Residues and Duality: Lecture Notes of a Seminar on the Work of A. Grothendieck, Given at Harvard 1963 64 (Lecture Notes in Mathematics 20)

Lecture Notes in Mathematics An informal series of special lectures, seminars and reports on mathematical topics Edited by A. Dold, Heidelberg and B. Eckmann, Z0rich

20 Robin Hartshorne Junior Fellow, Harvard University

Residues and Duality Lecture Notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64

1966

Springer-Verlag- Berlin 9 Heidelberg-New York

All rights, especiallythat of translation into foreign languages, reserved. It is also forbidden to reproduce this book, either whole or in part, by photomechani~a/ means (photostat, mlcrofi/m and/or microcard)or by other procedUre without written permission from Springer Verlag. @ by Sprlnger-Vedag Berlin 9Heidelberg 1966. Library of Congress Catalog Card Number (~'r~7~6~Printed in Germany. Title No. 7~0.

Preface

In the spring of possibility of duality

1963

I suggested

of my running a seminar at H a r v a r d for coherent

P,

and in his talk to the I n t e r n a t i o n a l

1958 [9],

agreed,

Congress

1963,

[IO] w h i c h were to be the basis

I quote

the summer of

from the preface

"Les presentes

notes

sur les pr~sch~mas. connues

d~s

point d'etre combl6e principe

de la dualit~

mais

d'Alg~bre H o m o l o g i q u e d'ensemble.

Cette

le manque

m'avait

assez d ~ t a i l l ~ e

des M o d u l e s

de fondements

emp~ch~

d'aborder

lacune de fondements

satisfaisant.

coh4rents

de la th~orie

par la those de VERDIER,

possible un expose

for the seminar.

of the pr~notes:

Les idees p r i n c i p a l e s

1959,

notes of

he w r o t e a series

donnent une esquisse

d'une th~orie c o h o m o l o g i q u e

He

of the material,

fill in the details and w r i t e up lecture

of "pr4notes"

redaction

of M a t h e m a t i c i a n s

provide an outline

During

ad~quats

195Z [8],

in

systematically.

the seminar.

m'etaient

on his theory

Bourbaki

but had never been d e v e l o p e d

saying that he w o u l d

if I w o u l d

the

sheaves - - a theory w h i c h had b e e n

hinted at in his talk to the S e m l n a l r e

in

to G r o t h e n d i e g k

une

est sur le

ce qui rend en

II est d ' a i l l e u r s

apparu

depuis qu'il existe

P

des theorles

cohomologiques p

dualite

formellement

tr~s analogues

de

p

a celle developpee

ici

/

dans toutes

sortes d'autres

contextes:

f

sur les espaces analytiques, espaces TATE),

topologiques

topologie

~tale,

avec

modules

le yoga g6n~ral

coherents

sur les

galoisiens

(VERDIER,

sur les schemas munis de leur

corps de classe

me semble une raison assez

.

faisceaux abellens

(VERDIER),

faisceaux de torsion

faisceaux

en tous genres

ser~euse

de la dualite

...

Cela

pour se familiariser dans un cas type,

comme

W 0

la th~orie c o h o m o l o g i q u e

des resldus.

La theorle consiste

pour

l'essentiel

dans des questions !

de variance:

construction

d'un

foncteur

f" et d'un

!

homomorphisme-trace iciest

compliquee

des c o n d i t i o n s attendre. apporter

> id.

et indirecte

aussi g e n e r a l e s

La c o n s t r u c t i o n

des s i m p l i f i c a t i o n s

Lichtenbaum,

pour

substantielles." fall and winter

of David Mumford,

John Fogarty,

sous

qu'on est en droit de s'y

took place in the

the a s s i s t a n c e

donnee

et n'est pas v a l a b l e

II faudra sans doute une idle nouvelle

The seminar with

Rfwf"

and others,

John Tare,

of

1963-64,

Stephen

and gave rise to a

series of six exposes which were circulated to a limited S

0

audience under the title "Seminalre Hartshorne".

The present

notes are a revised, expanded2 and completed version of the previous notes. I would like to take this opportunity to thank all those people who have helped in the course of this work, and in particular A. Grothendieck, who gave continual support and encouragement throughout the whole project.

R.H.

Cambridge~ May

1966

CONTENTS

page

III VI 1

Preface Contents Introduction Chapter I. ~O.

81. 82.

w B4. 85. 86. w

The Derived Category

Introduction Triangulated categories K(A) is triangulated Localization of categories Qis and the derived category Derived functors Examples. Ext and R Hom" Way-out functors an~ isomorphisms

Chapter II. Application to Preschemes 81. Categories of sheaves 82. The derived functors of f. and r 83. The derived functor of Hom" 84. The derived functors of | and f* 85. Relations among the derived functors 86. Compatibilities among the relations of 85 Injective sheaves on a locally noetherian w prescheme Chapter III. w 82. 83. w176 85 . 86. 87. 88. 89. w 811.

Duality for Pro~ective Morphisms

Differentials f~ for a smooth morphism f ~ Recall of the explicit calculations The trace map for projective space The duality theorem for projective space Duality for a finite morphism T~e fundamental local isomorphism f" for embeddable morphisms The residue symbol Trace for projective morphisms Duality for projective morphisms

19 19 20

25 28 35 ~9 62 68

85 85 87 9o 93 lOO 115 12o

137

137 i~5 i~8 154 16o 16~ 176

18~ 195 200 21o

page Chapter IV. 81. w w

Dualizing Complexes and Local Duality

252

Introduction Example: duality for abelian groups Dualizing complexes Uniqueness of the dualizing complex Local cohomology on a prescheme Dualizing functors on a local noetherian ring Local duality Application to dualizing complexes Pointwise dualizing complexes and f~ Gorenstein preschemes Existence of dualizing complexes

252 254 257 266 272

Chapter VI. w 81. 82.

83. 84. 85.

Residual Complexes

275 276 282 286 293 299 302

Introduction Residual complexes F~nctorial properties f" for residual complexes Trace for residual complexes Behavior with respect to certain base changes

Chapter VII. 81. w 83. 84.

215

Local cohomology groups, sheaves, and complexes 215 Depth and the Cousin complex 229 Generalization to complexes 240

Chapter V. 80. 81. 82. w w 85. w w 88. 89. 810.

Local Cohomology

302 304 311 318 335 349

357

The Duality Theorem

Curves over an Artin ring The residue theorem The duality theorem for proper morphisms Smooth morphisms

357 369 37~ 388

Index of Definitions

39~

Index of Notations

396

Bibliography

4oi

Appendix. Foncteur Errata

Cohomologie

~ Support

f!, par P. Deligne

Propre,

et Construction

du 404 422

1

Introduction (

9

%

oSos

~d~w u ~

~vw

- -

The main purpose duality

theorem

of these notes

for cohomology

to a proper m o r p h i s m

preschemes.

Various

curve

such theorems

is the duality X

theorem

I

is to prove a

of locally

sheaves,

noetherian

are already known.

for a non-singular

over an a l g e b r a i c a l l y

{

w~.

Heraclitus

of quasi-coherent

with respect

Typical

~

closed

complete

field k, w h i c h

says

that h~

where

D

= hl(K-D)

is a divisor, hi(D)

,

K is the canonical

= dim k Hi(x,

for any i, and any divisor

D.

divisor,

and

L(D))

(See e.g.

[16,Ch.

II]

for

a proof. ) Various to varieties

attempts were made of higher

out in his report Enriques-Severi

to g e n e r a l i z e

dimension,

and as Zariski

[20], his g e n e r a l i z a t i o n

[19]

is equivalent

this theorem points

of a lemma of

to the statement

that

2

for a normal projective variety h~

group

[FAC

of dimension n over k,

= hn(K-D)

for any divisor D. of Serre

X

w

Thm.

Hl(x,L(-m))

related theorem

This is also equivalent to a theorem ~] on the vanishing

of the cohomology

for m large and L locally free.

[FAC w

deduce on a non-singular

Thm. 3], Zariski

Using a

shows how one can

projective v a r i e t y the formula

hi(D) = hn-i(K_D) for O ~ i ~ n.

In terms of sheaves,

to the fact that the k-vector Hi(X,F)

and

spaces

Hn-i(X,FV|

are dual to each other, w h e r e

F

is a locally free sheaf,

F v is the dual sheaf Hgm(F,~x) , and of n-differentials

on X.

this result corresponds

Serre

same theorem by analytic methods

n w = ~/k

is the sheaf

[15] gives a proof of this for a compact complex

analytic m a n i f o l d X. Grothendieck theorems

[8] gave some generalizations

for non-singular

projective varies

of these and then

3

in [9] announced the general theorem for schemes proper over a field, with arbitrary singularities,

w h i c h is the

subject of the present lecture notes. To motivate the statement of our main theorem, US consider the case of projective algebraically

closed field k.

n space X = IPk

let

over an

Then there is a canonical

isomorphism

(I)

Hn(x,~) = k

where

w = ~/ k

Combining

is the sheaf of n-differentials.

this with the Yoneda pairing

(2)

Hi(X,F)

• Extn-i(F,w) X

------->Hn(X,w)

we obtain a pairing

(3)

Hi(x,F) x ~xt xn-i (F,~)

>k

w h i c h one shows easily to be a perfect pairing expose 12].

This generalizes

[SGA 62,

the statements above, because

for a locally free sheaf F,

Ext n-i (F, w ) = Ext n-i (~x'F v |

= H n-i(x,F ~|

.

Another way of looking at our duality pairing is as an isomorphism

> .os(Ri(x,F), k). Since everything is linear over k, we may introduce a k-vector space G, and have an isomorphism

(5)

xS -i(F, G k.)

> omk(Hi(x,F), G).

Before proceeding further, we must introduce the derived category.

It will be discussed in detail in

Chapter I, but for the moment it will be sufficient to know the following:

For each abelian category A, there

is a category D(A), called the derived category of A, whose objects are complexes of objects of A.

If

F: A

> B

is

an additive functor from one abelian category to another, then under reasonable conditions there is a right derived functor any

RF:

D(A)

> D(B)

w i t h the property that for

X E Ob A, if X denotes also the complex which is

in degree zero, and zero elsewhere, where

RiF

Finally,

X

then Hi(RF(X)) = RiF(X),

is the ordinary i th right derived functor of

if F: A

> B and

G: B ----> C

are two functors

F.

5

then

R(GoF)

spectral

= RGoRF

sequence

.

This replaces

of a composite

functor.

Now we can jazz up our duality as follows. X = ~ yn . D(Y)

We replace

We consider

derived f: X

of

of the direct

> Y is the projection.

Y,

categories

~X-mOdules

Then c o h o m o l o g y

functor

for projective

by a prescheme

the derived

of the categories

respectively.

k

the o l d - f a s h i o n e d

Hi image

and

space

so that D(X)

and

~y-modules,

becomes

~f.,

funntor

f., w h e r e

Ext b e c o m e s

the

the derived

!

functor

R Horn of Hom.

G 6 D(Y),

We define

and we replace

Then the isomorphism

f'(G) = f*(G)~w,

F by a complex

for

of sheaves

F 6 D(X).

(i) gives us an isomorphism

!

(6)

_Rf~f'G

~

>G

w h i c h we call the trace map.

The Yoneda

pairing

reappears

as a natural map !

(7) which,

~ Ho~(F, ~'G)

J

> =R ~o~(Rf~,= Rf~f'Q)= ,

c o m p o s e d w i t h the trace map

(6) gives us the duality

morphism !

(8)

~ HO~x(F,f'G)

> ~ Ho~(=RfJ, G)

6

which generalizes isomorphism Y,F,G.

(5).

This is easily proved to be an

([III 5.1] below)

under suitable hypotheses

on

In fact, the proof is nothing but "general nonsense"

once one has the isomorphism

(4).

Having examined the case of projective

space, we can

state the following ideal theorem, which is the primum mobile of these notes, although it may never appear explicitly in this form. Ideal Theorem.

(a)

finite type of preschemes,

For every morphism

f: X

>Y

of

there is a functor

I

f':

D(Y)

> D(X)

such that I)

if l

then

(gf)"

2)

if

> Z is a second morphism of finite type,

g: Y l

|

= f-g"

f

is a smooth morphism,

then

$

f'(G) = f*(G) e w , n w = ~/y

where 3)

if

is the sheaf of highest order differentials is a finite morphism,

f

then

|

f'(G) = HOm~y(f.~v,G) ~ ~

.

