent:

(i) E x t A ( N , M )

such that

But to say that an

so w e a r e d o n e .

A,

an integer.

which is the same as to say that there is

w h i c h i s i n n o n e o f t h e ~ i"

Proposition

ideal of

I,

soby f

b y i n d u c t i o n on

So s u p p o s e

the lemma

there

n > 0.

n.

If

n_< 0,

Then in particuhr

exists an

f s I

which is

gives rise to an exact sequence as follows:

40

0 - " M f-~ M - - ~ M / f M - - ~

where

the first map is multiplication

and from the exact sequence for

i < n - 1.

elements

f2'

Then clearly of

Therefore, ....

' f

by

of Ext's,

f.

Now from

our hypothesis Exti^ (N, M / f M ) = 0

it follows that

by t h e i n d u c t i o n h y p o t h e s i s ,

fn ~ I

f, fZ . . . .

,

0

n

there

which form an M/fM-regular is an

M-regular

are

sequence.

sequence

of

n

n<

0,

elements

I. (if) => (i).

We p r o c e e d

by i n d u c t i o n

is nothing to prove.

So s u p p o s e

M-regular,

is an exact sequence.

so there

n > 0.

on

n.

If

Then in particular

there

fl

is

0 --~ M ~ M --~ M/f IM-- 0 Now of

fZ . . . . . I,

fn

is an M/riM-regular

so by the induction hypothesis

for all A-modules integers

of finite type

i < n - 1.

N

sequence

of

it follows that with support

By the exact sequence

in

of Ext's,

n-

1

elements

i E x t A ( N , M / f 1 M) = 0 V(I), and for all this implies

that the natural m a p

E x t A (N, M)

is injective

for all

i < n.

map is also the zero map.

But since Hence

f l ~ I,

Ext~(N,M)

fl = 0

kills for all

so this

N, i <

n.

41

Remark.

Note that the lemma

is used only in the implication implication N, M

(ii} - > (i}

of finite type.

"Supp N _C V{l}"

of

A,

depthiM, fl ....

"fl'

Let M

A

M.

and Buchsbaum

Corollary

sequences namely,

Corollary M-regular,

3.4.

ring,

N"

let

of elements

of

I

{The I-depth

and I is

of M is what

o f M. }

Then all maximal

have the same number

of

depthiM.

3.5.

Let

A,I,M

be as above.

then depthiM = depth I M/fM

+ 1

of M,

{Note t h a t

is a local ring,

be as above.

A,I,M

be a n i d e a l

exist elements

[1} c a l l t h e I - c o d i m e n s i o n

Let

I

Then the I-depth

sequence.

we say simply depth

nor

the hypothesis

such that there

A

ideal,

elements,

n

The

Noetherian

of finite type.

f i n i t e . ) If

the maximal

A

are nilpotent in

n

which form an M-regular

and is therefore

M-regular

f

be a Noetherian

integer

n < dim M,

Auslander

one may replace

be an A-module

is the largest

' fn s I

....

hypotheses)

of the proof.

is valid without assuming

by

and let

( i b i s ) => {ii)

Moreover,

Definition.

{with i t s N o e t h e r i a n

If

f e I

is

42

Corollary

3.6.

Let

depthlM =

where

M~

be a s above.

A,I,M inf

(depth h~

is considered

)

as a module

over the local ring

and its depth is the (usual} depth with respect

Proof.

fl . . . . .

If

f

~ I

Then

to the maximal

A~

,

ideal

f o r m an M-regular sequence,

n

then the canonical images

f- , . . . ,

in

M[

~/~

, and f o r m an

1

n

-regular

of t h e

f.

in

1

A

are

sequence f o r a n y ~ s

Therefore

depthlM < depth M i for each

~

s V(1).

For the converse, we will prove the following statement: n'

is an integer such that for each

n' < depth 1M".

i

~ Viii,

W e proceed by induction on n' > 0.

M~

is not associated to

__> 1, so

~A~

is not associated to an

f ~ I

M.

which is M-regular.

induction hypothesis, n'

-

<

the case

Then for each M;

~

n' < 0

s V(1), ,

, then

depth

i.e.,

Then by L e r n m a B. I, t h e r e e x i s t s N o w using Corollary 3.5 and the

we find

1

n' _< depth M I

n',

being trivial. So suppose

"If

depth (MIfM)

43

for each

s V(I).

Hence

n' - 1 < depthiM/fM which implies

that

n' < d e p t h i M .

Definition.

Let

X

be a locally Noetherian

Y a closed s u b s e t

of

T h e n the Y - d e p t h of F,

or depthyF_ , is the

inf (depth Fx )" xzY

is an affine scheme,

say

R em_ar k. Y = V(1)

and

If F =

X

X,

A4, t h e n

and

depthyF

F

prescheme,

a coherent sheaf on

thus that the notion of I-depth depends

of the ideal

I.

Restating

X = Spec A, by C o r o l l a r y

= depthlM,

We o b s e r v e

X.

3.6.

only on the radical

t h e a b o v e t h e o r y i n t h e language o f p r e s c h e m e s ,

we have Proposition let

Y

and let

be a closed n

3.7.

X

Let

let

subset,

be an integer.

(i) E x t ~ v ( G , F ) = 0

F

for all coherent

(ibis) E x t ~ v ( G , F) = 0

sheaves

G

sheaf on X are equivalent: with

i < n.

for s o m e coherent sheaf

and for all integers

prescheme,

be a coherent

Then the following conditions

Supp G C Y, and for all integers

Supp G = Y,

be a locally Noetherian

i < n.

G

with

44

(ii)

depthyF

> n.

(iibis) depth F

for all

> n

x s Y.

x

Now we have a theorem the local cohomology

Theorem

let

Y

which relates

sheaves:

3.8.

Let

X

be a locally Noetherian p r e s c h e m e ,

be a closed subset,

and let

n

be an integer9

i (i) H y ( F )

(ii)

= 0

depthyF>

Proof.

(i)

for all

G

and for all

by i n d u c t i o n o n n > 0.

So s u p p o s e

Therefore,

G ~

X,

n ;

We proceed

in the category Y,

be a coherent sheaf on

n

is satisfied for

> n - 1.

F

T h e n the following conditions are equivalent:

(i) => (ii).

depthyF

support in

let

for all i<

there is nothing to prove. condition

t h e n o t i o n of d e p t h t o

n - 1,

i <

3.7,

of c o h e r e n t n -

E x t n" 1 ~X(G, F)

i.

If

n<

0,

in particular,

so by the induction hypothesis,

by p r o p o s i t i o n ~.y

Then,

n.

Ext,,

sheaves

(G, F) = 0

X on

X

with

In other words, the functor

45

from

~

Y

of i d e a l s of

to sheaves Y,

on

~rn

Let

~y

be the sheaf m

_-

m,

are in the category

~ r n ' --"

m' > m

is left-exact.

and for each positive integer

Then the sheaves

for

X

let

~m

Y

so the surjections

_~ y ,

0"m - - - 0

give rise to injections

0 -'Ext..

In o t h e r w o r d s ,

(

, F ) --" E x t

all the homomorphisms

1 ( -X

,,F)

in the direct

system

(Ext~ 'I (~rn ,F))m are injections. By Theorem ~-.8 the direct limit X of this system is --_~y-l(F), which is zero by our hypothesis. Hence e a c h of t h e t e r m s

of o u r d i r e c t

system

is zero,

y ( ~ / ~ Y' F) : 0. We have a l r e a d y ]Bxt n-,,,~..I for

f<

n - 1,

so f r o m

(ii) => (i). notation,

Proposition

Suppose that

by T h e o r e m

By Proposition "m

3.7,

these

h a v e s u p p o r t i n Y,

EXt~x ( ~ /

seen that

3.7 we conclude that

depthyF

> n.

Z. 8 w e h a v e f o r a n y

m

in particular,

d e p t h y F _ > n.

Using the above

i,

vX Ext's

hence

are

0 f o r i < n,

i Hy(F)

= 0

for

since the sheaves i<

n.

y, F) = 0

QE.D.

46

Corollary prescheme, Then

3.9.

and

X - Y

Y

Let

X

a closed

be a connected subprescheme

locally Noetherian

such that

d e p t h y 0 ~ x _ > Z.

is connected.

Proof.

By the theorem, for

i = 0, 1.

our hypothesis

By Proposition

implies

1.11,

that

this implies

that

the m a p H O(x, {~X) - ~ H 0 ( x -

is bijective. observe

To complete

t h e p r o o f of t h e c o r o l l a r y ,

that a prescheme

considered

as a module

Y, ~ X )

X

is connected

over itself,

connected => H O(x, ~ )

we need only

if and only if

is indecomposable.

= H O(x - Y, ~ )

H O ( x , ~ X ), For then

X

is indecomposable => X - Y

2 ~

is connected. If

X

H 0 ( U i ' (~X

is disconnected, where

is decomposable.

Ui

are nontrivial

elements

e I , eZ

e l e Z = 0;

e I = e l,

z

is indecomposable,

sets where

different z

e Z = e Z. hence

disconnected.

eI

suppose

idempotents from

so

H 0 ( X , (~X~

For each

i s decomposable.

and such that x ~ X,

e. 1

X, h e n c e

is zero and where

),

that is,

1 = e I + ez;

the local ring

is zero at X

sum of

H 0 (X, ~'X )

e I , e z e H0(X,~x

zero,

one of the

c a n b e z e r o a t e v e r y p o i n t of closed

is the direct

are the components,

Conversely,

Then there

H0(X, ~X)

x.

Neither

i s t h e di sj o i n t s u m o f t h e e Z is zero.

Thus

X is

47

Remark. codimension

The Corollary applies in particular if

at least two and

as it does for example

if

X

intersection

in a non-singular

Hartshorne

[8], w h o p r o v e s

of depth,

and applies

is normal, ambient

or if

X

prescheme.

the corollary

is of

Sz

of Serre,

is a complete For details,

directly

from the definition

intersections.

case of a local Noetherian

3.10.

maximal

ideal

Let

be a module

m

Let

A

and residue

ring,

be a local Noetherian field

of finite type,

Then the following conditions

the above

are

k.

Let

and let

ring with

X = Spec

n > 0

A,

U = X-{m}

be an integer.

equivalent

(i) depth M > n. (ii) Exti(k, M) = 0

for

(iii) HilX, ~i) -* HilU, ~i) injective

see

give the following

Corollary

M

has the property

it to the study of complete

In t h e p a r t i c u l a r results

~X

Y

for

i < n. is bijective

i = 0.

(iv) H~ (M) = 0 l m}

for

i < n.

for

i < n

and

.

w 4.

In this section, ring. of

We w i l l d i s c u s s

A-modules

will discuss

F u n c t o r s on A - m o d u l e s

A

will denote a commutative

Noetherian

various

left-exact

functors

from the category

to the category

of abelian

groups.