7

(b)

f: X

For every proper morphism

> Y of preschemes,

there is a trace morphism !

Trf:

~f~f"

of functors from

D(Y)

I)

g: Y

if

9 > id

to

D(Y)

such that

> Z is a second proper morphism, then

Trgf = TrgTrf . n if X = ~ y ,

2)

then

the canonical isomorphism 3)

if

f

Trf

is the map deduced from

Rnf~(w) ~

~y

is a finite morphism, then

Trf

is obtained

from the natural map "evaluation at one" H~

c)

If

>G

( f.~x ,G )

f: X

.

> Y is a proper morphism, then the duality

morphism !

f

R= O x(F,f'G)

obtained by composing the natural map (7) above with an isomorphism for

F 6 D(X) and G 6 D(Y).

Tr

f,

is

8

It should be noted that we have deliberately left certain technical details out of the above statement, ease of reading.

for

Thus it seems reasonable to make these

statements only for complexes of quasi-coherent

sheaves, or

rather for complexes of arbitrary sheaves, whose cohomology sheaves are quasi-coherent. Dqc(Y) or Dqc(X ).)

(We denote this category by

In fact, we give an example in the case

of a finite morphism to show that the duality theorem (c) fails if

G

[III 6.7]).

is not quasi-coherent

(see example following

Secondly one must expect certain boundedness

conditions on the complexes involved, namely F should be bounded above (we write F 6 Dqc(X)) and G should be bounded below (G 6 D + (Y)) for the duality theorem. qc

Finally, questions

of variance will be very important in the proof, and so the equals signs in al and bl must be taken with a grain of salt. We must preserve very carefully the distinction between "equals" and "is canonically isomorphic to".

Hence we will

have more precise statements below. As to proving the ideal theorem, we succeed only partially.

Certainly the conditions under which we can prove

it will suffice for most applications, but it is unsatisfying

9

to have restrictive hypotheses which are apparently not essential to the truth of the theorem.

I mention four sets of hypotheses

under which the theorem can be proved. (i)

For the category of noetherian preschemes of finite

Krull dimension, and morphisms

f: X

> Y

which can be

factored through a suitable projective space al-3 c

for

for

we have

D + (Y) [III 8.7], bl-3 for D + (Y) [III 10.5], qc qc

F 6 Dqc(X)

(ii)

N

~y,

and

and

G 6 D+qc(Y) [III ll.1].

For the category of noetherian preschemes which admit

dualizing complexes (see [V w

this implies in particular

that the preschemes have finite Krull dimension) and morphisms whose fibres are of bounded dimension, we have for

D~(Y), and

the subscript

c "c"

for

F 6 D[qc(X), G 6 D~(Y)

al-3 and [Vll 3.4].

bl-3 Here

denotes complexes with coherent cohomology

sheaves. (iii)

For the category of noetherian preschemes of finite

Krull dimension and smooth morphisms, we have bl-3 for G 6 D b (Y), and qc [VII 4.3].

c

for

F 6 D- (X) qc

al-3 for and

D + (Y) qc

G 6 qDDc(Y)

Here the exponent "b" denotes complexes which are

bounded in both directions, i.e., finite.

10

(iv)

Recently P. Deligne has shown (unpublished)

for the category of noetherian preschemes,

that

one can construct

I

f" and Trf satisfying and

G 6 D+(Qco(Y)),

al,bl, and c, working with F 6 D(Qco(X)) the derived categories of the categories

of quasi-coherent sheaves on X and Y, respectively.

The principal difficulty has been the lack of a suitable i

construction of the functor it locally, and then glue.

f'.

Our procedure is to define

For a finite or a smooth morphism,

we have it given to us, by a2 and a3 [IIl w167

Thus by

composition we can obtain it for any morphism which can be factored into a finite morphism followed by a smooth morphism [III w

However,

the derived category is not a local object, I

so we cannot glue these local determinations of f" to obtain a global one.

W e resort to a clumsy, round-about procedure I

of defining f" for a special class of complexes, complexes

[VI w

called residual

and then pulling ourselves up by our boot-

straps to get it for arbitrary complexes

[VII w

But our

result has the unpleasant hypotheses of (ii) above. I

Deligne's construction of f" is entirely different,

and

proceeds essentially by representing the functor (for F 6 D(X))

F

,

> R

HOrny (Rf.F,G).

ii

This approach has the advantage of giving the duality theorem !

immediately. locally on X.

He also has a method for calculating f'G However~

it is not immediately clear from his

construction that the properties

a2,3 and b2,3 hold.

In fact,

their proof will probably require some knowledge of the duality theorem as we have proved it here.

At least one can hope that

some combination of the two approaches will provide substantial simplifications of the theory at a later date. A second difficulty, which lurks on the fringes of these notes, is that the derived category seems to be a little too big when it comes to unbounded complexes. can have two unequal morphisms A is an abelian category)

f,g:

X

For example, one > Y in D+(A)

(where

such that their restrictions to each

truncation of the complex X to a bounded complex, are equal. This gives rise to some trouble with unbounded complexes

(see

the problem after [II 5.7], the boundedness hypotheses in [IV 3.1] and [VII 4.3a], and the remark following Perhaps one will have to replace

D+

[VI

1.1c]).

by the categories ind D b

and pro D b, which however may not be triangulated categories~

12

Thirdly,

some discussion of our noetherian hypotheses

is in order.

In the present state of the theory, the noetherian

hypotheses are well entrenched.

We have used them in [II w

for the structure of the injective objects in the category of sheaves on a locally noetherian prescheme, and its consequence that of categories.

D+(Qco(X))

> D~c(X )

is an equivalence

We have used them in the construction of the

trace map for projective space [III w

in the construction

!

of

f"

for a finite morphism

dualizing complexes complexes

[VI w

[III w

in the whole theory of

[V ~2], in the definition of residual and so forth.

Our finite Krull dimension

hypotheses are often needed only to make possible the definition of ~f~ for unbounded complexes

~

a problem which will disappear

when the second difficulty above is solved, and the relation between D(Qco(X)) and Dqc(X ) is better understood.

It is certainly

clear that our methods of proof rely heavily on noetherian hypotheses.

I expect, however,

that once a suitable statement

is obtained,

e.g., in case (iii) above, one could expect to prove l

the theorem without noetherian hypotheses, by reducing to the noetherian case.

More satisfactory,

of course, would be a

13

treatment where eliminated

noetherian

hypotheses

on the base w e r e

from the proofs as w e l l as from the statements.

W h e n that is achieved, duality t h e o r e m

it w i l l be r e a s o n a b l e

for a proper m o r p h i s m

fibres are n o e t h e r i a n

schemes...

of r i n g e d topos, w h o s e

At the present,

this must remain a dream of things

Chapter

of the

I gives the language of derived categories,

the d e f i n i t i o n

needed

description

of these notes.

is used c o n t i n u a l l y

essential,

however,

to come.

Now we w i l l give the reader a b r i e f organization

to state the

in the sequel.

of derived

while

in proofs

sections later on.

categories

Sections

1-5, c o n t a i n i n g

and derived

6 and ? are r e f i n e m e n t s This chapter

treatment

of the subject,

geometry,

so can be used i n d e p e n d e n t l y

functors, w h i c h are

to a l g e b r a i c

as an i n t r o d u c t i o n

Sections

notion of localizing

and the c o r r e s p o n d i n g

DA,(A),

of Verdier,

etc.)

are taken almost w i t h o u t

and should appear

to

1-6 (except for the

the notion of d e r i v e d category.

KA,(A),

are

is a s e l f - c o n t a i n e d

and makes no r e f e r e n c e

subcategory

which

in his thesis

change

[18].

categories from notes

14

Chapter applications category

II is a fairly systematic of the language

relations

, H __o m

, | , f*

There

functors,

However,

are needed.

of i n j e c t i v e

p r e s c h e m e X,

of certain

indecomposable

points x s p e c i a l i z i n g

them,

i n v o l v e d here, of k n o w n

new material.

~X-mOdules

morphisms,

on a l o c a l l y

injectives

J(x,x'),

sums

for pairs of

This extends r e s u l t s

of

and if the reader

be noted that the following

chapters,

for the proof of the general also give much new insight

sheaves.

e v e r y t h i n g we can say about d u a l i t y cares only for

he may stop at the end of this chapter.

p r o j e c t i v e morphism.

Here we

[5] for the case of q u a s i - c o h e r e n t

III contains

for p r o j e c t i v e

such as a s s o c i a t i v i t y

showing that they are all direct

to x' of X.

[13] and G a b r i e l

Chapter

and

some care is taken to see w h a t h y p o t h e s e s

noetherian

Matlis

the functors

into a new language

O n l y section 7 is notable

give the structure

to the

functors,

is r e a l l y no new m a t h e m a t i c s

since it is m e r e l y a t r a n s l a t i o n results.

We consider

, their d e r i v e d

b e t w e e n these derived

formulae.

of the

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

of sheaves on a prescheme.

F , f. , Hom

treatment

besides

d u a l i t y theorem

However,

it should

giving us the tools in C h a p t e r VII,

into the nature of d u a l i t y

for a

15

There are so many situations in which we have a functor |

f" like the one mentioned in the ideal theorem above, that we use different notations for them.

Thus we have

smooth morphism in section 2,

for a finite morphism in

f~

f~

for a

!

section

6,

and

f" for an embeddable morphism in section 8.

Later we will also have

fy

fz [VI w

and

f~[Vl w

for

residual complexes. In sections 3, A, and 5 we recall the explicit calculations of cohomology for projective space, and give the old duality for projective space in the new language of derived categories. Sections 6 and ? give the corresponding formalism for finite morphisms, and its relation to the case of smooth morphisms,

so

that in sections 8, iO, and ll we can prove the ideal duality theorem for projective morphisms.

Section 9 gives the formalism

of a residue symbol w h i c h generalizes the classical residue of a differential on a curve.

Even over the complex numbers, this

important concept of residue for varieties of dimension greater than one was not k n o w n before. In Chapter IV we study "local cohomology",

or cohomology

with restricted supports, of abelian sheaves on a locally noetherian topological space, generalizing results of [LC w167

In particular, we discuss various cohomological

16

properties which a sheaf or complex of sheaves may have with respect to certain families of supports.

This gives rise to

the notions of depth, Cousin complex, Cohen-Macaulay complex, and Gorenstein complex.

The results of this chapter are

independent of all other chapters of these notes, and so may be of use elsewhere, although their only application for the moment is to the theory of duality on preschemes. Chapter V discusses dualizing complexes

(read section 0

to find out what one is) and gives a duality theorem for modules over a local ring, generalizing results of [LC w167 In particular, the dualizing functor

D = R Hom( 9 ,R') treated

here will be useful for our bootstrap operation |

(construction |

of f') later, because it interchanges | and Hom , f" and ~fe , and commutes with

Rf~

(duality theorem:).

In Chapter VI we prepare for the final duality theorem !

by giving the construction of f" and Trf for residual complexes. This is accomplished by a delicate glueing procedure which is the most difficult part of the theory, some detail.

so we have given it in

Perhaps some day this type of construction will

be done more elegantly using the language of fibred categories and results of Giraud's thesis

[6].

17

Chapter VII contains residue

theorem,

which generalizes

the sum of the residues and w h i c h

two m a i n results.

the proof of the duality

theorem

now that all the functorial more than putting

It remains

the classical

of a d i f f e r e n t i a l

is proved by r e d u c t i o n

together

The first is the theorem that

on a curve

to that case.

is zero,

The second is

for a proper morphism,

machinery the pieces

to give the reader

has been

set up, is little

of a Jigsaw puzzle.

some p e r s p e c t i v e

some topics w h i c h have a logical place

which,

by listing

in these notes,

but

w h i c h are not here. i.

The c o h o m o l o g y

be proper

class a s s o c i a t e d

and smooth of dimension

it is shown how to a s s o c i a t e of codimension

n over a field k.

to each n o n - s i n g u l a r

p of X, a c o h o m o l o g y

Px(Y) c HP(x,

the remarks

One defines

> k is the projection.

-y

subvariety

Wy = H-n+P( g :k )

Then there |

ExtS"+P(wv , _

X

In [8, w

subvariety Y

)

for an a r b i t r a r y

g: Y

Let

class

N o w this can be done in [9].

to a cycle.

g'k) .

of X, using , where

is a c a n o n i c a l

element

18

One defines a natural map n-p

%Ik

>

'

which, together w i t h ~ and the construction of [8, w the cohomology class Px(Y).

gives

One proves the fundamental theorem

that formation of the cohomology class associated to a cycle takes an intersection of cycles into the cup-product of their cohomology classes. 2.

The theory of Poincare duality and the Gysin homomorphism

can be developed as in [8, w 3.

A Lefschetz-Verdier

fixed point formula for coherent

sheaves on a scheme proper over a field, to generalize [21, Thm. 2].

In particular,

the determination of the local

contribution at a non-simple fixed point presents an interesting topic for future investigation. 4.

The still lacking theory of duality for complexes

with differential operators as boundary operator, ordinary singular

its ties w i t h

homology theory and w i t h vector bundles with

integrable connections,

as suggested in [22].

A remark on references:

Theorem 5.1 of Chapter III is

referred to as "Theorem 5.1" in Chapter III, and as [III 5.1] elsewhere.

References to the bibliography at the end are given

by square brackets with an arabic numeral or some capital letters e.g.,

[14,(31.I)]

or [EGA III 2.1.12].

19

CHAPTER

w

THE

DERIVED

CATEGORY

Introduction. Let

A

and

be a functor.

B

be abelian

Our purpose

X E O b A , a complex right

derived

we will

D(A)

The derived considers elements classes

of

F

cohomology

acting

A,

to the derived D(A)

A, and w h o s e m o r p h i s m s

of m o r p h i s m s

of complexes.

by "localizing"

induces

K(A),

an isomorphism

This

X"

process

as

objects

for each

are the In general,

of objects

D(B)

follows:

of

The category

one

first of

equivalence D(A)

so that every morphism

of localization

A.

B.

are complexes

are h o m o t o p y

on cohomology,

of

from the derived

category

is obtained

K(A), whose

groups

X, RiF[X)." "

on

w i l l be a functor

the category

D(A).

whose

and let F: A --->B

functorially

for any complex

RF

category

of

obtained which

RF(X')

precisely,

category

of

categories,

is to define,

RF(X),

functors

define

More

in

I.

becomes

is in

K(A)

an isomorphism

w i l l be explained

in general

below.

A ithough categories, abelian. enough

A

information

categories.

B

above are a s s u m e d

the categories

However,

triangulated

and

K(A),

D(A)...

to be abelian are in general

they can be given a structure for our purposes,

category.

which

namely a structure

not

carries of

Thus we will be led to study t r i a n g u l a t e d

20

~i.

Triangulated

Categories.

Definition.

A triangulated

category

is an additive

category C, together with a)

an automorphism

called the translation b)

a collection

triangles C, and w: Z

of C,

u,v,w >T(X).

T: C

functor,

> C and

(X,Y,Z,u,v,w),

of sextuples

where

of the category

in each sextuple,

are morphisms A triangle

as follows:

called the

X,Y,Z

are objects of

u: X

> Y,

v: Y

> Z,

is usually written

W/u Z

X

A morphism

of triangles

is a commutative

x

u

X'

U

a

>Y

(X,Y,Z,u,v,w)

> (X',Y' ,z',u',v',w' )

diagram

> y

v

> y.

V'

> z

w

W

> T(X)

0

>TCX')

This data is subject to the following axioms:

21

(TRI)

Every sextuple

to a triangle,

(X,Y,Z,u,v,w)

is a triangle.

imbedded in a triangle

as above, isomorphic

Every morphism

(X,Y,Z,u,v,w).

u: X ---->Y

can be

The sextuple (X,X,O,i~,O,O)

is a triangle. (TR2)

(X,Y,Z,u,v,w)

(Y,z,T(X),v,w,-T(u)) (TR3)

is a triangle if and only if

is.

Given two triangles

(X',Y',Z',u',v',w'), commuting with

u,u',

(X,Y,Z,u,v,w)

and morphisms

f: X

there exists a morphism

(not necessarily unique:)

so that

(f,g,h)

the first triangle into the second. (TR4)

> X',

(The octohedral axiom).

y

ZI

(

X

vu

Y

!

"~X'

and g: Y h: Z

>y, >Z'

is a morphism of

22

Suppose

given

triangles

(X, Y, Z', u, j, .) (Y, Z, X', V,

., i)

(X, Z, Y', vu, ., .) Then there

exist m o r p h i s m s

f: Z'

> Y'

and

g. Y'

> X'

9

such that (Z', Y', X',

f, g, T(j)i)

is a triangle,

and the two other

f,g

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

as edges,

Definition. triangulated

An additive

category

if it commutes w i t h into triangles. triangles functor

with

faces of the octohedron

diagrams. functor

to another

A contravariant

F: C --->C'

is called

the t r a n s l a t i o n

the arrows

with

reversed,

a (covariant)

functor

~-functor

from one

and takes takes

~-functor

triangles

triangles

into

and sends the translation

into its inverse 9

Definition. triangulated c ovariant

An additive

category

functor

to an abelian

cohomologica ! functor,

a triangle,

the long sequence

H: C

category

if whenever

> A

from a

is called

a

(X,Y,Z,u,v,w)

is

23

9 ..

is exact

> H(TiX)

(the morphisms being

cohomological

>H(Ti+Ix)

> H(Tiy) ----> H(TiZ)

H(Tiu)

etc.).

functor, we often write

Hi(x)

One defines a contravariant

cohomological

If

H

for

~

. . .

is a H(TiX),

i 6 ~..

functor by reversing

the arrows. Proposition morphisms b) then

then

h

The composition

of any two consecutive

in a triangle is zero. If

C

HOmc(M," c)

a)

1.1.

is a t r i a n g u l a t e d

)

and

HOmC(.,M )

category,

M

and

are cohomological

If in the situation of (TR3)

f

and

g

an object of C, functors on

C.

are isomorphisms,

is also an isomorphism.

Proof.

a)

Let

(X,Y,Z,u,v,w)

sufficient to show that is a triangle.

(TR3) to the maps there is a map follows that

Also by

vu = O.

By (TRI),

h: T(X) --->O

and

By (TR2) it is

(Y,Z,T(X),v,w,-T(u)) is a triangle.

(Z,Z,O,idz,O,O)

v: Y ----> Z

T(v)(-T(u))

be a triangle.

>9Z ,

idz: Z

We apply

and conclude that

giving a morphism of triangles.

= O, or, since

T

It

is an automorphism,

vu = O. b) show

Let

M 60b

HornC (M, 9 )

C,

and let

(X,Y,Z,u,v,w)

is a cohomological

by (TR2) to show the sequence

functor,

be a triangle.

To

it will be sufficient

24

~Omc(M,x) -----~HOmC(M,X) is exact. given

By a) we know the composition

g 6 HOmc(M,Y ) such that

(TR3) to the triangles the map an

> "OmC(M,Z)

g: M

f: M

>Y

>X

and

O: O

such that

> Z

c)

(X,Y,Z,u,v,w)

and

and conclude that there exists

HOmC(',M ) is a (contravariant)

functor.

In the situation of

isomorphisms.

(TR3) suppose that

We apply the cohomological

the whole situation,

Hom(Z' ,X)

and

We apply

uf = g.

A similar proof shows that cohomological

is zero.

vg E HOmc(M,Z)

(M,M,O,i~,O,O)

So suppose

is zero.

functor

f

and

g

are

HOmc(Z' , .)

and obtain an exact commutative

to

diagram

>Hom(Z',Y) >Hom(Z',Z) >Hom(Z',T(X))--->Hom(Z',T(Y)) Ig.

~h.

l,(f).

~T(g).

Hom(Z',X') > Hom(Z'~' ) > Hom(Z',Z') > Hom(Z',T~'))--~Hom(Z',~(Y' )) where

f~ = Hom(Z' ,f) etc.

in C, it follows that abelian groups.

Now since

f~,g~,T(f)~,

f

and T(g)~

Hence by the five-lemma,

We conclude that there exists a

and

h~

g

are isomorphisms

are isomorphisms

of

is an isomorphism.

~ E HOmc(Z',Z)

such that

h~(~) = h~ is idz. ~ Hom(Z',Z'). Similarly using the cohomological there is a = #

and

# 6 Hom(Z',Z) h

such that

is an isomorphism.

functor HOmC(',Z ) ~h = idz,.

we find

It follows that

25

w

K(A) Let

is triangulated. A

be an abelian category.

is a collection dn:

Xn

X" = (X n)

> X n+l

f of complexes

n6~

such that X"

to

Y"

A complex of objects of

of objects of

A

A, together with maps

dn%Id n = 0 for all

n 6 Z.

is a c o l l e c t i o n of maps

fn:

A morphism Xn

~>yn

which commute with the maps of complexes: f n+l_ndx for all n.

d~fn

=

T w o maps

f,g: X" --->Y"

there is a collection of maps do not necessarily fn _ gn

for all n.

are said to be h o m o t o p i c if

k = (kn),

k n: X n ---> yn-I

(which

commute with d) such that =

~-~n

+ kn+l~

H o m o t o p y is an equivalence relation,

and the compositions

of homotopic maps are homotopic. We define

K(A)

to be the category w h o s e objects are complexes

of objects of A, and w h o s e morphisms are h o m o t o p y equivalence of morphisms if

Xn = O

of

K(A)

define

of complexes. for

n X"

is homotopic to O.

as described above.

Indeed, the mapping cone is

The matrix

27

01

k = dx

is a h o m o t o p y

operator.

Definition. which

of c o m p l e x e s for

W e define

takes a c o m p l e x

ker d%~im d -I

HoT l

X"

H

into its 0 th

functor

for any

that

K(A)

group,

to

A

namely

because homotopic We w r i t e

maps Hi

i 6 ~.

H

is a c o h o m o l o g i c a l to check

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

from

cohomology

(This is indeed a functor,

it is s u f f i c i e n t

is exact.

to be the

induce the same map on cohomology.)

Observe Indeed,

O/

functor

the long exact

cylinder

from

sequence

of a m o r p h i s m

and there one can check d i r e c t l y

K(A)

u: X"

to

A.

for t r i a n g l e s .>Y"

that the sequence

28

w

Localization

of Categories.

Definition.

Let

arrows

of

C

C

be a category.

A collection

is called a m u l t i p l i c a t i v e

system

S

of

if it satisfies

/

the

following (FRI)

X E ObC,

axioms

If i~

(FR2)

(FRI)-(FR3):

f,g E S, and

fg

exists,

then

fg E S.

For any

E S. A n y diagram Z

Is u X

with

s E S

>Y

can be completed v

W

u

X

with

t E S.

Ditto

to a commutative

diagram

>Z

>Y

for the opposed

statement

(i.e., w i t h all

arrows reversed). (FR3) conditions

If

f,g: X

>Y

are m o r p h i s m s

in

C, the following

are equivalent:

(i)

There

exists an

s: Y

>Y'

(ii)

There

exists a

t: X ' - - > X

in S such that

sf = sg.

in S such that

ft = gt.

29

If

Definition. morphisms

of

is a c a t e g o r y a)

Q(s)

b)

Any

C

is a category,

C, then the l o c a l i c a t i o n CS,

functor

F:

C

for all s E S factors u n i q u e l y Remark. hypotheses

S

such that

such that

F(s)

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

t h r o u g h Q. exists w i t h o u t

on S, but we w i l l not need this result.

system in C.

3.1.

Let

C

be a category,

and

Then we can obtain the l o c a l i z a t i o n

Ob C s = Ob C, and

for a n y

HOmcs(X,Y ) =

Ix

to

of

s 6 S, and

One can show that such a l o c a l i z a t i o n

Proposition

where

Q: C ---->C s

for every

> D

a collection with respect

C

of

together w i t h a functor

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

S

and

li~ Ix

X,Y

60b

X'

f

X

Cs

a multiplicative as follows:

C,

HOmc(X',Y )

is the c a t e g o r y w h o s e objects

in S, and w h o s e m o r p h i s m s

S

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

> X"

are m o r p h i s m s diagrams

s: X' --->X

30

Furthermore, Proof.

if

system

observe,

IX

satisfies

I, w

using

for t a k i n g

limits.

category, (FRI),

so is

(FR2),

the a x i o m s

and hence behaves

is r e p r e s e n t e d

CS

is an a d d i t i v e

First

the c a t e g o r y [GT, Ch.

C

LI,

as w e l l

Thus a morphism

C s.

and

L2,

(FR3)

that

L3 of

as an i n d u c t i v e X

of

to

Y

in

by a diagram X' X

with

s E S.

one with

This

diagram

Y

defines

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

as a n o t h e r

t 6 S X~i

X

Y

if a n d o n l y if t h e r e is a m o r p h i s m morphisms

f: X"'

sf = u = tg

and

To compose X

> X'

and

u: X'"

g: X " '

>X"

> X

morphisms y!

and X

Y

Y

S

s u c h that

a f = bg.