In particular,

dualizing functors

we

which will be used later in the section

on duality. As a matter (Ab)

of notation,

let

= the category of abelian g r o u p s = the category of A - m o d u l e s

d If

4~

= the category

is anidealof C~

If

A,

of A-modules

,

of finite type

let

= the category of A - m o d u l e s

with support in

= t h e c a t e g o r y of A - m o d u l e s support in V( e~t )

of finite type with

is any category,

we will denote by

~

the opposed

category.

Lemrna fr om

to

4.1. (Ab).

Let

T

be a contrav~riant

Then there

: T ~Hom

is a natural

A( ., T(A))

additive functor

functorial

morphism

V(

49

Proof.

For any

m

s M,

let

: A -~ M

be the morphism

m

which sends

1 into

T ( M ) -~ T ( A ) .

m.

Then

T(~m)

For a fixed element

we have an element

of

is a morphism T{M),

of HomA(M , T(A)).

as

of

m

s M

varies,

Thus we have defined a

morphi sm. ~b(M) : T ( M ) -* H o m A ( M , T ( A ) )

One checks and

t h a t it i s f u n c t o r i a l

HomA(M,T(A))

morphism

~f

the natural

to

4.2.

(Ab).

Let

In f a c t ,

A-module

is an isomorphism

Proof.

If

since the functor

T

structures,

T(M)

it i s a

if and only if

~

be a contravariant

additive functor

9 , T(A))

T

is left exact.

is an isomorphism,

Horn is.

then

On t h e o t h e r h a n d ,

T

is left-exact

suppose

T

is left-

since

T

is

T h e n in the f i r s t p l a c e

T(A r) -~ H O m A ( A r ,

T(A)) = T(A) r

is an isomorphism

for any positive integer

additive.

M

exact

if one gives

Then the morphism

~b : T - * H O m A (

exact.

M.

of A-modules.

Proposition fr om

in

Now if

sequence

is any A-module

r,

of finite type,

there

is an

50

A r - ~ A s --~ M - ~ 0

Letting

s t a n d for t h e f u n c t o r

T'

HOmA( 9

T (A)),

we h a v e a n

exact commutative diagram

0

--

T (M)

-~

T (A s')

-~

T (A r )

N

0

-~

T'(M)

w h e n c e by t h e 5 - l e m m a ,

Definition. denote b y

Sex(

functors from

to

T ' ( A s)

T(M) -~ T I(M)

~., 8

~,

~

explained thus:

If

-~

N

-~

T'(A r)

is an isomorphism.

a r e two a b e l i a n c a t e g o r i e s , )

B

we

t h e c a t e g o r y of l e f t - e x a c t ( c o v a r i a n t ) ( T h i s n o t a t i o n i s due t o G a b r i e l , a n d i s

s = s i n i s t e r , ex = exact. )

C o r o l l a r y 4.B

The categories

Sex( ~ f ~

Ab) a n d

~

m a d e e q u i v a l e n t by t h e f u n c t o r s

T ~

T(A)

I

HOmA(',

for

T e Sex( ~ f

and ~

I)

for

I s

9 , Ab)

are

51

Remark. exact functors injective, T(A)

correspond

HOmA{ 9 ,I)

is injective,

injective for all

Note that in the equivalence to the injective

is exact.

A-modules

Lemma an additive is a natural

modules.

M

4.4.

functorial

M ~

if

T

I

is

is exact,

a module

HomA(M,I)

the

I

is

i s exac_t

of finite type.

Let

contravariant

ring,

4.3,

Indeed,

On the other hand, if

since over a Noetherian

if and only if the functor

of C o r o l l a r y

~

be an ideal in

functor from

A,

~f

and let

to

(Ab).

T

be

Then there

morphism

: T --~HornA( ", I) where

I = l~n T(A/ n ) . n

Proof. some power

of

Let ~

of finite type over

Then

M s n

, say A/

n

is annihilated

In other words,

So b y L e m m a

T(M} -->Horn

M

M

4. 1 t h e r e

by

is a module is a morphism

n(M,T(A/4~n} : H o m A ( M , T ( A / n ) }

A/ Furthermore,

there

is a natural

morphism

Horn A (M, T ( A / ~ n)) -... lim Horn A (M, T ( A / ~ n)) n

then

SZ

a n d t h i s l i m i t i s e q u a l to and therefore

H o m A ( M , I ),

HomA(M , ")

these morphisms,

since

IV[ i s of f i n i t e t y p e ,

c o m m u t e s with d i r e c t l i m i t s .

Composing

one obtains a morphisrn

T (M) --~ Horn A (M, I)

w h i c h i s e a s i l y s e e n t o be i n d e p e n d e n t of t h e a n d f u n c t o r i a l in

The morphism

a n i s o m o r p h i s m if a n d only if

If

for each

M s

n' > n,

T

of L e m r n a 4 . 4 i s

~

is left exact.

is an isomorphism,

~

since the functor Hom is. exact, and let

chosen above,

M.

Pr0positiDn 4.5.

Proof.

n

then

T

On t h e o t h e r h a n d , s u p p o s e

~f

is left exact, T

be a m o d u l e a n n i h i l a t e d by

is left 4M n.

Then

the morphism n I

T(M) --"H o r n A ( M,T(A / ~

is an i s o m o r p h i s m ,

by P r o p o s i t i o n 4. Z.

g o e s to i n f i n i t y , we f i n d t h a t

Corollary 4.6.

~

))

Taking the limit as

n I

is also an isomorphism.

The categories

Sex( ~

fo

, Ab)

and

a r e m a d e e q u i v a l e n t by t h e f u n c t o r s T ~

lirn T(A/,~M n) n

for

T zSex(

~fo

Ab)

53 and I ~

Proof.

HOmA(" ,I)

of those elements

annihilated by

Proposition

4.7.

If

I

suppose

n

~

I

i s t h e u n i o n of t h e s e s u b -

to h a v e s u p p o r t in

In the e q u i v a l e n c e

correspond

Proof. Conversely,

~

l i r n H o m A ( A / ~ n , i ) ~ I. n t o t h e s u b m o d u l e of I c o n s i s t i n g

is isomorphic

s i n c e it w a s a s s u m e d

exact functors

I ~

We n e e d o n l y c h e c k t h a t

N o w H o r n ( A / r i M n , I)

modules

for

V( ~

).

of C o r o l l a r y

4.6,

the

to the i n j e c t i v e m o d u l e s .

is injective,

then

T

and let

isexact,

H O m A ( 9 , I)

is exact.

I= lira T(A/

,~n).

We

n

wish to show

I

is an ideal in

A,

injective. and

f:

extends to a morphism,

I s

~

theorem, of

A,

Thus

Since ,

f

~

A,

is of f i n i t e t y p e n

annihilates

the ~-topology

also annihilate s

as follows:

then

~

f

f : A -~ I.

of

and

(A

01, -~ I

f:

n.

M r r-~

such that ,

and

By Krull's

i s i n d u c e d by t h e r

a

being Noetherian),

for some

i.e. , there is an integer f

a morphism,

-~I

i s a n i d e a l of

So suppose

m o r p h i sin.

It wiI1 b e s u f f i c i e n t t o p r o v e t h a t i f

- topology

M r e-~ ~ C

and therefore

f

not, factors

54

P

fl r

But now we have an injection r

of m o d u l e s in

,

fl : and

T

we have a rnorphism

(it/

is exact.

r

r

~

~

Therefore,

~I,-"

fl

I

;

extends to a morphism

m

fl

If

p : A --" A /

f=fl

0

p

: A/Mr

~

r

--" I

is the canonical projection morphism,

is the desired

e x t e n s i o n of

f.

Hence

then

I

is

be an injective A-module,

let

inj e c t i v e .

Corollary 4.8. be of

i d e a l of

an

I

A,

and let

with support in

Proof. H o m A ( 9 , I). i s in~ective.

Let

Let

I In

= H~

V( ~w ).

Then

T s Sex ( ~f

9

Then by Proposition

4.7,

(I) In

Ab) I~

be the largest

submodule

is injective.

be the exact functor = lirn

T(A/~

n)

55

We n o w c o m e t o t h e s t u d y o f d u a l i z i n g f u n c t o r s .

Proposition A

(i. e . , s u c h t h a t ~i

T a Sex(

(i}

e

Ab)

,

Let

4.9.

For all

444

A/~ 9

be an ideal of finite colength in

is an Artin ring), and let

Then the following conditions are equivalent:

M a

~f

,

T(M)

is an A-module

of

finite

type, and the natural morphism M

TT(M)

-~

(defined via the isomorphism (ii) where

T

~

.

is exact,

of L e m m a

4.4) is an isomorphism.

and for each field

is a maximal

k

ideal containing

~

of the f o r m

A/

~.,

1

, there is some

1

isomorphism

T ( k ) ~- k.

(i) => (ii).

Proof. above.

Then

T(k)

Since

k

be a field of the form

is of finite type by hypothesis;

~*l., h e n c e m u s t be i s o m o r p h i c 1 Z

annihilated by n > 0.

Let

T

is additive,

TT(k) = kn ,

A/~t i

moreover,

to

kn

whence

as

it is

for some n = 1.

Thus

T ( k ) ------ k. It r e r r ~ a i n s t o s h o w faithful, i.e., non-zero,

if

O #0,

T then

is exact. T ( Q ) ~ 0.

then there is a surjection Q -~ k --.- 0

,

We f i r s t Let

show that

O s

~L

T be

is

56

where

k

i s a f i e l d of t h e f o r m

A/~

Since

..

T

is exact,

1

0 -~ T ( k ) -'~ T ( Q ) . But

T ( k ) -- k / 0, Now

T

T ( Q ) / 0.

hence also

i s l e f t e x a c t by h y p o t h e s i s ,

t h a t it i s r i g h t e x a c t .

so we need only. show

Let

O - - ~ M , --,.M be a n i n j e c t i o n of m o d u l e s

f

in

Apply

T,

and let

Q

be the cokernel" T(M)

Since

--

T(M')

is left exact,

T

--

--

0

we have an exact sequence

0 -* T ( Q ) --'- T T ( M ' ) --" T T ( M ) But

is isomorphic

TT

T ( Q ) = 0,

Therefore,

to the identity functor by hypothesis. and since

T

is faithful,

Q = 0, i . e . ,

T

i s exact.

(ii) ---> (i). If series above.

s

~..~f

, then

whose quotients are all fields Therefore,

composition Hence

M

series

T(M)

length of

M).

since

T

k

is exact,

M

has a finite composition

of the f o r m T (M)

, as

will have a finite

w h o s e q u o t i e n t s a r e of t h e f o r m

is of f i n i t e l e n g t h

A/4~ti

T ( k ) ~ k.

(in f a c t , i t s l e n g t h i s e q u a l t o t h e

57

For the second assertion where

M

is a field

k

of

(i), w e first c o n s i d e r the c a s e

of t h e f o r m

in some way, and the natural map be z e r o ,

M

k

is easily s e e n not to

-~ T T ( M )

so is the middle one.