!

in

Z

and

31

we use

(FR2)

to find a c o m m u t a t i v e

diagram

X"

ty X'

Y'

X

with

Y

t' 6 S, and then take

One v e r i f i e s

Z

X"

st' , bc

easily that the r e s u l t i n g

independent

of the r e p r e s e n t a t i v e s

and

Z

Y

diagram

to

chosen,

to be the composition. morphism

X

of the m o r p h i s m s

and is also i n d e p e n d e n t

to

of

Z

X

is

to

Y

of the c o m m u t a t i v e

chosen.

One can also v e r i f y has the p r o p e r t i e s

easily that the functor

r e q u i r e d and that

Cs

Definition. multiplicative

Let

C

the t r i a n g u l a t i o n (FR4)

be a t r i a n g u l a t e d

system of morphisms.

S

if the following

s 6 S T(s)

6 S, w h e r e

Q: C

is a d d i t i v e

(again using LI, L2, L3 to show that the li B

with

of

> Cs if

The

same as

f,g 6 S, and r e q u i r e

(TR3),

that

c a t e g o r y and

S

a

is said to be c o m p a t i b l e two axioms are satisfied: T

is the t r a n s l a t i o n

but w h e r e we assume

h 6 S.

is

is a group).

functor. (FRS)

C

that

32

Proposition S

3.2.

If

is a m u l t i p l i c a t i v e

then

CS

that

Q

above

for 8-functors

C

system compatible

has a unique

structure

is a ~-functor,

Proof.

is a t r i a n g u l a t e d

and

Q

Hom~

category

has the u n i v e r s a l

into t r i a n g u l a t e d

and

the triangulation,

of triangulated

Left to the reader.

can also calculate

with

category

such

p r o p e r t y b)

categories.

It helps

to observe

that one

(X,Y) as

c(x,Y') Jy where

Jy

s: Y

~> Y'

is the category w h o s e

objects

are m o r p h i s m s

in S, and w h o s e maps are commutative

diagrams

Y

Y'

Proposition

3.3.

a multiplicative

system

of

E Ob D

that

C (i.e., X , Y S n D

~ Y"

Let

C

in C.

be a category, Let

S

be

be a full s u b c a t e g o r y

> H o m D ( X , Y ) = HOmc(X,Y))

is a m u l t i p l i c a t i v e

that one of the following

D

and let

system

two c o n d i t i o n s

in

D.

holds:

Assume

and assume furthermore

33

(i) X 60b X"

Whenever

s: X'

D, then there

60b

D and

(ii)

is a m o r p h i s m

is a m o r p h i s m

Ditto with the arrows functor

DSN D

can be identified

Proof.

Straightforward.

Proposition system in C, and be another

f: X"

in S, w i t h

> X'

such that

sf 6 S.

Then the natural i.e.,

>X

3.4.

Let

Q: C

category,

reversed.

DSN D

>C s

is fully

w i t h a full subcategory

be a category,

C

and let

F,G:

Cs

of

C s.

S a multiplicative

the localization

> CS

faithful,

> D

functor.

Let

D

be two functors.

Then the natural map ~: of m o r p h i s m s Proof. for each X

> Y

Hom(F,G) of functors

> Hom(FQ,GQ) is bijective.

To give a m o r p h i s m

X 60b

CS, a m o r p h i s m

is a morphism,

F(X)

then

>

l G(X)

of functors

F(Y)

l >

G(Y)

F

>G(X),

> G

means

such that if

to give,

34

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

diagram.

is injective.

Thus, a

To show

Ob C = Ob CS, the m a p

surjective,

suppose

morphism

FQ

for each

X E Ob C = Ob CS, and c o m m u t a t i v e

X

>Y

in

>GQ.

since

C.

T h e n we have a m o r p h i s m

A morphism

in

CS

given a

F(X)

>

diagrams

is r e p r e s e n t e d

G(X)

for m o r p h i s m s

by a d i a g r a m

X I

X of m o r p h i s m s

in C, w i t h

are isomorphisms,

Y s 6 S.

But for

so we get the r e q u i r e d

s 6 S,

F(s)

commutative

and

diagram.

G(S)

35

w

Qis

and the Derived Category.

Let

A

be an abelian

triangulated isomorphism induces

category

described

to be a m o r p h i s m

an isomorphism

collection

category,

K(A )

and let

in w

We define

f: X"

on cohomology.

> Y"

in

Let

Qis

be the

a quasi-

K(A )

which

be the

of all quasi-isomorphisms.

Proposition Proof.

4.1.

This

Qis

is a m u l t i p l i c a t i v e

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

system

of the following

in K(A).

more general

proposition. Proposition be an abelian to

C

that

H(Ti(s))

A.

multiplicative

(FR4)

Let

To prove

C

be a t r i a n g u l a t e d

and let

S

H

for all i 6 ~ .

system in C, compatible

We must v e r i f y (FRS)

functor (FR2),

let a diagram z

Is >Y

functor

in

Then

(FRI)-(FRS).

from the long exact

and the five-lemma.

x

s

let A

C S

such is a

w i t h the triangulation.

the axioms

follows

category,

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

be the set of m o r p h i s m s

is an isomorphism

are trivial.

cohomological

Let

category,

from

Proof.

4.2.

(FR1)

and

sequence

of a

36

be given,

with

s 6 S.

(Z,Y,N,s,f,g). Then

(u, idN)

the first,

Complete

Complete

fu

s

to a triangle

to a triangle

(W,X,N,t,fu,h).

is a map of two sides of the second triangle

so there is a map

v: W

>Z

into

g i v i n g a m o r p h i s m of

triangles.

V

Z

t/u idN X

>Y U

Now

sv = ut,

so it remains = 0

to prove t 6 S.

s 6 S, we have

H(Ti(N))

for all i 6 ~

sequence

of the

first triangle.

sequence

of the second triangle,

Indeed,

since

by the long exact

Applying

this to the long exact

we

H(Ti(t))

find

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

for all i 6 ~. The opposed To prove showing

statement

of (FR2)

(FR3), we consider

the following

is a morphism):

is proved

similarly.

the m o r p h i s m

two p r o p e r t i e s

equivalent

f-g, and reduce (where

f: X

to > Y

37

(i)

There

exists an

(ii)

There

exists a

Suppose

s:

Y

t: X'

> Y'

6 S

such that

sf

~ X

6 S

such that

ft = O .

X

W

Z

''

S

> y.

By (TR1) and

(TR2) we can find a triangle

suitable

Now

Z. >Z

Proposition that

s 6 S,

H(Ti(Z))

One checks

for suitable X'. triangle,

this implies

there is a map (TR2) we can

By the same

the existence t 6 S.

that

of v

Since sequence

t E S.

(ii)----> (i) is analogous.

The derived

S i m i l a r l y we define

category

of D(A),

of A, D(A),

D+(A ) = K+(A )Qis'

easily using P r o p o s i t i o n

subcategories

(TR1) and

for

for all i 6 ~ , by the long exact

In turn,

Definition.

l.l b),

We need only show that

= 0

The implication

A g a i n by

to this second

ft = O.

of cohomology.

(A )Qis"

f = vg.

(X',X,Z,t,g,w)

applied

implies

(Z,Y,Y',v,s,u)

sf = O, so by P r o p o s i t i o n

such that

find a triangle

K

O

(i) holds.

X'
X

in Qis,

2

operator

for

ft,

g f = O, b u t

and

sees easily.

then

induces

f

>2Z

the z e r o m a p on

For e x a m p l e

take

>O

> ZZ~ ----->o 9 on c o h o m o l o g y , such t h a t

follows:

the single

g 6 Qis,

is false.

the z e r o - m a p

exist

Then

as o n e

(~),

X:

induces

.

to zero,

cohomology,

f

X

b e the z e r o map.

is n o t h o m o t o p i c

Now

,

to b e the i d e n t i t y i~:

Let

>0

2

ft N O.

take a cycle

cohomology show that

but

there

(The r e a d e r

x E X'

g r o u p Z 2 o f X. 2k(x)

d o e s not can

such t h a t If

= i, w h i c h

k

is a

is i m p o s s i b l e . )

40

So for two maps

f,g: X

> Y of complexes, we see that the

following implications are all strict:

f = g ~ -----> f

and

g

give the same morphism in

-----> f

and

g

give the same map on cohomology.

Proposition 4.3. object

X

and

elsewhere,

O

f homotopic to g

of

A

The functor

D(A), which sends each X

in degree O,

gives an equivalence of the category

Hi(X ") = 0

Proof.

>

into the complex consisting of

the full subcategory of such that

A-

D(A)

D(A) for

A

with

consisting of those complexes

X"

i 40.

Left to reader.

We will now give three lemmae, and another description of the derived category D+(A) when Lemma 4.4.

Let

A

A

has enough injectives.

be an abelian category,

be a morphism of complexes of objects of A.

Then

i)

Z"

2)

Each

3)

I"

f

is acyclic Ip

is injective

is bounded below.

is homotopic to zero.

and let

Assume

f: Z"

>I"

41

Proof. Lemma s:

I"

Well known 4.5.

> Y"

Let

(and easy).

A

be an abelian

c a t e g o r y and let

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

of objects

of

A.

Assume

i) 2)

s induces an i s o m o r p h i s m

on c o h o m o l o g y

each I p is i n j e c t i v e

3) i p is b o u n d e d below. Then

s

has a h o m o t o p y

Proof.

Suppose

Z" = T ( I ' ) ~ Y "

given

s: I"

be the m a p p i n g

and so the triangular conditions

inverse.

of Lemma

call the h o m o t o p y

(k,t):

morphism

> Y" as above.

cone of s. v: Z"

Then > T(I')

4.4, and so is h o m o t o p i c

operator

T(I') 9 Y"

i

Then we have the e q u a t i o n

V = (idi. , O) = (k,t) Separating

the components,

we

d Z + d I (k,t) find

Let

Z" is acyclic, satisfies

to zero.

the

Let us

42

id I = dk + kd + ts

and dt - td = O . Thus

t: Y"

homotopic

> I"

to

1).

ts, so

4.6.

Lemma

Let (i)

element of

is a m o r p h i s m

Let P

is a h o m o t o p y

A

be an a b e l i a n category.

A

E v e r y object of

Ob A

X"

Assume

(ii)

and assume

admits an i n j e c t i o n

I"

of o b j e c t s of

o

> Y

>x

P

that

into an

X 6 P, then

x~

> xI

is an exact

sequence,

Then every X" I"

of objects

Let

A'

of A'.

integer

>X n

n, such that if

> O

X n E P.

admits a q u a s i - i s o m o r p h i s m

of

into a

P.

be a thick

A' has enough A - i n J e c t i v e s . quasi-isomorphism

is a short exact

X ~ ,... ,X n-1 6 P, then

and

6 K(A)

satisfies

Y 6 P z E P.

> X n-I

> 9

into a

P.

> Z ---->O

(iii) T h e r e exists a p o s i t i v e

objects

is

s.

admits a q u a s i - i s o m o r p h i s m

6 K+(A)

furthermore

If

sequence, w i t h

3).

of

id I

P.

bounded below complex

complex

inverse

and

t

be a subset of

Then every

2).

of complexes,

subcategory Then every

of A, and assume that X"

6 K~.(A)

into a b o u n d e d b e l o w c o m p l e x

admits a

I" of A - i n J e c t i v e

43

Proofs. X~ ----> I ~ choose

1).

with

We may assume I~

in P.

X p = O for p < O.

Having

defined

Embed

I~ 9I1 ,...,I p ,

I p+l to be an element of P c o n t a i n i n g iP/im

i p-I

X p+I

~

xp and define the maps

I p ----> I p+l

and

obvious way.

One checks

isomorphism.

Note that in this c o n s t r u c t i o n

X p ----> I p

> I P+I

X" ----> I"

We p r o c e e d i

o

isomorphism

in several

be an integer. of the t r u n c a t e d >o

into a c o m p l e x injective.

all the maps

I"

steps.

X" o

X"

>x

> ...

of P, w i t h each

io-I

io

>x

---->I

Then we have a q u a s i - i s o m o r p h i s m

X"

>X" o

for

is injective.

Suppose

> X~

X i ---->X i o

given a c o m p l e x

i 2 < i I.

X[

with

i 6 P X2

for

>....

such that X i 6 P o

i X1 6 P

Then we w i l l c o n s t r u c t

such that

> Ii

io+l

>I

--

Xi

to be the c o m p l e x

io-2

i > io, and each

be a complex,

complex

of elements

Define

Let

io+l

--->x

>x

X1

is a quasi-

Then by i) we can find a quasi-

io o

and let

in the

are injective.

2). and let

easily that

X p+I

for

i >_ il,

a quasi-isomorphism

i ~ i2, and

i Xi 1 = X2

for

44

i ~ il+n.

(Here

n

is the integer of condition (iii) above.)

Indeed, by the first step above, we can find a quasi-isomorphism X i ---->X'"

is injective.

Let

acyclic complex, and above.

X .i 6 P

such that

Hence

Now define

X~

i ~ i2, and each

for

yi = coker(X~ yi 6 P

Bi(y ~ ) 6 P

>x'i).

for

for

Y"

> X 'i

is an

i ~ il, by condition (ii)

i ~ il+n , by condition (iii).

by

x,i i X2

Then

i

X1

i

Bi(x '" )e i_ixx

=

for

i < il+n

for

i = il+n

for

i > il+n .

x1

i x1 One sees easily that

Xi

> X~

is a quasi-isomorphism.

It

follows from (ii) and the exact sequence

O

>xli

"

i

>0

is as

that the middle term is in P, for i ~ il+n , so required. Now, given a complex integers

io > il > "'"

X" 6 K(A), choose a sequence of tending to

as in the first step, and choose

-oo.

Choose

Xl, X~,...

successively as in the second step.

for

X" o

for

i o

il,i 2, 9 . .

Then we have quasi-isomorphisms

~5

x"

>X'o

Xi

> Xi ~ o

in P. of

>x[

>

i Xl

Hence

...

> "'"

and for each i, the sequence is eventually constant, and eventually

I" = li~ X" r

is the required complex of objects

P. 3).

We may assume

X i = O for i < O.

Embed H~

an A-inJective of A'

which is possible since

Extend this to a map

fo:

X ~ ----> I ~

I ~ is A-inJective.

Having define

fi: X i

i = O,...,p,

> Ii

for

A-inJective of

(*)

A'

H~

") in I ~ ) 60b

A'

which is possible since

I~

p, and

choose

I p+I

to be an

containing

IP/im i p-I ~

zP+I(x ") . Xp

We must check that this latter is in A'. subcategory,

Indeed, A' is a thick

so it is sufficient to note that IP/im I p-I 6 A'

(one shows by induction that

Bi(I" )

and

zi(I" )

for all i), and that the quotient of (*) by HP+I(x'), which is in A' by hypothesis. zP+I(x ")

> I p+l

to a map

easily that the resulting map isomorphism, as required.

fp+l: f:

xP+l X"

are in

IP/im I p-I

A' is

Extend the natural map > I p+I. > I"

One checks

is a quasi-

46

Let

Proposition 4.7. let

I

be the (additive)

of A.

A

be an abelian category, and

subcategory of injective objects

Then the natural functor K+(I)

is fully faithful.

> D+(A)

(Note that the results of section 3 carry

over to additive subcategories of abelian categories. ) Furthermore, object of then

~

A

if

A

has enough inJectives

(i.e., if every

admits an injection into an inJective object)

is an equivalence of categories.

Proof. system in

We note that

K+(I) N Qis

is a multiplicative

K+(I), by Proposition 4.2, and we observe by

Lemma 4.5 that condition

(ii) of Proposition 3.3 is satisfied

for

Qis.

K+(I) _c K+(A)

and

D§ is fully faithful.

Hence the natural functor

> D+(A) But on the other hand, Lemma 4.5 shows

also that every quasi-isomorphism hence

K+(I) = D+(I).

in

K+(I)

is an isomorphism,

47

Now if

A

has enough injectives,

applying Lemma 4.6 in the

case A = B and P = the injective objects, we see that every object of

D+(A) is isomorphic to an object in K+(I),

e~ivalence

so

a

is an

of categories.

Proposition 4.8.

Let

A

be an abelian category, and let

be a thick abelian subcategory. A-injectives,

Assume that

i.e~ every object of

A-inJective object of

A'.

A'

A'

A'

has enough

can be injected into an

Then the natural functor

D+(A ' )

> D~, (A)

is an equivalence of categories. Proof. Clearly

Qis

We apply Proposition 3.3 to the inclusion K+(A ' ) --->K+(A). is a multiplicative

quasi-isomorphism with

system in each.

X" 6 K+(A'),

then

Y"

each

Ip

injective in

A.

X" --->Y"

is

has cohomology in A',

and so by Lemma 4.6 admits a quasi-isomorphism Y" I" 6 K+(A'),

If

> I"

Hence condition

with (ii) is

satisfied, and so the functor

D+(A ' ) is fully faithful.

(A).

> D+(A)

The same Lemma 4.6 also shows that the image is

a

48

Exercise. of the last and

D-(A ).

We leave to the reader

five results

the analogous

in the case of projective

statements

objects

of

A

w

Derived Functors. We will treat only the question of right derived covariant

functors,

leaving the reader to make the obvious modifications

for

left derived covariant functors, and right and left derived contravariant Let

functors.

A,B

be abelian categories,

be a ~-functor

(see w

F

K(A)

F: A

> B.

K(A)~

into

to say that it does is to say that it localizes

and gives rise to a functor from if

F

D(A)

to

D(B).

That will be the

is an exact functor.

Thus we are led to ask if there is a functor from D(B)

J> K(B)

It extends to

will not take quasi-isomorphisms

quasi-isomorphisms

case, for example,

F:

Such is the case, for example, if we

are given an additive functor In general,

and let

which is at least close to

notion of derived functor below.

D(A)

to

F, and this gives rise to the Before giving the definition

w e generalize slightly. Definition.

Let

A

be an abelian category, and let

be a triangulated subcategory of that

K~(A) n Qis

that

K~(A)

functor

K(A).

K~(A)

Note by Proposition 4.2

is a multiplicative system in

is a localizingg subcategory gory of

K(A)

K~(A). We say if the natural

50

K*(A)K.(~)n0is

-~ ~(A)0is = D(A)

is fully faithful, and in that case we write D*(A) for the first of these categories. Examples.

i.

Any intersection of localizing subcategories

is localizing. 2.

K-(A),

K+(A),

and Kb(A) are localizing subcategories of

K(A) (see section ~). 3. KA, (A)

If and

+ (A), is a thick subcategory of A, then KA,(A), KA,

A'

KAb, (A) are localizing subcategories of K(A) (see

section ;4). *4.

The complexes of finite inJective dimension form a

localizing subcategory K+(A) fid of K(A) (see Corollary 7.7).. Definition.

Let

A

and

B

be abelian categories,

let K*(A)

be a localizing subcategory of K(A), and let

F: be a ~-functor.

K*(A) to D*(A) is a ~-functor

K*(A) Let

Q

> K(B) denote the localization

resp. K(B) to D(B).

functor from

The right derived functor of F

51

R'F:

D*(A)

> D(B)

together with a morphism of functors

~:

K*(A)

from

to

QoF

> R*FoQ

D(B), with the following universal property:

If G:

D*(A)

> D(B)

is any ~-functor, and if ~:

Q0F

>GoQ

is a morphism of functors, then there exists a unique morphism

D:

R*F

> G

such that =

Remarks.

I.

If

R*F

exists, it is unique up to unique

isomorphism of functors. 2.

If

K*(A) is K+(A), K-(A), KA.(A), etc., we will write

R+F, R-F, RA.F etc. we w i l l

write

simply

for RF

R'F, and when no confusion can result, for

all

of

these.

52

3.

We will write

RiF

the results below that if F: A

>B,

and if

usual derived ~. K(B),

If

and if

morphism

A

F

F

RF

Hi(RF),'" and it will

comes

from a left-exact

has enough injectives,

functors of 9:

for

> G and

follow

from

functor

then these are the

F. is a morphism of functors

RG

both exist,

from

K~"(A)

to

then there is a unique

of functors ~:

RF ----~RG

compatible with the ~'s. This

follows

immediately

from the

de fini tion. 5.

If

K~(A)

~ K~(A)

are two localizing

subcategories

of

K(A), and if

F: is a ~-functor,

K~(A) ----> K(B)

and if both

there is a natural morphism

R**(FIx**CA) )

R*F

and

R~*(FIK'X'~'(A))

below).

then

of functors

). R*FID**(A)

We do not know if it is an isomorphism in all the applications

exist,

we have in mind

.

in general,

but it w i l l be

(see e.g. Corollary

5.3

53

Theorem 5.1. A, B,

K ~

(Existence of derived functors).

(A), and F be as in the definition above.

there is a triangulated

subcategory L ~ K~(A)

l) Every object of

K~(A)

Let Suppose

such that

admits a quasi-isomorphism

into an object of L, and 2) If Then

I" 6 0 b

F(I') Then

for any

L

has a right derived functor

I" 6 0 b L , ~(I'):

> I

>~F.Q(I')

First observe that the restriction of into quasi-isomorphisms.

is a quasi-isomorphism

is also,

so

F(s)

F:

LQi s

Then

is a quasi-isomorphism.

to the quotient to give a functor > D(B)

F

to

L

Indeed,

if

of objects of L, let

the third side of a triangle built on s. F(J')

Furthermore,

in D(B).

takes quasi-isomorphisms 9

(R~F,~).

the map

Q.F(I')

is an isomorphism

s: I 1

(i.e., Hi(I ") = O for all i),

is also acyclic. F

Proof.

is acyclic

J"

be

J

is acyclic, Hence

F

so

passes

54 w i t h the property the morphism

~.Q = Q.F

from a category

on

L.

for L, K*(A), T:

is an equivalence quasi-inverse

and

Q

to its localization.)

Second note that the hypotheses satisfied

(We denote as usual by

of Proposition

3.3,

Qis, and so the natural

LQi s

> D*(A)

of categories,

using

i) above.

(ii) are

functor

Let

U

be a

of T, i.e., a functor U:

together with

D*(A)

functorial

u: and

> LQi s

isomorphisms

ILQis

:

> U.T

ID* (A)

> T .U.

Then define R*F

=

~.U

.

We define a morphism of functors : Q.F

as follows.

Let

Q(I') = U.Q(X').

>R*FoQ

= ~.U.Q

X" E Ob K*(A),

and let

We have an isomorphism

I" 6 0 b L in

D*(A),

be such that

55

~(Q(x')).

Q(x')

~

>

T.u(Q(x'))

-- T(Q~')

.

This isomorphism can be represented by a diagram of morphisms

X ~

I"

y"

in

and

s,t 6 Qis.

Furthermore,

i) above, we may assume

Y" s Ob L.

Now applying the

K~(A), with

hypothesis

Y" 6 0 b

K~(A)

by

functor F, we get a diagram in K(B)

F(X')

F(I')

~(Y')

where

F(s) is also a quasi-isomorphism,

as we remarked above.

This in turn gives a morphism in D(B),

~(X'):

QoF(X')

>Q.F(I')

= F.Q(I')

= F~

= R~F~

One can now check without difficulty that ~(X') does not depend on the choice of the diagram functors

from

Q.F

to

derived functor of

F.

(~) above,

ReF.Q,

that

~

gives a morphism of

and that the pair

(R~F,~)

is a

56 Now if

X" E Ob L,

then

is also a quasi-isomorphism, in

and so

~(X') is an isomorphism

D(B), as required. Proposition 5.2.

and let K(A).

A, B, K~(A)

Let

K**(A) ~ K*(A)

and

F

be as above,

be another localizing subcategory of

Suppose there is a triangulated subcategory

satisfying hypotheses furthermore, R*F

F(t) in the construction above

and

that

i)

and

2)

L D K**(A)

R**(FIK**(A))

L

of

K*(A)

of the theorem, and suppose,

satisfies

l)

for

K**(A).

Then

both exist, and the natural map

is an isomorphism. Proof.

The existence of the two derived functors follows

from the theorem. X" 6 0 b

D**(A)

To prove the isomorphism,

since every

is isomorphic to one coming from an object of L,

we may assume that

X" = Q(I')

with

I" 6 0 b ( L

n K**(A)).

Then

the statement follows from the last part of the theorem. Corollary 5.3.

K+(A)

> K(B)

additive functor injectives.

Then

u.

Let

A,B

be a ~-functor

be abelian categories,

(defined for example by an

F o : A --~ B) J and assume that R+F

exists.

let

A

has enough

57

8.