(since any

M s

-~ O,

of t h e f i r s t e x a c t s e q u e n c e i n t o

If the two outside arrows

5-1emma,

' and

--- T T ( M " )

with a canonical morphism

the second.

~L

then w e d e d u c e a n exact s e q u e n c e

0 --- T T ( M ' )

M

TT(k) ~

--" M ~" M " "-" 0

is a n exact s e q u e n c e ,

Of

Then

k --" T T ( k )

is any module in

0 -"M'

the

i

h e n c e it is a n i s o m o r p h i s m . Now if

together

A/~

are isomorphism,

then by

T h u s by i n d u c t i o n on the l e n g t h

has finite length), we see that the

~

natural morphism

M

-~ T T ( M )

is an isomorphism

Definition. a maximal

for all

Let

ideal.

M s

A

f ~ ~,~

be a c o m m u t a t i v e

Then a functor

the equivalent conditions of Proposition functor for

A

at

~

Q.E.D.

Noetherian

T z Sex( 4.9,

ring,

~L ~ , Ab),

and

satisfying

is called a dualizing

An injective module

I

with support in

58

V( , ~

)

is called a dualizing module

T = Horn( 9 , I)

Proposition

is a dualizing functor for

4.7,

these dualizing

there

is a natural

functors

ring with maximal or

ideal

,%~

Remarks.

1)

ideal.

A

at

one-to-one

Let

A

,%~

,%~

)

If

be a Noetherian

ring, A

is a local

ideal

~f

the categories

and

at

AM

Indeed,

between

functor"

understood.

Then a dualizing functor for

A~

A

"dualizing

thing as a dualizing functor for the local ring ~

(By

correspondence

, we say simply

with

, if the functor

,~

and dualizing modules.

"dualizing module, "

maximal

for A at

,~

~e~

a

is the same

at its maximal involved are

isomorphic. Z) A-module.

Similarly, Then

is dualizing for categories 3) and

I

A/~

n- m o d u l e

,W~ I

n

I A

~i Let

let

A

be a local ring,

has a natural

structure

and of

I s ~ ~-module,

i f a n d o n l y if i t i s d u a l i z i n g f o r

~.

an and

I

Again the

involved are isomorphic. A

be a Noetherian

and A-module.

For each

n

HOmA (A/~

I).

if and only if for all large is dualizing for

A/~

n

,~

ring,

a maximal

n-

1, 2,

Then

I

is dualizing for

enough

n

(and hence for all

at the maximal

....

ideal

let

I

ideal,

~

n

/

be the A

n

at n), Indeed,

59

observe

that if

HomA(M,I) all

~- H o m f ~ ~

M s

Hom

M s

n

A/

is annihilated

(M, In).

n

Thus

is exact for

A/

HOmA(k,I) ~ Hom

k =A/~

an injective

essential

extension

M r~ N = 0,

then

Definition. is an essential language,

is exact for

I)

n,

n modules

of f i n i t e t y p e .

for any

f o r t h e s a k e of c o n v e n i e n c e ,

An injection

M-~

if whenever

N

n,

where

If

we also call

M

is an

hull,

P

is a submodule

I

where

an injective

of

is an P

(Eckmann-Schopf

such that

is injective.

of

By abuse

[3, C h . II, p . 2 0 f f . ] )

A-module

has an

isomorphism.

A

be a Noetherian

I is dualizing for

A

if and only if it is an injective

field

k.

M

M.

[2]; s e e a l s o Then every

hull

Let

hull of the residue

4.10.

I

an injective

hull of

unique to within (non-unique)

Proposition

of A - m o d u l e s

A-module,

M --~ I

be a ring with identity.

an A-module

the definition of

N = 0.

extension

Theorem.

injective

then

hull.

Definition.

A

,

Thus the result follows from criterion (ii) of Proposition 4.9. We n o w r e c a l l ,

Let

n

~

enough

(k,I) n

A/An

by

HomA(-

if and only if for all large

(M, In)

Moreover,

~f

local ring.

Then

of

6O

Proof. Hom(k,I)

~ k,

show that of

I

First

k --~ I

such that Hom(k,Q)

V( ~

), I

is an essential k ~Q = 0.

s o Q = 0.

= 0.

But

I,

Then any homomorphism because

k --~ I

Hom(k,I)

~

k,

Corollary ideal.

a prime. Z oo'

Then

A

let

(3,

implies

in

~ k,

has that

I.

To

be a submodule

(3

Hom(k,I)

also

I

we must

support I

in

is injective,

into

is an injective I

must

extension,

is a dualizing

A

hull of

k.

have

its image

in

k

is a field.

and

Thus

module.

be a Noetherian

has a dual/zing

ring,

module

at

= pZ

,

and ~

, unique

A =Z

automorphisms

,

andlet

module

for

are

~ A

at

~

where

p

is the group

the units in the ring of

p-adic

P i n t e g e r s.

Let functor.

A

k,

isomorphism

Then a dualizing

whose

k

Let

Let

since

suppose

4. 11.

(non-unique)

Example.

I

k

k.

is an essential and

extension,

dualizing

of

Then

is dualizing.

we can embed

hence

hull of

On the other hand,

to within

I

Then

Now I

is an injective

a maximal

that

and so in particular,

have

so

suppose

be a Noetherian

local ring,

We list without proof various

and

D

properties

of

a dual/zing D,

most

of

is

61

which follow f r o m the above d i s c u s s i o n .

see

For more details,

Matlis [9].* 1.

D

gives an isomorphism _f C~

its dual category gives

a

1-1

DZ Z.

systems,

D(M),

b e t w e e n s u b m o d u l e s of and vice versa,

with

length.

M,

for any

It

and

M e

i s i s o m o r p h i c to the i d e n t i t y .

D transforms

d i r e c t s y s t e m s of

correspondence

~f

into inverse D

T h u s , p a s s i n g to l i m i t s ,

a n d vice versa.

give s a

b e t w e e n d i r e c t l i m i t s a n d i n v e r s e l i m i t s of m o d u l e s

of f i n i t e l e n g t h o v e r 3.

~f

D is exact and preserves

correspondence

q u o t i e n t m o d u l e s of Finally,

O

of t h e c a t e g o r y

A.

In p a r t i c u l a r ,

the functor

D = Horn( ", I),

is a dualizing m o d u l e , gives an a n t i - i s o m o r p h i s m (ACC) = t h e c a t e g o r y of

R-modules

(DCC) = t h e c a t e g o r y of A - m o d u l e s

where

I

of th e c a t e g o r i e s

of f i n i t e t y p e , a n d M satisfying the following

three equivalent conditions: (i) M h a s t h e d e s c e n d i n g c h a i n c o n d i t i o n , (ii)

M i s of c o - f i n i t e t y p e , i . e . , H o m ( k , M ) i s a

finite-dimensional vector space over (iii) M in this correspondence, of

~

is a s u b m o d u l e D(~) = I;

k, a n d of

M

has

In , for s o m e

D(1) = Hom(I,l) = ~.

c o r r e s p o n d to s u b m o d u l e s

of

support

I

in

n. T h e ideals

by %.he m a p

*See also Macdonald [21], w h o e x t e n d s t h e r a n g e i z i n g f u n c t o r to " l i n e a r l y c o m p a c t " m o d u l e s .

of

the

dua i-

62

9,. " ~

D(.~/ Oi, )

, I

a n d the ideals of definition c o r r e s p o n d to the s u b m o d u l e s of

of

finite length.

Proposition

4. l Z .

of l o c a l N o e t h e r i a n

rings,

B.

B --" A

and let

be a s u r j e c t i v e

morphism

be the maximal

i d e a l of

Then 1) If

to

Let

D

A-modules Z) If

is a dualizing functor for

is a dualizing functor for I

is a dualizing module for

is a dualizing module for Proof.

The first statement

HomB(

then

I9

restricted

A. B,

then

I' = H o m B ( A , I )

A.

definition of a dualizing functor the functor

B,

9 , I)

follows immediately

(see Proposition 4.9 (ii)).

is a dualizing functor for

associated dualizing m o d u l e is

from the

lirn H o m B ( A / ( ~ A )

A,

and its

n, I) = I',

n

since any morphism

of

~

A.

of

A

into

I

annihilates

Hence

some power

63

We n o w g i v e t h r e e

specific examples

Example

A

a field

k 0,

extension

of

1.

be a Noetherian

and suppose that the residue k 0.

Then any module

dimensional

vector

functor

by

D

Let

of dualizing functors.

space over

k 0,

local ring containing

field

M s ~

f

k

is a finite

is a finite-

and we can define a dualizing

D(M) = HOmk0(M, k0) The correspondingd u a l i z i n g module is I = lirn Horn

{A/,~,~n, ko ) k0

w h i c h c a n be t h o u g h t of a s the m o d u l e of c o n t i n u o u s h o m o m o r p h i s m s of

A

with the

4~-adic

Example

2.

Definition. residue

field

k,

topology into

k 0.

T h e c a s e of a G o r e n s t e i n

A Noetherian

local ring

is called a Gorenstein

ring. *

A

of d i m e n s i o n

n,

ring if

EXtA(k'A) = I kO for ii/=nn *See also the paper of Bass [13], and [17, ch. V w 9] for more information about Gorenstein rings.

with

64

Examples.

Any regular

local ring is Gorenstein.

Gorenstein

ring is Cohen-Macaulay.

local ring

A

is called a Cohen-Macaulay

its dimension:

See Cor.

Proposition n.

where

Let

A

X-

i ~ n,

~

be a Gorenstein

r i n g of d i m e n s i o n

E x t n (M, A)

and its associated Spec A,

Proof 9

First

and Y = {~t

}

= 0.

is a n exact s e q u e n c e

in

~'k

9

Indeed,

i.

M e

M = k.

'

and for

~

-~ E x t I ( M ' , A )

If t h e t w o o u t s i d e g r o u p s a r e z e r o ,

~2

,

, then there is an exact sequence 9

ExtI(M,A)

~.,,f~

--~ 0

by induction on the length of

for all

M s

if

ExtI(M,,,A) -~ E x t I ( M , A )

for a n y

n

H e , (A) - H y ( X ,

is the closed point.

we show that for any

Exti(M,A)

n

dualizing module is

0 --~ M' --~ M -~ M "

for

if its depth equals

Then the functor

is dualizing,

Thus

ring

that a Noetherian

3.10.)

4. 13.

M

any

(We r e c a l l

Any

M,

,

so is the middle one.

one shows that

= 0

and all

i ~n,

s i n c e by h y p o t h e s i s

it i s t r u e

O'x),

65

It f o l l o w s n o w t h a t t h e f u n c t o r

M

is exact for

~

M s

Extn(M,A)

~~

On the other hand,

hypothesis,

so we have a dualizing functor.

by Theorem

2.. 8, i s

k

I = lirn Extn(A/ ~ k In particular,

Proposition dimension

n.

4.14.

Let

n

A

-- k

by

Its dualizing module,

,A) = H ~

H ~n

we have shown that

Extn(k,A)

(A)

(A)

is an injective

be a Noetherian

Then the following conditions

are

hull of

local ring of

equivalent:

(i) A i s G o r e n s t e i n . (ii) A i s C o h e n - M a c a u l a y ,

Proof.

(i) => (ii)

(ii) => (i). for

i < n,

Since

by Corollary

proof of the previous and

all

i < n.

is left exact,

A

is dualizing.

proposition.

is Cohen-Macaulay,

3.10.

proposition

~

H~(A)

by the previous

Therefore,

T : M

n

and

Therefore, that

Extl(k,A)

we can show as in the

Extl(M,A}

= 0

for all

the functor

Extn(M,A)

and so by Proposition

= 0

4.5 and Theorem

2.8,

k.

66

E x t n ( M , A ) m& Horn (M, H ,n~

n

H ~

But

(A)

was assumed

an exact functor, a n d

to be

Therefore,

dualizing.

T

is

T(k) ~ k.

It r e m a i n s to s h o w that T

(A))

for

Ext1(k,A) = 0

i > n.

Since

is exact, the functor

M

~-': ~ Extn+l (M, A)

is left exact for

M z

~

f

, and so, as above,

E x t n + 1 ( M , A ) ~ H o r n (M, H ,n+ ~ 1 (A))

But

n+l H , ~ (/%) = 0

Extn+l (M, A) = 0

Thus funct

by Proposition I. IZ, since for all

M

z

f vt~

n = d i m (Spec A). , and therefore the

or

M is left exact.

~

P r o c e e d i n g thus by induction on

that

E x t i ( M , A) = 0

A is

Gorenstein.

Exercises. dimension

n,

E x t n + Z (M, A)

for all

i > n

and all

i > n, M s

~.f

one show

s

Thus Q.E.D.

1) and let

Let

A

be a Cohen-Macaulay

local ring of

be an ideal genera ted by a maximal

67

A - sequence.

n

Show that

H~(A)

is an essential

and therefore that the dimension is independent of ~ Z) n

H~

(A)

Let

A

e

3) S h o w that if q u o t i e n t of a r e g u l a r

module,

A

power

series

in

n

namely

connecting

B

~I

we know there

4. 1 4 ( i i ) ,

by an ideal A

~

(i.e., a

which can be

is a Gorenstein

ring.

x ]] b e t h e ring of f o r m a l n of b o t h

n H ~ (A).

So b y t h e u n i q u e n e s s

is some non-canonical

of the

isomorphism

them. in terms

of t h e t h e o r y

be the m o d u l e of continuous differentials of

this case will be the free Let

Show that

This falls under the situation

T h i s f a c t c a n be i n t e r p r e t e d Let

ring.

so we have at hand two dualizing modules,

Hom contk(A,k ) and

~njective hull,

Hom(k,A/~

a n d d e d u c e t h a t in P r o p o s i t i o n

A = k[[x 1 .....

variables.

of the above examples,

exercise.

is a complete intersection

local r i n g

Let

space

,

n H~a (A) injective.

generated by a B-sequence), then

3.

A/~

of

e = 1 for a Gorenstein

be as in the previous

it is sufficient to a s s u m e

Example

of t h e v e c t o r

S h o w also that

is anindecomposable

extension

f n = An(f~l)

called the module

A-module

by the

generated

nth exterior

by

power

of (continuous) n-differentials.

dx of

of differentials.

A/k , l

,

f~l

Then

9

9

9

which in ,

d~"~

9

n

which i s n

is the

68

free A - m o d u l e generated by a dualizing module.

Res:

dx I ^

... A dx n . So

H ~n

(~n)

is

N ow t h e r e s i d u e m a p * i s a c o n t i n u o u s h o m o m o r p h i s m

n

H,~(~n)

-- k

,

canonically defined, which gives rise to a canonical i s o m o r p h i s m

~(~n)

_~, Horn contk(A, k)

between the two dualizing modules.

*The d e f i n i t i o n of the r e s i d u e map is rather subtle. In the t e r m i n o l o g y of [17, ch. VI, w 4], it is the trace map, a p p l i e d to the last term of a residual complex, w h i c h can be i d e n t i f i e d w i t h ~ (~n).

w 5.

S o m e Applications

In t h i s s e c t i o n w e g i v e s o m e a p p l i c a t i o n s developed

so far.

Ext group";

In particular,

various

of d i r e c t i m a g e

p a r t on b a s e c h a n g e s ,

Proposition let

Y

that on

X

> n,

Let

and let

with support in

for all

Y.

X let ~.fy

be a locally Noetherian F

be a c o h e r e n t be the category

n

Proof.

Proposition

First

case

X

localization

X

of c o h e r e n t

such

sheaves

isomorphism,

n

Horn Cx(G, Hy(F))

note that since depth y F

= 0

for all

5.7.

Thus

may not vanish. proposition,

prescheme,

sheaf on

Then there is a functorial

E x t ~ X (G, F) ~

, F)

the

f ,,~y,

G s

_Ext _ ~ ,Iv ( G

especially

w i l l be u s e d i n S e c t i o n 6.

5.1.

--

of b a s e c h a n g e ; a n d t h e c o h e r e n c e

S o m e of t h i s m a t e r i a l ,

be a c l o s e d s u b s e t ,

depthyF

we study "the first non-vanishing

properties

sheaves.

of t h e t h e o r y

i < n, n

To establish

and for all

is the first integer

> n, --

G s

the sheaves

le ~..f y,

for which

the sheaf isomorphism

modules

and show that this definition commutes

to a smaller

affine.

Extn~x(G,F)

of t h e

we make a definition for the corresponding

is affine,

by

We t h e n i n v o k e p r o p o s i t i o n

with Z. Z

in

70

and L e m m a

2.12 to pass f r o m m o d u l e s to sheaves, and glue together

the i s o m o r p h i s m s

w e have defined over the affine pieces of

So suppose that ring

A,

A,

and let

and

N

seen ttRt

an

X

is the p r i m e s p e c t r u m of a Noetherian

Y = V(J), A-module

i ExtA(M,N)

X.

= 0

F=

N,

whose for

where

J-depth

i < n

J

is an ideal in

is

~ n.

and for

M s

Then we have f ~j.

Thus

the functor n

M ~

is left exact,

ExtA(M,N )

and so by Proposition

4.5 there is a canonical

functorial i somorphism

E x t A (M, N)

-~ H e m A (M, I)

where lim__~ E x t A k

I =

Now if B = AS ,

S

is a multiplicative

then the isomorphism

is the localization Ext's

of thisr one,

X

system

in

A,

obtained as above for since localization

and

if

B-modules

commutes

with

and direct limits.

Proposition let

(A/Jk, NI = H (NI

n > 0

be an integer,

such that

points

5.Z.

Let

be a locally Noetherian

and let

Exti~v(G, F) = 0

z s Supp G

X

be coherent

G, F

for

such that depth

i < n. F

z

= n.

Let

7.

Assume

prescheme,

sheaves

on

be t h e s e t of that

Z

71

has no embedded

points.

Then

n

Ass EXt~x(G,F

Lemrna let

Z

5.3.

Let

) = Z

X, Y, F

be the set of points

y z Y

be as in Proposition 5. i, and such that depth

F

= n. Y

Then

A s s Hn(F) C ~ y

and moreover, of

z

points,

is in

if

Z

~

z

t

i s any point of

A s s H~(F).

Z,

then s o m e

In particular, if

Z

specialization

has no e m b e d d e d

then

Hn(F)-- = Z --y

Ass

Proof. be the closure

Let of

z.

z

be any point of

Y,

Then by Proposition

and let

Y'CY

5.1,

n

Ext ~X (

Therefore,

,, F) -~ H o r n

using the expression

of a Horn g i v e n in s u b l e m m a

3.2,

(

,,

(F))

f o r t h e s e t of a s s o c i a t e d we have

H n A s s Ext n~ X ( ~y,, F) = Y' ~ A s s -- Y (F)

primes

72

Now if depth

X, xC

z s Ass F

z

= n,

and observe

and

z s Z.

that depth

F

z

0

(Use Corollary can only increase

3.6 and Proposition

3.7,

when one specializes

z.) S u p p o s e on t h e o t h e r h a n d t h a t

.x n xC

s o i t s s e t of a s s o c i a t e d

non-empty. and also

z ~ Z.

If

z'

z' s Y',

is one of its a s s o c i a t e d i. e. ,

z'

1~ > n,

primes

primes,

is a specialization

Proof of Proposition 5.2. Proposition 3.7, depth

Then

Let

m u s t be then

of

Y = Supp G.

z'

z. T h e n by

and, as in the proof of the lemrna,

we find that Ass Ext

n

'X

But since we assumed applies to show

Ass

Corollary Extn~ (G,F) v X

5.4.

that

(G, F ) = Y r~ A s s H n ( F ) = A s s H n ( F ) . --Y --Y

Z

has no embedded points,

Hn(F) = Z

--y

the 1emma

the proof.

U n d e r the h y p o t h e s e s of P r o p o s i t i o n

has no embedded associated

Next we study the behavior Ext's,

which completes

5. Z,

primes.

of t h e l o c a l c o h o m o l o g y g r o u p s ,

and dualizing functors under base change.

will be useful in Section 6 on duality.

These results

n

A s s --y H (F),

73

Proposition topological

5.5.

spaces; let

-l(y); Y' = f and let

Let Y

f : X' -* X

b e a c o n t i n u o u s m a p of

be a closed subspace

F' be an abelian sheaf on

of

X,

X'.

and let

Then there

is a spectral sequence Epq

Proof. functor Now

p n = Hy(X, Rqf,(F ')) => H~(X', F')

We i n t e r p r e t

l"y o f., f,

where

takes injectives

the functor f.

is the direct image functor.

into injectives,

exact functor

f

theorem

s t a t e d i n S e c t i o n 1,

A,

the spectral

Corollary q > 0,

[5, Ch. 0, 3 . 7 . Z].

sequence

then the spectral

of a c o m p o s i t e

functor.

R q f , ( F ') = 0

sequence degenerates,

of

for

and

H i (X', F')

i. Example.

Then

which gives the existence

If i n t h e p r o p o s i t i o n

Hiy(X, f,F') ~ for all

since it is adjoint to the

Thus we can apply the

of d e r i v e d f u n c t o r s

5.6.

l"y, a s the c o m p o s i t e

y ' = y r-~ X ' ,

supportin X',

Let

X'

be a closed subprescheme

and for any abelian

H cx', FIx'

sheaf

F

on

of X

X. with

74

J all

Corollary

5.7.

be an ideal in

B,

i

Let

B -~ A

and let

M

be a morphism be an

of r i n g s ,

A-module.

let

Then for

we have isomorphisms

Hij(M B) ~_ H i j A ( M ) B

where a superscript considered

B

X = S p e c B.

exact in this case since [5, Ch. I I I . ,

Corollary

Remark. remarkable

"by restriction

We a p p l y t h e p r e v i o u s

X' = S p e c A a n d

of s c a l a r s .

corollary

"

to the case where

an

is

f~

is

affine morphism

1.3.2].