Let

A,B

be abelian categories,

let

F: A

~ B

be an additive functor, and assume that there exists a subset P of

having the properties

Ob A

(i) and (ii) of Lemma 4.6,

and also

(iv)

F

carries short exact sequences of objects of

P

into short exact sequences. Then of

F

R+F

to a 7.

exists.

~-functor

Let

A,B

F

(We denote also by

K+(A)

>

the extension

)

be abelian categories,

let

F:

A

>B

be an additive functor, and assume that

a)

The hypotheses of

b)

F

8

has finite cohomological dimension on A, i.e.,

there is a positive integer all

by

above are satisfied, and

Y 60b

A

and

all

i > n.

8, so this makes sense.)

restriction of

Remark.

RF

u

to

such that

n

Then

D+(A)

RF

RIF

is equal to

P

for

exists

exists, and the R+F.

is a special case of 8, since if

enough inJectives, then the set properties

(Note that

RiF(y) = O

A

of inJectives of

(i),(ii), and (iv) for any additive functor

has A

has F.

58 Proof.

u.

Let

L _c K+(A)

be the triangulated

of complexes of injective objects of A. every of

X" 6 0 b

L.

K+(A)

is an isomorphism. satisfied

for any

8.

Then by Lemma 4.6~ I),

admits a quasi-isomorphism

Furthermore,

~-functor F, and we deduce that

exists.

~+F

L ~ K+(A) to be the triangulated

subcategorybecause

P

(Note

L

is a

is stable under direct sums

Condition I) of the theorem is satisfied as above.

For condition 2), let above repeatedly, D

ker d~ 6 P that

in L

Hence condition 2) of the theorem is

subcategory of complexes of objects of P.

by (ii).)

into an object

by Lemma ~.5, every quasi-isomorphism

In this case we take

triangulated

subcategory

Z"

be acyclic.

and the fact that

for every p.

We use condition

Z" 6 K+(A),

Then by condition

(ii)

to show that

(iv) of P, it follows

F(Z') is acyclic. 7.

We take

P' to be the collection of F-acYclic objects

of A, i.e., those X E Ob A such that Then

P'

has properties

take

L ~ K(A)

RiF(x) = O for all i > O.

(i),(ii) and (iii) of Lemma 4.6.

to be the complexes of objects of P'.

We

Then

using Lemma 4.6 and an argument similar to the one in B above, one sees that the hypotheses of the theorem are satisfied, RFexists.

so

59

One sees by Proposition 5.2 that the restriction of RF to

D+(A) is

R+F .

Proposition 5. #. K*(A) __CK(A)

and

Let

A,B,C

KT(B) ~ K(B)

be abelian categories, let be localizing subcategories,

and let

F:

K'(A)

> K(B)

G:

K~(B)

> K(C)

be ~-functors. a). and

Assume that

R~(G,F)

F(K~(A)) _c Kf(B), assume that R~F, RiG,

exist, and assume that

R~F(D~(A)) _c Dr(B).

there is a unique morphism of functors

~: R~(GoF)

> R~G,R~F

such that the diagram QoGoF

R~(G.F) oQ is commutative.

~G

~~

> R7 G,Q oF

> R~G,R~F.Q

Then

60

b).

Assume that

F(K*(A)) c KT(B), assume that there are

triangulated subcategories

L _c K~(A)

the hypotheses of Theorem 5.1 for

F(I)

assume that

=M.

F

and and

S c K~(B) satisfying G, respectively,

Then the hypotheses of

and

a) above are

which therefore exists, is an

satisfied, and the morphism isomorphism. Straightforward.

Proof. Remarks.

I~

If

F,G,H

then there is a commutative

are three consecutive

functors,

diagram of ~'s (provided they all

exist):

R(H-G-F) ~G.F,H> RH-R(G.F) I~F,H'G I~F,G R(H.G)oRF

2.

>

RH.RGoRF

.

This proposition shows the convenience of derived

functors in the context of derived categories.

What used to be

a spectral sequence becomes now simply a composition of functors. (And of course one can recover the old spectral sequence from this proposition by taking cohomology and using the spectral sequence of a double complex.)

61

Corollary 5.5. Proposition

Left to the reader:

in the style of Corollary

Proposition

5.6.

Let

A

that

and let

F: K+(A)

the

5.9.

be an abelian category,

be a thick abelian subcategory, category,

Illustrate

let

B

R+F and R+(FIA ,) both exist.

be another abelian be a ~-functor.

>K+(B)

A'

let

Suppose

Then there is a natural

morphi sm

+(FIA ) of functors

from D+(A ' ) to D(B), where ~:

D+(A ' )

is the natural functor. A-injectives,

and

A

> D+(A)

If furthermore

A'

has enough injectives,

has enough is an

then

isomorphism. Proof.

of the derived functor. A

follows from the definition

The existence of

has enough injectives,

If

A'

has enough A-injectives,

then we can use A-injectives

calculate both functors above, by Corollary an isomorphism. equivalence

(Recall by Proposition

of categories

in that case.)

to

5.3~, and so

4.8 that

~

and

is an

~

is

62 w

Examples.

Ext and R Hom.

Definition.

Let

X'3y"

be objects of

X',Y"

to be

1.

--

If

by taking Horn in D+(A), 2. ExtI(X,Y)

be an abelian category,

D(A).

Ext i (X',~') Remarks.

A

.th We define the i hyperext of

HomD(A) (X', Ti(Y ") ).

X',Y"

6 D+(A),

then we get the same Ext

for D+(A) is a full subcategory of D(A).

This definition gives us in particular for any

enough injectives

and let

X,Y 6 A.

a definition of

We will see below that if

(so that the usual Ext is defined)

A

has

then this

definition agrees with the usual definition of Ext. Proposition 0

>x"

6. i. >Y"

Let > z"

>o

be a short exact sequence of complexes of objects of A, and let V"

be another complex of objects of

A.

Then there are long

exact sequences

...--, Exti(v',x ") > Exti(v',Y ")

> Exti(v',z ")

> E x t i + l ( v ' , x ") > ...

and ...

>

Exti(z',v ")

> Exti(y',v" ) > Exti(x',v ") --~Exti+I(Z',V ') > ...

63

Proof.

Let

W"

be the third

W

0

>

Then by Proposition g:

W"

> Z'.

short exact

Using

we see that

g

there

>z"

is a m o r p h i s m sequence

and of the triangle,

is a quasi-isomorphism,

Hence we may replace follows

> Y"

the long exact

sequence,

Z"

by

X" --> Y'.

on

"

x"

l.lb,

side of a triangle

W"

>o

of complexes of cohomology

and using

i.e.,

of the

the five-lemma,

an isomorphism

in the conclusion 9

which

in D(A).

then

from the same proposition.

Remark.

It follows

0 is a short

>x"

exact

is a m o r p h i s m

from the proof

>x"

sequence

Z"

>z"

of complexes

> T(X" )

that w h e n e v e r

in

D(A)

>0 of objects making

of

A 9 then there

X" ,Y" ,Z"

into a

triangle. Now we will

define a functor whose

cohomology

gives

the

Ext groups. Definition. w e define

If

a complex

X"

and

Hom" (X" ,Y')

Homn(x',y" ) =

are complexes by

HOmA (xP,yp+n)

pEZ and

Y"

of objects

of A

6#

dn = ~

(~-i

+ ( _ 1) n+l ~ p+n ) .

Notice under this definition that the n-cycles of the complex Hom'(X',Y')

are in one-to-one

complexes of

X"

to

Tn(y'),

correspondence

and the n-boundaries

~ o s e morphisms which are homotopic

Hn(Hom

Now

Hom"

A

(X',Y')) ~=

to zero.

correspond to

In other words,

HOmK( A)( x "{~

.

is clearly a bi-~-functor Hom':

If

"

with morphisms of

K(A) O x K(A)

has enough injectives,

> K(Ab).

we can calculate its derived

functors. Lemma 6.2. Y" E Ob K+(A)

Let

morphism of

K(A) be a complex,

and let

be a complex of injective objects.

a) y. is acyclic, Proof.

X" 6 0 b

or

b) X" is acyclic.

Then

Assume either

Hom'(X',Y')

By the remark above, one has only to check that any X"

also satisfies

to

Tn(y ")

the hypotheses

any morphism of

X"

to

Y"

is homotopic to zero, or, since Tn(y ") of the lemma, it is enough to show that is homotopic

to zero.

In case

Y" is split exact, and it is easy to construct the homotopy reader).

is acyclic.

In case

b) the result is Lemma 4.4.

a), (left to

65 Now suppose

A

the triangulated

has enough inJectives,

and let

L ~ K+(A)

be

subcategory of complexes of inJective objects.

Then using the 1emma, part a), we see that for each

X" q Ob K(A),

L satisfies the hypotheses of Theorem 5.1 for the functor

Hom'(X','): K+(A)----->K(Ab). Hence this functor has a right derived functor.

It is easily seen

to be functorial in X', and so we have a bi-~-functor RiiHom" :

K(A) O • D+(A)

>D(Ab).

Now using the lemma, part b), we see that this functor is "exact" in the first variable, complexes,

i.e., takes acyclic complexes

and hence passes to the quotient,

into acyclic

giving a trivial right

derived functor ~I~

Hom':

D(A) ~ • D+(A)

(We will denote this functor by

> D(Ab). RHom-

Suppose on the other hand that

A

when no confusion can result.) has enough proJectives.

Then

by the usual process of "reversing the arrows" we see that there is also a functor RII R I Horn" :

Now if functors

A

D-(A) ~ • D(A)

> D(A)

has enough injectives and enough proJectives, RIRIIHOm"

and

RIIRIHOm"

are defined on

then both

D-(A)~215

66

and are canonically isomorphic,

as we see by the lemma below.

Thus we are justified in using the ambiguous notation R Hom'. Lemma 6.3.

Let

A, B, and C be abelian categories,

T:

K~(A) x KT(B)

be a bi-~-functor.

Suppose that

(where the subscripts or second variable,

> K(c) ~I~

T

and

R~I

T

both exist

I,II denote the derived functor in the first

respectivelyl.

Then there is a unique isomorphism

between them compatible with the morphisms ~2:

and let

~l: T --~ R I R ~ T

and

T --->R]I RIT. Proof.

Follows directly from the definition of derived

functors. Theorem 6.4.

(Yoneda)

enough inJectives.

Let

Then for any

A

be an abelian category having

x" ~ D(A), Y" ~ ~+(AI,

H i (R +Eom" (X', Y" ) 1 = Ext i (X', Y" 1. Corollary 6. 5 . X,Y 6 A , the Proof.

If

Exti(x,Y) Let

I"

the above theorem,

A

has enough injectives,

then for any

defined above i~ the usual Ext.

be an injective resolution

and Theorem 5. i, we have

Exti(x,Y)

= Hi(Hom "(x,I'))

which is the usual definition.

,

of Y~

Then using

67 Proof of Theorem 6.4. of

Y"

Let

s: Y"

into a complex of injectives

> I" I'.

be a quasi-isomorphism

Then

~xti(x',y "1 = ~xti(x',i "1 = HomD(A)(X',Ti(I')). But by using Lemma 4.5, one sees that every morphism in

D(A)

a complex

say Ti(I'),

X"

to a complex of injectives bounded below,

is represented by an actual morphism of complexes. is equal to

HomK(A)(X', Ti(I')) _- Hi(~om'(X'I')) -- H i(R Hom" (X" ,Y" ) ) .

of

Hence the above

68

iT

Way-out

functors

Definition.

F:

D(A)

way-out

Let

> D(B) (right)

whenever

X"

F: D(A)

of

way-out

i < n2

right

2. for

Y"

If

F

If

If

F

is

such that

n2 6 Z

for all

Hi(x" ) = 0

except

~-functor,

directions.

the definition

that we reverse

the inequality

o

: A

> B

is an additive

5.3, ~ or 8, then

satisfies

functor

R+F

the h y p o t h e s e s

of

satisfying

is a w a y - o u t W, then

RF

in b o t h directions: X"

~ D(A)

is an u n b o u n d e d

is in general 7.i.

categories,

F and G be ~-functors > G

exists

F

n2 .

of C o r o l l a r y

Proposition

7: F

We say that

left, w a y out in both

is a contravariant

is the same,

i.

6 D+(A),

be abelian

and let

for all i < n I.

similarly w a y - o u t

i>

functor.

is w a y - o u t

= O

categories,

~-functor.

is a complex with

right

the hypotheses

be abelian

D(A)

>D(B)

Examples.