The result

2.8 to express

of C o r o l l a r y

5.7 becomes

more

of N o e t h e r i a n

assumptions

when we can

the groups involved as direct limits

F o r t h e n it s t a t e s t h a t t h e r e is a n i s o m o r p h i s m

lirn E x t Bi ( B / J

n , M B ) - -~ ~

n

which is surprising isomorphic

denotes that module

The direct image functor

f : X' --~ X

in the presence

use Theorem of E x t ' s .

applied to an A-module

as a B-module

Proof.

,

lira EXtA

(A/(JA) n , M)

because the Ext groups are in general

b e f o r e t a k i n g t h e lknit.

For example,

the natural map Ex i

,

n

CB/J , B/J

-- Ext iB.j B/J,

B/J

if

not

A = M = B/J,

75

is in general term

is

not an isomorphism

0,

since the right-hand

However,

shows that the first non-vanishing

Proposition let

i > 0,

and the left-hand one not.

proposition

rings,

for

N

type, and let

5.8.

Let

B --~ A

be a B-module, n

be an integer.

iV[

the following

Ext's will be isomorphic.

be a morphism an

A-module,

of N o e t h e r i a n b o t h of f i n i t e

Then the following conditions are

equivalent:

where

(i)

E x t B~ ( N ,

M B) = 0

(ii)

i ExtA(N A,M) = 0

N A = N: ~ ) B

Furthermore,

for all

i< n

,

for all

i < n

,

A .

if the conditions are

satisfied,

then there

is

an i somorphi sm

n

n

ExtB(N, M B) -~ EXtA (NA, M)

Proof.

Let

J

be a n i d e a l of

T h e n it f o l l o w s t h a t Supp NA = V ( J A ) . Theorem for all i

HjA(M)

3.8, i~ = 0

the

n,

B

such that

Supp N = V(J).

Now by P r o p o s i t i o n

condition (i) is e q u i v a l e n t

to saying

B. 7 a n d

Hij{M B) = 0

and the condition ~ii) is equivalent to saying

for all

i < n.

equivalent by Corollary 5.7.

But these latter two conditions are

76

From

Corollary 5.7 and T h e o r e m

depthjM B = depthjAlVi. then this c o m m o n

3.8 one also deduces that

If the conditions (i) and (ii) are satisfied,

depth is

_> n.

Therefore by Proposition 5.1

there are i s o m o r p h i s m s n

E x t B ( N , M B) -~ H o r n B (N, H

(MB))

and n

,~

ExtA(N A, M)

n

-" H o m A ( N A, HjA(M))

But by Corollary 5.7, again, the modules are isomorphic, hence our if

B --"A

Extn's are isomorphic.

is a morphi.sm of rings; if

presentation; and if

M

H j ( M B)

N

H jnA (M)

and

(In general,

is a B - m o d u l e of finite

is an A - m o d u l e , then there is an i s o m o r p h i s m

H o m B ( N 'M B ) _~ HomA(NA~vi )B

Proposition J D J'

be two ideals

finite type. respect

5.9.

Denote

to the

of

A,

by a hat

J'-adic

A

Let

and let ^

i

topology.

~

-~

H i

^ ( J

M

)A

ring,

let

be an A-module

the operation

i s o m o r phi s m

Hj(M)

be a Noetherian

Then for all

of completion i

there

of with

is an

77

Proof.

Using Theorem

groaps as direct limits

Z. 8 t o r e p r e s e n t

o f E x t ' s,

the local cohomology

we need only show that the natural

maps

EXtA(A/jn

are isomorphisms. is of f i n i t e t y p e ,

But since

9

,

so to t e n s o r

^

i

Ext

Proposition 5. i0.

U.

to

U

Let is

F F 0,

M

,M) (~)A ~

of p r e s c h e m e s , be a p r e s c h e m e ,

and let

F0

be any coherent sheaf on

conditions are equivalent:

so tensoring with

of w h e n t h e d i r e c t i m a g e o f a

X

Let

Y = X - U,

and let

J'

it with

This proves the proposition.

sheaf, under a morphism

an open subset, let

n

since J D

Now we study the problem

on

and since

is the same as to tensor

it with

is an isomorphism.

coherent

A,

g r o u p on the r i g h t i s a n n i h i l a t e d by

= A/J n A

~I )

is flat over

M ) = Ext A ( A / J

On the other hand, the n

2k

( ~ / ~n

we have

Extt (~/~n A

J

M) --. E x t i

n

be an integer.

X

is coherent. let

U

be a c o h e r e n t

be sheaf

whose re striction

Then the following

78

(i)

RJ,(F0)

is coherent

for all

i

<

where

n,

j : U -~ X

is the inclusion map. H~(F)

(ii)

Proof.

is coherent for all

i <

n.

By Corollary I. 9 there is an exact sequence

0--'H~F)---__ F---j,(F0)--I-I!~F)---0

,

and t h e r e a r e i s o m o r p h i s i n s R q j , ( F O) -~ --y H q+l(F)

for

q > O.

kernel

Now the statement

and cokernel

an extension [5,

Ch.

of a map

of a coherent

of the proposition of coherent

sheaves

sheaf by a coherent

follows are

since the

coherent,

and

sheaf is coherent.

O, 5 . 3 ] Moreover,

there

is also the following

result,

which we will

not prove. Theorem. prescheme, equivalent (iii)

[16, Corollary VIII-II-3] A If X is locally embeddable in a non-singular

then conditions

Z

and

to the following

condition:

for all

,

depth whe r e

(i)

z e U

(Fo) z + codim

is the closure

of

z

in

(ii)

(Y r~ Z , X.

of proposition

Z)>n

are also

79

Examples. coherent _> 2

A necessary

is that for all

in

Z,

2) f:X

1)

and let

since for Suppose

"--'Y

be a proper

Rqf0,(F0)

Indeed, f, o j,

z e Ass

j,

Y

where

f to

are

Y

for all

on

U.

let

Noetherian,

Then the i < n.

as the composite

f0~ ~

injectives

satisfied,

is locally

of

carries

into injectives,

functor so there

is

sequence

Now the sheaves

o) are

Rqj,(F 0) E pq Z

Thus the

pr opo sition.

p,

since

f

It f o l l o w s t h a t t h e a b u t m e n t i<

5.12

be the restriction coherent

be

( F 0 ) z = 0.

of proposition

morphism,

j,(F0)

be of codimension

depth

EzPq--

and for all

that

Y :% Z

F 0,

we can interpret

and

a spectral

F 0,

the conditions

f0 : U - " Y

sheave s

z e Ass

condition

coherent

terms

are

for

q

coherent

is a proper m a p

by the

n,

sheaves

for

q < n

[5, Ch. III, T h e o r e m 3.Z. I].

i a f0,(F0)

terms

<

are coherent for

n'. Problem.*

closed

subset.

_~y(F) = 0

Let If

for all

n

equations,

be a prescheme,

is an integer,

i >

It i s s u f f i c i e n t

X

n,

find conditions

Y under

be a which

and for all quasi-coherent

sheaves

Y

by

that

but not necessary.

*This problem

and let

is studied

be locally For

in

describable

example,

[19].

if

X

n

is the cone

F.

80

over a n o n - s i n g u l a r the cone,

plane cubic curve,

a n d if

then the condition is satisfied for

necessarily

Y

n = I,

is a g e n e r a t o r but

Y

of

cannot

be d e s c r i b e d by one equation at the vertex of the cone.

w 6.

Local Duality

In t h i s s e c t i o n w e p r o v e a c e n t r a l cohomology theory, over a regular forms

projective if for

F

the duality theorem.

local ring (Theorem

of t h e d u a l i t y t h e o r e m

These theorems

is a coherent

It i s s i m p l e s t

6.3); however,

valid over more

[10, n ~

Theorem

s h e a f on p r o j e c t i v e

there is a perfect

Hi(X,F)- X

to state

we also give

general

local rings. of S e r r e ' s

1], w h i c h s a y s t h a t

r-space

X = IP r ,

then

pairing

r-i Ext ~ x ( F ,

n ) --" k

of finite-dimensional vector spaces over sheaf of

of t h e l o c a l

s h o u l d b e t h o u g h t of a s l o c a l a n a l o g u e s

duality theorem

i _> 0,

theorem

r-differential f o r m s on

k,

r

where

]P , i s o m o r p h i c to

~

is the ~ x ( - r - I).

E v e n the proof, which uses the axiomatic characterization of derived functors, is similar. W e will apply the duality t h e o r e m to determine the structure of the local cohomolog~r groups

i H~

(M)

over a local ring, and w e

end by giving an application of duality to a t h e o r e m on algebraic varieties.

We b e g i n b y r e c a l l i n g the Ext groups

briefly Yoneda's

(for d e t a i l s a n d p r o o f s ,

interpretation

s e e [7, e x p o s ~ 3]) .

of

82

Let

~

be an abelian category with enough injectives.

A complex in

~,

is a collection

together

with coboundary maps

all

d n+l 9

K

n, by

fn:

K n --. L n + s

such that for

We d e n o t e t h e c o h e m o l o g y

morphisms are

(pn)ns

s

of a c o m p l e x

K

into

L

gn

We d e n o t e b y

elements If complex

f e Horn A K,

S

~

the

in t h e

group of

such that for all

s n+l + (-i) p

n 9

dE

the g r o u p of h o m o t o p y c l a s s e s

of

(K, L).

such that

0

of m o r p h i s m s

ns

Two such morphisms

s.

pn 9

L)

o_ff K

if there exists a collection

n+s-I

i s a n o b j e c t of

c : A -~ K 0,

is exact.

of degree

= dL

HS(K,

(fn)

a morphism

p n : K n --- L n + s - 1

of m o r p h i s r n s

-

,

H o r n S ( K , L)

said to be homotopic

fn

~

with the coboundary maps

We d e n o t e b y

of

in

is a collection

which commute

t w o c o m p l e x e s.

map

d n : K n -~ K n + l

are two complexes

K, L

of d e g r e e

L

f, g

of objects of

Hq(K). If

into

d n = 0.

(Kn)n s

~

Kn = 0

,

a resolution for

n < 0,

of

A

together

is a with a

such that the sequence

A

~

A resolution

K 0 ~

K 1

(K, ~ )

of

A

i s s a i d t o be e x a c t i f

n~,

83

Hq(K)-- = 0 Ki

for

q > 0.

is said to be injective

if each

is injective. Now let

of

The resdution

A,

A

and let

the complex

b e a n o b j e c t of

L

be a n y c o m p l e x .

( H o r n (A, Ln))

we define an element

of

determines

Theorem

~

Corollary. and

L

of

f.

H o r n ( A , L)

f ~ HornS(K, L),

by composing Hem

c

(A, L ) ,

H S ( H o m (A, L ) ) ,

with and

which depends

Thus we have defined a natural

HS(K, L) -" H S ( H o m

E.

a n d if t h e c o m p l e x maps

Given an

be a resolution

s,

cs :

natural

class

of

(K, ~)

We d e n o t e b y

is a cocycle in

an element

only on the homotopy map for each

nzZ"

, let

Horn(A, L) s

One shows that this element hence

~

s

If L

(K, e ) consists

(A, L))

is an exact resolution entirely

of injectives,

of

then the

defined above are isomorphisms. If i n p a r t i c u l a r

i s an e x a c t i n j e c t i v e

B

resolution

is another of

B,

object of

then there are

canonical i somorphi s m s

HS(K, L)--" ExtS(A, B)

Now we apply this theory below.

A,

to the situation we will meet

f0

84

Proposition

6.1.

Let

~ ,

~'

having enough injectives, and let

T:

an additive covariant left-exact functor. of

~

,

be a b e l i a n c a t e g o r i e s ,

If

~"

~'

A

and

be

B

are objects

then there are pairings

RiT(A) X E x t S ( A , B) --- Ri+ST(B)

for all

i, s

Proof.

(L, U)

of

RIT (A)

and

and

Pick exact injective resolutions

A

and

RiT(B)

T (L). If f

then

B,

respectively 9

(K,~)

and

T h e n we c a n c a l c u l a t e

a s t h e c o h o m o l o g y of t h e c o m p l e x e s

is a m o r p h i s m of

T (f) is a m o r p h i s m of

T(K)

K into

into

L

T(L)

of d e g r e e of d e g r e e

T(K) s, s,

w h i c h on p a s s i n g t o c o h o m o l o g y g i v e s a m o r p h i s m

f , : RiT(A) --. Ri+ST(B)

for any c l a s s of

s. f.

One s e e s e a s i l y t h a t

f~

d e p e n d s o n l y on t h e h o m o t o p y

Thus using the isomorphism

of th e c o r o l l a r y a b o v e ,

we h a v e d e f i n e d a pairing

RiT(A) for all

i, s.

X ExtS(A, B) --- R i+s T(B)

85

6. Z.

Corollary ideal, and let

Let

A

be a r i n g , l e t

be A - m o d u l e s .

M,N

~

be an

Then there are pairings

i ( M ) X ExtJ(M'A N} -~ H i+j H~ ~ (N)

for all

i,j. Let

Proof.

of

T

be t h e f u n c t o r

M

~

H 0 (M)

The

H~(M)

for

i > 0

are right derived functors

by C o r o l l a r y Z. 7.

T h e o r e m 6.3. of dimension

n

(Duality) Let

A

(See proposition 4.13),

be a G o r e n s t e i n r i n g let

~

ideal, let I = H n (/%) be a d u a l i z i n g m o d u l e , a n d l e t functor

Horn ( ", I).

Let

M

Hi (M) X E x t A- i ( M , A ) - - ~ I

gives r i s e to i s o m o r p h i s m s

~i

:

i

H ~ ( M ) --~ D(ExtA'I(M,A))

and r

: ExtA -i(M'A)^

be t h e m a x i m a l D

be a m o d u l e of f i n i t e t y p e .

the pairing

(*)

from

A - m o d u l e s into i t s e l f , a n d a p p l y the p r e v i o u s

t h e c a t e g o r y of proposition.

T

-~ D(H~(M)) i

be t h e Then

86

In particular, {A p a i r i n g

if

A

is complete,

L X M -=~ N

L ==~ H o r n ( M , N )

and

isomorphisms.

then

{*)

is a perfect

is said to be perfect

if the maps

M -=" H o r n { L , N )

it induces

are

pairing.

both

)

Proof.

We first

consider

the case

i = n,

and show that

the map n

~n : H ~ (M) -~ D(Hom(M,A))

is an isomorphism. since

H ~ (A) = I = D ( A ) .

H n (M) Hence all

Indeed,

and

is an isomorphism

Considered

D 91

are

by a now familiar

M

Jn

as functors

right-exact

argument,

in

for

M = A,

M,

both

and covariant:

is an isomorphism

n

for

of finite type. To show that the remaining

4.

are

isomorphisms,

we

1

use the axiomatic is exact, derived form

characterization

the functors functors

D(EXtA-1 ( " , A))

of

a connected

of derived

D{Hom{.,

sequence

A)).

functors.

for

i < n

and

are

D

left ~

i H4~ { 9 )

The functors

of functors,

Since

~

is an isomorphism 9 n

Hence,

to show that the other

~

are

isomorphisms,

we need only

i show that the functors

of

H

In fact,

CM), since

i.e, A

that

Hi~ { 9 ~

for

i < n

H i . IP) - 0 for

is Noetherian

are

P

left-derived

p r o j e c t i v e and

and since we are

interested

functors

i < n. only

87

in

is of f i n i t e t y p e ; s i n c e

of f i n i t e t y p e , w e m a y a s s u m e

M

Hi

is additive,

i H ~ (A) = 0

we need only show that

This follows from the fact that

A

~. 1

i.

is an isomorphism

for all

D

i<

n.

Thus

is Cohen-Macaulay.

Applying the dualizing functor

for

to the isomorphisms

~i'

we obtain the isomorphisms

D(~bi): D D ( E X t A "i ( M , A ) ) -~ D(Hi~ (M)) A

But

DD(M) = M

for any module

M

of finite type.

(Indeed,

A

the functors

M ~

exact covariant; isomorphic

and

they are isomorphic

for all

i somor phi sm s

DD(M)

M ~

M

are both right

M = A,

for

hence they are

of f i n i t e t y p e . ) H e n c e we h a v e a l s o t h e

M

~ i" Q.E.D.

We n o w c o n s i d e r

an arbitrary

Noetherian

(subject however to the weak restriction regular

of t h e m o d u l e s

finite type, and are easier define

~

to express

i H~(M),

is the maximal in terms

Ti(M) = D(HL(M)),

where ideal.

of the duals,

where

D

A

t h a t it be a q u o t i e n t o f a

local ring) and prove two propositions

structure

local ring

which describe M

the

is a module of

M o s t of t h e s e p r o p e r t i e s so f o r e a c h

i

we

is some fixed dualizing

88

functor. TO

Then the

Ti

is right exact,

functors

of

T

and the

Ti

for

additive functors

i > 0

in

M;

are the left derived

0 ,

Proposition

6.4.

(quotient of a regular M

are contravariant

be an A-module

1)

Let

A

be a Noetherian

local ring) of arbitrary

dimension,

of f i n i t e t y p e a n d o f d i m e n s i o n

Hi~(M) = 0

and

local ring

Ti(M) = 0

for

and let

n > 0.

Then

i> n ^

Z)

The modules

3) For each

4)

dim

Ti(M)

i,

are

of finite type over

A.

dim^ T i ( M ) < i. A

T n ( M ) = n.

(In p a r t i c u l a r ,

HI

(M) / 0. )

i Proof. A/Ann

M,

1) M

which is of dimension

we may assume HI(M)

= 0

expres sing n

is also a module

that

for

A

elements,

by Theorem

of some dimension is a surjective

r

n.

5.7

Then we see that

either by using Proposition

in terms

For the remaining

Thus by Corollary

is of dimension

i > n

Hi~ (M)

n.

over the local ring

1. lZ , o r b y

of a l i m i t o f K o s z u l c o m p l e x e s

of

Z. 3. statements,

of which

map of Noetherian

A

fix a regular is a quotient.

local rings,

local ring Then

B

B --" A

and using Corollary

5.7

*In fact, the functors T i, w h i c h really have nothing to do w i t h local cohomology, can be interpreted as E x t l ( ' , R ") for a suitable dualizing complex R- on A. See [17, ch V w 2].

89

and Proposition calculated theorem

over 6.3,

Ti(M)

4. lZ w e s e e t h a t A

or over

Thus we may apply the duality

B.

and find

T i (M) ~

Ext r-i(M, B)

To prove

Z)

it is also over

B u t an

Ext

Ti(M)

B,

that since

and hence

of modules

A,

~f

is of finite type over

is of fini~ type over

t~

for all

i.

3),

we note that

and that the dimension over

or

Ti(M)

of

B .

t~

B . so

But it is also a

h e n c e i t i s of f i n i t e t y p e o v e r

T o prove

calculated

M

'

of finite type is again of finite type,

is of finite type over

m o d u l e over

ring,

observe

fi)

-----Ext r'1 ^ B

"

S

A~

is the same whether

A .

is also a regular is the same,

Thus we have reduced

local

whether the problem

to the following: Lernma r;

let

for all

M

6.5.

and

Let

N

be a regular

be modules

l o c a l r i n g of d i m e n s i o n

of finite type over

A.

Then

i,

dim

Proof.

Ext A- i (M, N) < i

To say that an A-module

is to say that for any prime L~

A

i szero.

If

~

~

is sucha

C A

L

i s of d i m e n s i o n

of coheight

prime,

then

A~

> i,

< i

the module is a regular

9O

local ring of d i m e n s i o n homological

dimension

< r - i. is

B y Serre's t h e o r e m

< r - i,

so

N)7 - ExtAr - t T hu s

its g~obal

,N 7 ) - 0

d i m ExtA-~{M, N) < i.

To prove assertion

4) o f t h e p r o p o s i t i o n ,

we have

IB By assertion

1)

non- vanishing that is

Z

which we have already

Ext

group.

Applying

i s j u s t t h e s e t of p o i n t s

n:

since

for any

]B

is regular,

y ~ Spec B .

proved,

this is a first

Proposition

y s S u p p ~4 depth

whose dimension

]By = d i m

In p a r t i c u l a r ,

Z

5. Z, w e f i n d

]By = d i m y

has no embedded

points,

SO

Tn(M)

Ass

which shows that dim

~f = n ,

dim

= Z

T n ( M } = n.

every point of

Z

But furthermore,

is associated

since

to

~f,

so

Z

n.

This proves

A

is the set of points of assertion

5)

ring

of dimension

of the next proposition.

Notation. Noetherian

A s s t /% M

If A,

0t

is an ideal of dimension

we will denote by

0I,

n

n

in a

the intersection

of

91

t h o s e p r i m a r y i d e a l s in a N o e t h e r i a n d e c o m p o s i t i o n of a r e of d i m e n s i o n p r i m e s of

~

( T h e s e p r i m a r y i d e a l s a l l b e l o n g to m i n i m a l

, h e n c e a r e u n i q u e l y d e t e r m i n e d . ) If

of p r i m e i d e a l s of 7.

n.

A,

we w i l l d e n o t e by

w h i c h a r e of d i m e n s i o n Proposition 6.6.

ring

9, which

Z

n

Z

is a set

t h o s e p r i m e s in

n. Let

A

be a c o m p l e t e l o c a l N o e t h e r i a n

( q u o t i e n t of a r e g u l a r l o c a l r i n g ) of a r b i t r a r y d i m e n s i o n ,

and let

M

be a n

A - m o d u l e of f i n i t e t y p e of d i m e n s i o n

n > 0.