B

n I 6 Z, there

Hi(F(X'))

to be

and

if given

One defines If

A

be a (covariant)

60b

i < n2, then

and isomorphisms.

not a w a y - o u t

(Lemma on W a y - o u t let from

be a m o r p h i s m

A'

complex,

D+,(A)

of functors.

functor

Functors)

be a thick

R Horn'(X" ,Y" ) ,

then in Let

subcategory

(or DA,(A))

Y'. A of A,

to D(B),

and let

and let

B

69

(i) Then

Assume

n(X') (ii)

that

F

that

and

G

is an i s o m o r p h i s m (iii) and that

and

(iv) A' admits

Let

Definition. integer

~(X" )

for all

X 60bA',

Then

~(X" )

into an object

of P.

X 6 P, and that

F

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

the following > O

> X n+l

- - - > X n-I

> O .7 xn-i

such that every object

of Ob A'

n E Z~ we define

~>n(X" ) :

Then

E DA, (A).

be a c o m p l e x

9 ''

functors.

in both directions.

X"

9>n(X') :

Let

right

for all X ~ Ob A',

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

be a subset

D(X)

E Ob D~,(A).

E D~,(A).

are w a y - o u t

for every Then

X"

for all X"

X"

for all X E Ob A'.

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

that ~(X)

an i n j e c t i o n

isomorphism functors.

P

for all

are both w a y - o u t

G

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

~(X)

for all

Assume F

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

is an isomorphism Assume

and that

D(X)

Assume

of objects

is an

are w a y - o u t

for every

right

X E Ob A'.

of A.

For an

truncations:

----> X n + 2

> xn---->O

>

G

and

~(X)

of

>

im d n ---->X n+l ~i ker d n

~> O

...

> X n+2 ____> ...

...

.

70

There are natural morphisms of complexes giving rise to the following exact sequences:

(i)

o

> T>n(X" )

> X" ------>T_ O>n(X" )

>0

The truncation functor

o

has the property

Hi(O>n(X')) = ~ H i ( X ") for 0 Hi(On(X') 7>n (X')

,>

-

Xn

>n x')

(4) Hn(x')

~%n(X')

Then

71

Proof.

The triangle (3) is deduced from an exact sequence of

complexes O

> %n (x')

> %n (X')

>X n

(see Remark following proof of Proposition let

~

I

9

)

> 0

-

6.1).

For the second,

be the complex > O

> xn/im d n-I

Then there is a natural map quasi-isomorphism,

%n(X')

->Xn+l

> .'' .

---->C>n(X" )

which is a

and there is an exact sequence of complexes !

O

~>Hn(X ") ----> % n ( X ' )

> 0>n(X" )

> O

.

Thus by the same remark we get a triangle in D(A). Proof .Of Proposition.

(i)

Let

X" 6 0 b

DA b ,(A).

We prove,

by descending induction on n, that D(%n(X')):

F(a>n(X'))

is an isomorphism for all n. has zero cohomology,

since

the zero object of D(A), and

>G(%n(X'))

If X"

n

is large enough, then % n ( X ' )

has bounded cohomology. of it is an isomorphism.

Hence it is The

induction step follows from the hypotheses and Proposition l.lc, applied to the triangle (4) above.

72 Now for

n

small enough, the natural map

X" --> a>n(X" )

is a

quasi-isomorphism, hence an isomorphism in D(A), and we are done. (ii)

Let

X" 6 0 b

D;.(A).

To show

n(X')

is an isomorphism,

it is sufficient to show that

HJ(D(X')):

HJ(FCX'))

is an isomorphism for all j 6 ~. n2

Given j, let n I >_ j+2, and choose

as in the definition of way-out functors, to work for

both.

o

x"

for

Hi(F(%n2(X')~

for

> HJ(G(X'))

i < nl, in particular for

i

< n2,

.

we have

= O = Hi(G(%n2(X')) )

i = j,j+l.

Therefore from the long

exact sequence of cohomology we get isomorphisms

HJ(F(a

~_n2

(X')))

HJ(G(O HJCGCX'))

73

But

o

(X') E D~,(A),

so

D

is an isomorphism on it by what we

2

have just proved. (iii)

Hence

Given

HJ(D(X'))

is an isomorphism,

X" 6 DA,(A), we treat first

as required.

~ HP'(c "')

resolutions

in the ordinary

this is called an injective

We recall

for convenience

and C a r t a n - E i l e n b e r g Lemma

7.5.

X" of objects b)

If

Cartan-Eilenberg is a map c)

A

> Y"

> D'"

f,g: X"

then

of double

injectives,

has a C a r t a n - E i l e n b e r g

w

complexes

> Y"

F

of

X"

and

are h o m o t o p i c

is homotopic

to G.

and if

C",D'"

Y', respectively,

complexes

resolutions

then every complex

resolution.

is a map of complexes,

of double

are maps of C a r t a n - E i l e n b e r g respectively,

[M, Ch. XVII,

of X').

some properties

has enough

resolutions

F: C'" If

If

A

f: X"

resolution

(cf.

resolutions.

a)

of

sense

lying over maps,

then there

f.

and F,G: C'"

lying over

are

f and g,

> D'"

77

d) then

If

F,G:

C'"

s(F),s(G):

> D'"

s(C")

are homotopic

> s(D")

maps of double complexes,

are homotopic maps of the associated

simple complexes. e)

If

F,G:

C'"

> D'"

are homotopic

and if we define truncation

II aWn

the restrictions

G

of

F

and

maps of double complexes,

as above, with respect to d2, then to be maps

G s(C ~

and

of

X"

q > no

into a double for suitable

is a resolution,

into the associated

no,

then the

simple complex is a

quasi-isomorphism. Proofs. in [M, Ch. spectral

a), b), and c) are in [M, Ch. XVII,

IV, ~4].

sequence

e) is easy, and f) follows

Prop.

either

1.2].

d) is

from the

[EGA O,,, 11.3.3 (ii)] or by an

of a double complex

easy direct calculation. Proof of Proposition, be a Cartan-Eilenberg enough inJectives. C ,pq =

continued.

resolution

Let

Cp q

C ''" be

for

ker d~ n f o r 0

for

Given

of X', w h i c h the truncated q< q

q>

n =

~

n .

X"

E Ob L,

exists

let

since

complex

C'" A

has

rr (e "') ~

9

i e 9 .,

78 Then

for each p 6 Z

O

>xP-->C

we have an exact sequence

'pO ---> C 'pl

C ,pO ,... ,c,P,n-1

Now

is F-acyclic. R F ( X p)

are all injective,

and

> C 'pn -->O

xp 6 P, so

.

C 'pn

Hence this r e s o l u t i o n may be used to calculate

(see end of T h e o r e m 5.1) and we have an isomorphism

~(tP): where

> ... -->C 'p'n-I

G(X p) = R n F ( X p)

tp

. ~

>

F(c'Pn)/Im F(C 'p,n-l)

is the q u a s i - i s o m o r p h i s m

Xp

> C 'p.

,

This can be used

to construct a map eP:

F(C ,pn)

> G ( X p)

whence a map

F(c'") -->G(x') of double complexes, Taking a s s o c i a t e d

u: X"

in the n

th

simple complexes we have

s(~): But now

w h e r e the second is concentrated

F(sC' "') > sC''"

F-acyclic objects,

> G(X" )[-n].

is a q u a s i - i s o m o r p h i s m

by the lemma,

part f), and so there is an

isomorphism

)

into a complex of

> F(sc"')

row.

79

But also X" is made of G - a c y c l i c

objects,

~(id x.):

~G(X')

G(X')

>

so t h e r e is an i s o m o r p h i s m

and c o m p o s i n g we can define a m o r p h i s m

~,(x'):

~(x')

I do not care w h e t h e r Cartan-Eilenberg that w h e n e v e r ~(X') This

and

resolution

f: X" -->Y"

%(Y')

>

m(x')

~(X') C''.

depends

. on the choice of the

It w i l l be s u f f i c i e n t

is a m o r p h i s m

d e f i n e d as above,

of c o m p l e x e s

to v e r i f y in L, then

fit into a c o m m u t a t i v e

is not hard to show u s i n g the results

diagram.

of the lemma above,

and

can safely be left to the reader. Thus we have a m o r p h i s m :

To show that are w a y - o u t functors,

RF

>

of functors LG[-n]

~ is an isomorphism, in both directions,

we reduce to showing

But that is clear

.

we note that RF

and

LG

and so, by the lemma on w a y - o u t ~(X)

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

from the construction,

for any X 6 P. q.e.d.

80

Proposition injectives,

7.6.

and let

Let X"

A

60b

be an abelian category w i t h enough K+(A).

Then the following conditions

are equivalent : (i)

X"

admits a q u a s i - i s o m o r p h i s m

X"

> I"

into a b o u n d e d

complex of injective objects of A. F = R Hom'(',X')

(ii) The functor way-out

left (and hence w a y - o u t

from

D(A) ~ to D(Ab)

is

in both directions) 9

(iii) There is an integer n

such that

Exti(y,X" ) = O for all

O

Y E Ob A

and all i > n

Proof.

O

(i)--> (ii)

We may assume that

complex of injectives, and calculate it is clear that

F

(ii) -~. (iii)

is w a y - o u t Choose

F(Y')

X"

is a b o u n d e d

as

Hom'(Y',X').

in both directions. such that w h e n e v e r Y"

n

Then

E D(A)

and

O

Hi(x 9 ) -- 0 possible

for

since

i < no, then F

is w a y - o u t

H i ( F ( Y ")) = O left.)

for

Then this

i > O. n

(This is Indeed,

will do. 0

-n

let

Y 60b

Hi(y '" ) = 0

Hi(F(y',))

A, a n d l e t for

i < n

Y'' O'

= Hi(F(T-no(y)))

ExtJ(Y,X ") = O for

(iii)-->(i).

so

be the complex Hi(F(Y''))

= Hi(Tn~

T

O(y).

Then

= 0 for i > O.

But

= Hi+no(F(y)).

This says

j > no, as required. Let

X"

satisfy condition

(iii)

Since

for n O

A has enough injectives, w e can find an injective r e s o l u t i o n of

X"

81

(Lemma 4.6),

i.e., a q u a s i - i s o m o r p h i s m

| 9

s: X"

> X '"

where

X

is a complex of injective objects of A, b o u n d e d below. I claim that contrary that

HI(X '' ) = 0

H m ( x '" ) ~ O

for

for some

i > n . o m > n . o

Suppose to the Then from the

exact sequence O

> Bm(X '')

> zm(x '9

> Hm(x '9

we see that the first inclusion is strict. Y E Ob A

(e.g., Y = zm(x'')) Hom(Y,

is strict.

Hence there exists a

such that the inclusion

Bm(x '')) - -

However,

> 0

> Hom(Y,

zm(x ''))

there is a commutative

Bm(Hom" (Y,X'')) --

>

Hom(Y, Bm(X''))

diagram

Zm(Hom" (Y,X'") )

Hom(Y,

zm(x ''))

w i t h an i s o m o r p h i s m on the right since Hom is left exact. hypothesis

Extm(y,x ") = Hm(Hom'(Y,X''))

is an isomorphism, for i > n

o

It follows that the map i:

= O, so the a r r o w on top

which gives a contradiction 9

as claimed 9

a< n (x' ") - - - +

x' "

But by

Hence

Hi(X '" ) = 0

82

is a q u a s i - i s o m o r p h i s m introduced

show that

n > n . o

showing that

is injective

(using the notation Now I claim that

for any

This will

n > n . --

o

the exact sequence of complexes

> T!n(X'")

> O 0 D(X)

exists. Problem. sheaves on

X

Without the hypotheses of the Exercise,

study

which are acyclic for Hom in the first variable.

Find out whether there are enough of them, and hence whether ~iHom"

exists.

Definition.

If

F" 6 D(X)

and

G" 6 D+(X), we define the

local hyperext

Exti(F ",G') = Hi(R Hom'(F',G'))

Lemma 3.2. Let

F

Let

be a coherent

X

.

be a locally noetherian prescheme. ~x-mOdule,

and let

G

be a coherent

92

(resp. quasi-coherent)

~X-mOdule.

Exti(F,G) is coherent (resp. Proof.

Then for all i ~ O

quasi-coherent).

The coherent case is [EGA Oiii12.3.3] , and the

quasi-coherent case is proved similarly. Proposition 3.3. prescheme.