Then 5) A s s Tn(M) = (Ass M)n 6) If ~

s A s s Tn(M), t~

w h e r e for any N~

(Tn(M)) = t ~

A-module

N,

over the local ring

then (M) ,

iN)

d e n o t e s t h e l e n g t h of

AI

7) A n n Tn(M) = (Ann M)n 8) T h e r e i s a n a t u r a l m a p :

M

"-~

TnTn(M)

w h o s e k e r n e l i s t h e s e t of e l e m e n t s of dimension

<

n.

M

whose support has

9a

5) w a s a l r e a d y

Proofs. let

~

s Ass Tn(M).

proved above.

To prove

Then,

letting

B

be a complete

r

of which

A

is a quotient,

l o c a l r i n g of d i m e n s i o n

r-n

Tn{M) ~ E x t B

6),

regular

(M, B)

and so Tn(M)~

But b y

5),

~

_--__

EXtrB-?{MI ,

i s of d i m e n s i o n

l o c a l r i n g of d i m e n s i o n

r - n.

n,

B~ )

so

B7

is a regular

H e n c e by P r o p o s i t i o n

4.13,

the

functor r-n

is dualizing,

and so preserves

proves

and also

6)

7),

lengths and annihilators. since the primes

the only ones which could be associated 8)

Tn(M)

by P r o p o s i t i o n

~ = Tn(A).

the natural map

since by x s ker ~.

< n, 5)

are

functor in

M,

so

4.2 there is a canonical isomorphism n

dimension

Ass Tn(M)

Ann Tn(M).

is a left exact contravariant

T (M)

where

to

in

This

-*

By means

~r.

If

x

of t h i s i s o m o r p h i s m

is an element

then any morphism

above,

,

HomA(M,~)

Ass ~

of

M

we can define whose support has

f : M -* ~

i s of p u r e d i m e n s i o n

annihilates n.

Hence

x,

93

G o n v e r s o l F, l e t is of dimension

n.

x

be an element

is a morphism

Or,

equivalently,

letting

by

x,

N

M

(x) / 0,

To show that

to show that there

of

whose

it w i l l b e s u f f i c i e n t

f : M-~

such that

be the submodule

of

where

j : N -~ M

is the canonical

then there

will be an

restricted

to

From

N

generated

map

,

injection,

f 8 Tn(M) = HomA(M,

is not zero,

f(x) ~ 0.

M

it w i l l b e s u f f i c i e n t t o s h o w t h a t t h e n a t u r a l

T n ( j ) : T n ( M ) ---- T n ( N )

support

i.e. ,

is not zero. ~)

For

such that

f

f(x) / 0.

the exact sequence,

M--~ M / N - - ~ 0

j

0-~N

we deduce an exact sequence

T n (M) T AJ) T n

But by dim

3)

above,

Tn(N) = n,

Tn0)

must

dim since

contain an isomorphic of their whose

dimensions.

~ = Tn(A)

only associated

A

4),

andby

Therefore,

n.

T n- 1(M/N)

which is impossible

Tn(N)

is just those elements

ker

would

because of

M

< n.

is a complete

is torsion-free prime

n-l,

is of dimension

c o p y of Hence

If

M/N)

since if it were zero,

support has dimension Example.

then

T n-I(M/N)< N

be non-zero,

-* T n" 1

local domain

of rank

is the zero ideal,

1.

Indeed,

of dimension b y 5),

hence it is torsion-free.

n,

its

94

Moreover,

to be the zero ideal in

taking

j u s t t h e r a n k of = rank

for any

N,

Let

(N)

is

Hence rank

complete be aAlocal ring of dimension

A

be a dualizing functor, and let

D

N .

~

A = 1.

Definition. let

A-module

A,

Then

n,

n

~ = Tn(A) = D(H,m

is called a m o d u l e of dualizing differentials for

(A)). A.

It is determined up to i s o m o r p h i s m . If

Exercise. the map

~

M

8)

of

is a Cohen-Macaulay

above is an isomorphism.

A -~ T n T n ( A ) = H O m A ( ~ , Problem. * Let in

A,

and

M

module,

e)

show that

In particular,

is an isomorphism.

A

be a Noetherian ring,

a m o d u l e of finite type.

J

an ideal

Consider the local

cohomology module s i i n H j I M ) = lirn E x t A ( A / J , M ) n

Find finiteness

statements

assertion

of Proposition

2)

without using duality. projective

limits

Lefschetz

relations

of a n A r t i n - R e e s 6.4.

type,

to generalize

But they should be stated

One should also have commutativity

(cf. t h e i n v a r i a n c e of c o m p l e t i o n s

theorem

of Z a r i s k i ,

with and the

along a subvariety).

*This p r o b l e m is s t u d i e d in [18]. See also [16, e x p o s e XIII] for a m o r e p r e c i s e s t a t e m e n t of the p r o b l e m .

95

Now we c o m e to d u a l i t y t h e o r e m s

for non-regular

local rings.

F o r a c o m p l e t e C o h e n - M a c a u l a y l o c a l r i n g , it w i l l be s u f f i c i e n t to replace

Extn-l(M,A)

Extn-i(M, ~),

where

i n t h e s t a t e m e n t of T h e o r e m ~

6.3

by

i s a m o d u l e of d u a l i z i n g d i f f e r e n t i a l s .

If o u r l o c a l r i n g i s not C o h e n - M a c a u l a y ,

h o w e v e r , t h e r e w i l l be a

f i n i t e n u m b e r of m o d u l e s

...,

to i s o m o r p h i s m

~ = ~0,

~1,

~n

d e t e r m i n e d up

by t h e l o c a l r i n g , a n d w e r e p l a c e

Extn'l(M,A)

in T h e o r e m 6 . 3 by t h e a b u t m e n t of a s p e c t r a l s e q u e n c e w h o s e terms are

ExtP(M, ~q).

Let n

E pq

A

be a c o m p l e t e N o e t h e r i a n l o c a l r i n g of d i m e n s i o n

( q u o t i e n t of a r e g u l a r l o c a l r i n g ) .

t he a s s o c i a t e d d u a l i z i n g f u n c t o r

D,

m o d u l e of d u a l i z i n g d i f f e r e n t i a l s f o r type over

A

by

of r a n k 1

if

A

Z)

Fix a dualizing module I and let A.

~ = D ( H ~ (A))

Then

~

and be a

i s of f i n i t e

of P r o p o s i t i o n 6 . 4 , a n d i s t o r s i o n - f r e e

is a d o m a i n .

Since the functor

r i g h t e x a c t on t h e c a t e g o r y of A - m o d u l e s ,

H n~a~( 9 )

is

t h e r e is a c a n o n i c a l

i somorphi sm

M

(A)

for all

M

of f i n i t e t y p e

n

n (M)

(the u s u a l a r g u m e n t

n

H ~ (~2)----- H ~ ( A ) ( ~ A H ~

n

.

. . ).

(A), I)

Therefore,

96 and we see that there

is a natural

map

(n) -- i C o m p o s i n g this m a p with the pairings of Corollary 6. Z, w e can define pairings for all

i n-i

(*)

Hi~(M) X Ext A

Theorem (where

k > 0

6.7.

i<

The following conditions are equivalent

is an integer): (i)

n-k<

the pairings

= 0

In particular, the pairings A

are perfect

for all

n - k <

for alI

($)

i < n.

are perfect for all

i

if and only

the proof of Theorem

6.3.

is Cohen- Macaulay.

First first consider

Proof. the case

4n

Moreover, ~

n

We m i m i c i = n,

for

(M, ~))

M - A,

since

both sides are right exact covariant is an isomorphism

We

and the morphism

: H n (M) -~ D ( H o m

This is an isomorphism

so

(~c)

n. (ii) H ~ ( A )

if

(M,e)--~I

for all

M

D(~)=

functors

of finite type.

To show that i " H i ~ (M) - - D ( E x t A ' i ( M , ~ ))

DD(H~a (A))= H i (A). in

M,

97

is an isomorphism

for

i H4m ( 9 )

functors

n - k ~ i < n,

n

are left derived functors

that

HiMt(A} = 0

(ii).

That the pairing is perfect for

for

n - k<

the proof of Theorem Conversely, n-

k~

i < n,

H~(A) = 0

assume

n-

( 9 ),

n - k < i < n

condition

k~i<

(i).

i.

e.

our condition

f o l l o w s a s in

follows from Theorem

Therefore,

S. 8, s i n c e

l o c a l r i n g of d i m e n s i o n

of w h i c h

r for

Let

B A

is a quotient.

T h e n by T h e o r e m

_C J).

directly,

be a c o m p l e t e

(we m a y a s s u m e ,

B

A

n.

Instead of proving the theorem 6. S.

I = HomB(A,J)

i < n.

for

for

n.

we deduce it from Theorem

b e a dualizing m o d u l e

H i (A),

Then

if and only if its depth is

Second Proof.

that

H~

This is precisely

n-i E x t A (A, ~ ) = 0

The last statement is Cohen-Macaulay

i < n.

of

6.3.

is dual to

for

we need only show that the

regular Let

J

by Proposition

4. 1Z,

6.3 and Corollary

5.7

t h e r e is a p e r f e c t p a i r i n g

i

{1)

H~(M)

for all If

M

i,

I

and for all

is an A-module,

A-modules,

r-i

X Ext B

M

(M, B) -* J

of f i n i t e t y p e o v e r

then both partners

sub-B-module

of

J

(or

B).

to t h i s p a i r i n g a r e

s o t h e i m a g e o f t h e p a i r i n g w i l l be in

is the largest

A

I.

(In f a c t ,

which is also an A-module.

)

98

Thus to obtain a duality the modules

express

intrinsically

over

There

rings

statement ExtB'I(M,

intrinsic B)

over

in terms

A,

we need only

of things

defined

A.

is a spectral

sequence

associated

with a change

of

[IZ, Ch. XVl, w

E zpq : Ext,(M, Ext,(A, This reduces

the problem

to calculating

which (except for their

numbering)

of

dual to

B,

since they are

B))

i M xtB(,B)

->

the groups

do not depend

H~q(A):

Extq(A,

B),

on the choice

In our situation

we

have 0

for

q < r - n

for

q = r - n

for

q > r

Ext q (A, B)

0 Assuming r - n < q< partially,

condition

(ii) o f t h e t h e o r e m ,

r - n + k.

Therefore,

Extq(A, B) = 0

the spectral

sequence

for degenerates

and we find

Ext rB- i (M,B) ------Ext A-"I(M, ~)

for

i > n - k.

theorem.

Substituting

in

(1)

gives

condition

(i)

of the

99

The converse

(i) - > (ii)

is proved as in the first proof.

N o w we c o m e to t h e m o s t g e n e r a l proof will generalize Theorem dimension ~i

n n-i

= D(H~

whose

duality theorem,

t h e s e c o n d p r o o f of t h e l a s t t h e o r e m .

6.8.

Let

A

be a c o m p l e t e

( q u o t i e n t of a r e g u l a r

(2%.)) for

l o c a l r i n g of

local ring).

i = 0, I, ..., n.

Let

Then there is a spectral

sequence

(M)) Proof. let

B

which

A s i n t h e s e c o n d p r o o f of T h e o r e m

be a c o m p l e t e A

regular

is a quotient.

6.7 above,

l o c a l r i n g of d i m e n s i o n

T h e n by T h e o r e m

r,

of

6.3,

ExtB'i(M,B ) = D(Hi~(M))

for all

i.

Furthermore,

there is a spectral

sequence

i EZPq= ExtAP(M, Extq(A, B)) => E x t B ( M , B )

Since

Extq(A, B)

is dual to

Hr'q(A),

I

0

for q for

Ext B (A, B) = 0

for

we h a v e

i<

r-

n

i = r - n +q i > r

(see above),

I00

Now by s u b s t i t u t i n g a n d j u g g l i n g t h e i n d i c e s one o b t a i n s the s t a t e m e n t of t h e t h e o r e m . Remark. EXtB(h/l, B),

One c a n g i v e a n o t h e r i n t e r p r e t a t i o n

intrinsic over

A,

e x a c t i n j e c t i v e r e s o l u t i o n of Let

KA

be the c o m p l e x

to t h e l e f t .

Then

B

as follows.

Let

K(B)

in t h e c a t e g o r y of

Horn B ( A ; K ( B ) ) ,

of the g r o u p s be a n

B-modules.

shifted

KA

is an injective c om pl e x over , d e t e r m i n e d to w i t h i n h o m o t o p y , a n d

r - n A,

places unique ly

i<0

f

0

for

f~ i f o r (In p a r t i c u l a r ,

KA

if a n d only i f

A

i>0

i s a n i n j e c t i v e r e s o l u t i o n of

is Cohen-h4~caulay. )

~=~0

Now

r-i n-i Ext B (M,B) = H (HOmA(M,

w h i c h i s , by d e f i n i t i o n , t h e h y p e r e x t g r o u p

KA) )

n-i Ext A (M,KA).

A s a n a p p l i c a t i o n of the t h e o r y of d u a l i t y , we p r o v e the following theorem, Theorem

c o n j e c t u r e d by L i c h t e n b a u m .

6.9.

over a field

k,

F

H n ( x , F)

k.

on

X,

Furthermore,

Let

X

of d i m e n s i o n

be a q u a s i - p r o j e c t i v e n.

scheme

Then for any c o h e r e n t sheaf

is a finite-dimensional vector space over

the following conditions are equivalent:

*K A is a d u a l i z i n q [ 1 7 , c h . V w 2].

complex

for

A,

in t h e

sense

of

101

(i)

All irreducible

components

of

X

of dimension

n

are non-proper.

(ii)

Hn(x,F)

(iii) H n ( x , is a very ample

ffx(-m))=

Since

as a locally closed

X

X'

for all quasi-coherent 0

for all

sheaves

m >> 0,

sheaf induced by some projective

Proof.*

Let

= 0

X

is quasi-projective,

subscheme

be its closure

is the restriction

IP r .

where

sheaf

X.

fix(l)

we can embed it space

Then any coherent

of a coherent

on

embedding.

of s o m e p r o j e c t i v e

in

F

F'

on

]pr.

sheaf X',

F

on

and there

an exact sequence

Hn(X ',F') -.Hn(X,F) -~ Hx'-x(n+l "X',F')

The last termis dimensional the middle

zero,

vector term

by Proposition

space over Hn(X, F)

k

1. l Z , a n d t h e f i r s t i s a f i n i t e [10, n~

is also.

Theorem

TI~ proves

1], s o

the first

assertion. (i) of

X.

Let

irreducible Then

=> (ii). Y

X'

be a projective

b e a f i n i t e s e t of c l o s e d p o i n t s ,

component

X' - Y

immediately

Again let

of

X',

also satisfies

t h a t it i s s u f f i c i e n t t o t r e a t

(i),

one from each

X C X' - Y C X'.

and such that condition

completion

a n d one sees

the case

X=X'-Y.

*Another proof, not using local cohomology, has since b e e n given by Kleiman [20], and a third proof, using purely local methods, is g i v e n in [19].

is

10Z

Since

H~

commutes

i t is s u f f i c i e n t t o p r o v e

(ii)

with direct limits for coherent

(Proposition

sheaves

F.

Z. 11)

There is

an exact sequence

n

!

Hy (X , F) ~-~ H n ( X ', F) -" Hn(X, F) -~ 0

where

F

is any coherent

is the same as showing

s h e a f on

~r

X'.

surjective,

To show

H n ( X , F ) = 0,

or equivalently,

that the

map

H O m k ( H n ~ X ',F), k) ~

of the dual vector spaces is injective.

H O m k ( H n (X' F), k)

We i n t e r p r e t

these vector

spaces by m e a n s of local and global duality theorems. be e m b e d d e d in projective space s h e a f of

r-differential

forms

V = pr V.

on

and let

Let

X' be the

Then by Serre's global

duality theorem,

HOmk(Hn(X', F), k)

=e ~

Ext r-n

(F,~)

V For the local cohomology group, o b s e r v e t h a t b y t h e e x c i s i o n formula (Proposition i. 3),

II '

i

. Yi

F):/Ii Hti( .. Yi )

103

where V;

Y = {Yi } ' 4~M i

andwhere

i s the l o c a l r i n g of

ffYi

its maximal ideal, and

Fyi

the s t a l k of

F

Now by the l o c a l d u a l i t y t h e o r e m 6 . 3 , a n d the f a c t t h a t i s a d u a l i z i n g f u n c t o r f o r m o d u l e s with s u p p o r t in

Yi at

in Yi"

HOmk( 9 , k)

V(4Ad.) 1

(See E x a m p l e

3 of

w4) n

!

HOmk{Hy(X , F ) , k )

~

(9 , @'Yi Yi

I I Extr-n

i Moreover

~

) Yi

(and the j u s t i f i c a t i o n of t h i s s t a t e m e n t r e q u i r e s the

e x a m i n a t i o n of the r e s i d u e m a p , a n d of the r e l a t i o n b e t w e e n l o c a l

and globalduality),the transportedmap ~c

I I Ext[-n

(~

i

, ~

Yi

)

~

Yi

Ext r-n(F,~)

0' v

i s the one i n d u c e d on the d e r i v e d f u n c t o r s by the n a t u r a l m a p (for a n y two s h e a v e s

F, G

on

V)

HOm~v(F,G)-~ HOmyy (Fyi, Gyi)-'H~ i To show t h a t

~

is injective, observe that

i s t h e e n d i n g of a s p e c t r a l s e q u e n c e

(Fyi,6yi) Yi Ext

(F, ~ )

(see P r o p o s i t i o n 2. 10) w h o s e

E pq- t e r m i s

EPq= HP(v, EXt~v(F,~)) *The compatibility of local and global duality is proved in [17, ch VIII, Prop. 3.5].

104

But since the support Ext%(F,~)

= 0

of

for

Ext

F

is contained

q<

r-n

r - n,

in

of d i m e n s i o n

n,

so

H0(V, Extr~ n

e )

X'

(F, ~))

--V E x t ~:v' i n ( F , ~ )

Now

to prove, global

is either

or a first

must

be zero.

support

of pure

Ext.

~

In the latter

dimension

at all the points Thus

(ii) < = >

in which case there

non-vanishing

section has

thus if it vanishes

zero,

Yi

under

n

is nothing

case,

any

by Proposition

the map

~ ,

5. Z:

it

is injective.

(iii). O n e implication is obvious.

follows f r o m the fact that for

F

T h e other

coherent a n d for all

m

large

enough, one can find surjections (- m ) P

-~ F -" 0

X

( i i i ) =>

that if then in

X

(i).

One sees

is projective

Hn(X,

~X(-m

V = ]pr

this

that it is sufficient

and irreducible

)) / 0 Hn

easily

for all

m

of dimension >> 0.

to show n

over

Embedding

X

is dual to

But this sheaf m

Ext

is n o n - z e r o by Proposition 5. Z, hence for all

large enough its global sections are non-zero. Q.E.D.

k,

105

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M . A u s l a n d e r and D. Buchsbaurn: "Codirnension and Multiplicity, " Annals of M a t h e m a t i c s , vol. 68 no. 3 (1958) pp. 625-657.

Z.

Ecl~xnann and Schopf: " U e b e r injektive Moduln, " A r c h i v der Mathernatik 4 (1953). P. Gabriel: " D e s Cat4'gories A b e h e n n e s , " B u l l . Math. F r a n c e 90 (1962) 3 2 3 - 4 4 8 .

Soc.

~'

.

R. G o d e m e n t : "Topologie Algebrlque et Th6orie des Faisceaux, " Herrnann, Paris (1958).

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A. Grothendieck: "Sur Quelque Points d'Alg~bre Hornologique, " T o h o k u M a t h e m a t i c a l Journal, vol. IX (1957) pp. IIg-ZZl. A. Grothendieck: "Serninaire d'Alg~bre Hornologique, " m i m e o g r a p h e d notes, Paris (1957). R. Hartshorne: " C o m p l e t e Intersections and G o n n e c t e d n e s s , " A m e r i c a n Journal of M a t h e m a t i c s , 84 (1962) 4 9 7 - 5 0 8 .

.

E. Matlis: "Injective M o d u l e s over Noetherian Rings, Pacific Journal of M a t h e m a t i c s , vol. 8 no. 3 (1958) pp. 511-5Z8.

10.

J.-P. Serre: " F a i s c e a u x A l g ~ b r i q u e s C o h e r e n t s , " A n n a l s of M a t h e m a t i c s , vol. 61 no. Z (1955) pp. 197-Z78.

11.

J. -P. Serre: "Alg~bre Locale--Multiplicit~s, " Springer-Verlag, "l.ecture N o t e s in M a t h e m a t i c s , " n o . l l (1965). S. Caftan and S. Eilenberg: "Hornological Algebra, " Princeton University P r e s s (I 956).

IZ.

106

(Bibliography, 13.

14.

continued)

H. Bass, "On the ubiquity of Gorenstein Math. Zeitschr. 82 (1963) 8-28. / Seminaire Cartan, 1948-49, alg~brique," mimeo notes,

rings, "

"Semlnalre de topologie Secr. Math. Paris.

/

15.

Seminaire Cartan, 1950-51, "Cohomologie des Groupes, Suites spectrales, Faisceaux," mimeo notes, Secr. Math. Paris.

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17.

R. Hartshorne, "Residues and Duality, " Springer Verlag, "Lecture Notes in Mathematics, " no. 20 (1966).

18.

R. Hartshorne, to appear.

19.

R. Hartshorne, "Cohomological varieties," to appear.

20.

S. Kleiman, Amer. Math.

21.

I.G. FMcDonald, "Duality over complete Topology I (1962) 213-235.

I

"Affine duality

9

J

and cofiniteness,"

dimension

of algebraic

n

"On the vanishing of Soc. (to appear).

H

Offsetdruck: Julius Beltz, Weinheim/Bergstr.

(X,F),"

Proc.

local rings,"

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