Let

Let

F" 6 Dc(X)

X

be a locally noetherian and

G" E Dc+(X) (resp. D + (X)). qc

Assume either a)

F ~

b)

G"

Then

E

Dc(X)

D+(X)fid

R Hom'(F',G') -

6 D (X)

(cf. [I.?]) 9

(resp. Dqc(X )1.

c

Proof. twice.

or

This follows from the 1emma and [I.7.3] applied

The details are left to the reader.

93

w

The derived functors of | and f~. Let

X

be a prescheme,

we define the tensor product

and let F'QG"

associated to the double complex

(F'eG")n =

Z

F',G"

6 K(Mod(X)).

Then

to be the simple complex

(FP~Gq),

i.e.,

FP~Gq

p+q=n and d = d F + (-l)ndG

Homotopies

.

carry over to the tensor product,

so we have a

functor |

K(Mod(X))

Lemma 4.1. G"

Let

X K(Mod(X))

F"

be a complex of flat

> K(Mod(X)).

be a complex of ~X-mOdules,

~X-mOdules,

bounded above.

that either a)

G"

is acyclic,

b)

F"

is acyclic,

or

and assume also that either

Then

1)

F"

is bounded above, or

2)

G"

is bounded in both directions.

F'~G"

is acyclic.

and let Assume

94 Proof.

Let

K'" be the double complex

K pq = FPoG q.

Then there are spectral sequences [EGA O~i 11.3.2]

'Epq = HIP HqII(K")---~

" E pq = H p H q ( K " ) ~

and

En = Hn(F'~G ")

E n = Hn(F'~G').

Our hypotheses I) or 2) imply that these spectral sequences are biregular.

In case

flat for each q, and so implies that

a), one shows that Bq(G ") = Zq(G ") is FP~G"

This

'E2Pq = O for all p,q, and hence E n = O, and

F'eG"

is acyclic.

since

Gq

again

E n = O, and

In case b), F'~G q

is flat, and so

Now let

is acyclic for each p.

F'~G"

is acyclic for each q,

''E2Pq = O for each

p,q.

Hence

is acyclic.

F" E K-(Mod(X)), and let

L c K-(Mod(X)) be the

triangulated subcategory of complexes of flat

~X-mOdules.

by Proposition 1.2 and the lemma, part al,

satisfies the

hypotheses of [I.5.1] for the functor

K-(Mod(X)) and hence

> K(Mod(X)),

L

Then

95

~=| exists.

K-(Mod(X)) • D-(X)

(It is clearly

> D(X)

functorial in F').

By the lemma, part bl, this functor is exact in the first variable,

hence passes to the quotient

~i~=|

D-(X)

X D-(X)

Of course we can derive

|

in the second variable,

to give

> D(X).

in the first variable

first,

then

and by [1.6.3] we get the same result.

We will therefore use the ambiguous notation

F'|

for

~I~=| Definition.

If

F',G"

6 D-(X), we define the local hyperTor

Tori(F',G" ) = H-i(F'~ ") . Proposition F" 6 0 b

4.2.

Kb(Mod(X)).

(i)

Let

right

There is a q u a s i - i s o m o r p h i s m

The functor

and let

There is an n

s: G"

> F"

where

G"

is

~X-mOdules.

F'|

(and hence w a y - o u t (iii)

be a prescheme,

Then the following conditions are equivalent:

a bounded complex of flat (ii)

X

from

D-(X)

to

D-(X)

is w a y - o u t

in both directions). o

such that

~X-mOdules G, and all i > n . o

Tor. (F" ,G) = O for all 1

96 Proof.

The proof is entirely analogous to the proof of

[I.7.6] and will be left to the reader. I) an all

x-mOdule YX -m~

F G,

Important points are

is flat if and only if 2) if Z

> B

> O

e = ~X) such that Z@G ~ B~G, and

[I.4.5]

for flat modules fails.

for

is a surJection of

~x-mOdules with Z ~ B, then there exists an example

TorI(F,G ) = O

~x-mOdule e (for 3) the analogue of

Instead, we use the commutative

diagram F'"

>

a>n(F'')

F"

>a>n(F')

!

where

F "

Since

F"

O~n(F" )

is a (not necessarily bounded) itself is a bounded complex,

is equal to F'o

Hence

flat resolution of F'.

for n small enough,

a>n (F'')

> F"

is the required

quasi-isomorphism. Remark.

We see from the proof that it is sufficient in (iii)

to consider only quasi-coherent to consider only the sheaves Similarly,

sheaves G.

~X' and

k(x)

Indeed, it is sufficient for every point

x q X.

in (ii) it is sufficient to consider the restriction of

the functor F~174 "= to

Dqc(X)o

97

Definition and Corollary

4.3.

If

F" 6 0 b

Kb(Mod(X))

satisfies the equivalent conditions of the Proposition, we say that

F"

has finite Tor-dimension.

localizing subcategory of Kb(Mod(X))fTd . denoted by

K(Mod(X) ), which we denote by

The corresponding subcategory of

Now let

F" 6 0 b

Kb(Mod(X))fTd

modules.

is

Cf. [I.7.7]. K(Mod(X)), and consider the functor

Kb(M~ L

D(X)

Db(x) fTd"

Proof.

Let

These complexes form a

> be the subcategory of complexes of flat

Then by Proposition 4.2 and Lemma 4.1, part a2,

satisfies the hypotheses of [I.5.1 ] for this functor.

L

Hence we

can take the left derived functor in the second variable, and obtain

~=~:

K(Mod(X))

• Db(X)~d

> D(X).

By Lemma 4.1, part b2, this gives rise to a functor L I Lvr@:

D(X) X Db(x)fTd

which we will also denote by @.

> D(X)

98 Problem.

|

Does the functor

K(Mod(X)) • Kb(Mod(X))

> K(Mod(X))

d

admit a left derived functor in the first variable? G" E Ob Kb(Mod(X))fTd,

L

noetherian prescheme),

Then

does there exist a subcategory

K(Mod(X)) satisfying the hypotheses of [I.5.1] for "~G~ Proposition ~.3.

let

Given

Let

X

and let

G ~ E Dqc(X) (resp~ Dc(X)).

a)

F" E D-(X), o r

b)

G~ E Db(X)fTd.

F" _@_ G" E Dqc(X) Proof.

Using

be a prescheme (resp. a locally F ~ E Dqc(X)

(resp. Dc(X))

Assume either

(resp. Dc(X))~

[I.7.3] as before, it is enough to show that

are quasi-coherent

(resp~ coherent)

&-modules,

Tori(F,G ) is quasi-coherent

(resp. coherent)

for all

if

F,G

and

i > O~

The question is local on X, so we may assume X affine. has a resolution by quasi-coherent

(resp~ coherent)

Then

F

~x-mOdules,

namely direct sums (respo finite direct sums) of copies of We can use this resolution to calculate the

then

Tori(F,G).

~X o

99

Suppose now that

f: X

> Y

is a morphism of preschemes.

Then we have

f*: Mod(y) ---~Mod(X), and we can take its left-derived functor

Lf*

D-(Y)

> D-(x)

since there are enough flat If f

f~

~y-modules, and they are fW-acyclic.

has finite cohomological dimension on Mod(Y), then we say

has finite Tor-dimension, and we can extend the domain of

definition of

Lf ~

to

r.,:~: D(y) Proposition ~.~.

Let

> D(X) f: X

> Y

(resp. locally noetherian preschemes).

Then

If

takes

into

Tor-dimension,

the same is true for unbounded complexes.

Left to reader.

D~(X)).

Lf ~

(resp. D~(Y))

Proof.

Dqc(X ) (resp.

be a morphism of preschemes

f

Dqc(Y )

is of finite

1OO

~5.

Relations among the derived functors. In this section we will make a list of various natural

homomorphisms and isomorphisms among the derived functors discussed in the previous sections. P r o p o s i t i o n S . i.

Let

morphisms of preschemes.

~+:

R+(g~.f,~)

of functors from

f: X

>Y

and

g: Y

> Z

be t w o

Then there is a natural isomorphism

~

>,__+g..R§

D+(X) to D+(Z).

Suppose furthermore that

X

and

noetherian of finite Krull dimension.

Y

are Then there

is a natural isomorphism

a(g,~.f~) of functors from Proof.

D(X)

~ > to

We use [I.5.~b].

L _~ K+(Mod(X))

Rg, 9 Rf,

D(Z). For the first statement, let

be the complexes of injective sheaves, and let

M c K+(Mod(Y)) be the complexes of f.(L) _c M

(indeed, f.

g.-acyclic sheaves.

Then

of an inJective sheaf is flasque

[EGA O7~12.2.4 ] and a flasque sheaf is g.-acyclic [EGA OiiI Then the hypotheses are satisfied, so we have ~+.

12.2.l]).

101

For the second statement,

let

L c

K(Mod(X))

be the

complexes of sheaves which are f.-acyclic and g.f.-acyclic. Let

M ~

K(Mod(Y))

be the complexes of

g.-acyclic

sheaves.

Then by the first statement of the proposition we see that f.(L) ~ M. then and

Indeed,

if

F

is both f.-acyclic and g.f.-acyclic,

~f.(F) ~ f.(F), so for f.F

is g.-acyclic.

the collection g.f.-acyclic

P

satisfies

Rlg.(f.F)

~ Ri(g.f.)(F)

= O,

On the other hand, one sees easily that

of sheaves on

satisfy the hypotheses by

i > O,

(i)

X

which are

f.-acyclic and

and (ii) of [I.~.6],

so L and M

of [I.5.1] , and we have the isomorphism

[I.5.4b].

Proposition 5.2.

Let

f: X

> Y

be a morphism of preschemes.

Then there is a natural isomorphism

~+: _R+r(x,.) ~ > ~+r(Y, ~f.(. )) of functors

from

D+(X)

If furthermore dimension,

to

D+(Ab).

X and Y are noetherian of finite Krull

then there is a natural isomorphism

r of functors

St(x,.) from

~ >

~r(y, ~f.(. ))

D(X) to D(Ab).

102

Proof.

Similar to proof of previous proposition.

Proposition

Let

5.3.

X

be a prescheme.

Then there is

a natural isomorphism

R Hom'(F',G') of bi-functors Proof.

Gj then

N

>

D-(x)" • D+(x)

from

Use [I.5.~].

Hom(F,e)

from

~:

~Z-mOdules

> Y

G

and

g: Y

> z

from

to

D-(x).

f and g have finite Tor-dimension

N

D-(Z)

> to

Left to reader. into flat

be

~-f* . ~-g*

>

Lf*

(see w

. Lg*

D-(X). (Note that

~y-mOdules.)

and

and hence F-acyclic.

Then there is a natural isomorphism

D-(Z)

L(g.f)*

F

is injective,

g.f, and there is a natural isomorphism

then so does

Proof.

f: X

_L-(f*.g*)

If furthermore

of functors

[G, II 7.3.2],

Let

morphisms of preschemes.

C:

D(Ab).

Also note that if

is flasque

Proposition 5.4.

to

Note that for any two sheaves

= F(Hom(F,G)).

Hom(F,G)

of functors

RF(X, R H o m ' ( F ' , G ' ) )

g

takes flat

lO3

Proposition 5.5.

Let

f: X

> Y

be a morphism of

preschemes, with X noetherian of finite Krull dimension.

Then

there is a natural functorial homomorphism af.= R= HOm'x(F',G')

for

F" E D-(X) Proof.

and

>

R= Hom'y(R=f.F', R_f.G')

G" 6 D+(X).

Note first that our hypotheses on f,Y,F', and G"

ensure that both objects above are defined (cf. w

and ~3 above).

We wish to define a morphism between two functors from D-(X)" • D+(X)

to

D(Y).

Let

of f,-acyclic objects, and let of injective objects.

L ~ K- (Mod(X)) be the complexes M G K+(Mod(X))

be the complexes

Then, as we have seen before ([I, m5] and

~2 above) the natural functors

and

Lqi s

> D-(X)

MQi s

> D+(X)

are equivalences of categories.

Hence it will be sufficient to

define a morphism between the extensions of our functors to functors from

LQi s

x MQi s

to

D(Y).

applied to the triangulated categories

But now, using [I.3.~] L

and

it is enough to define a morphism of functors

M, we see that

lO4

Rf~ R HOm'X (QF~

(where or

Q

M

to

QG')

> R HOm'y(Rf~QF',= Rf~QG'= ) ,

denotes the localization functor from u-M-is) for

F"

6 L

and

G"

L

to

LQi s

6 M.

We now make explicit the morphisms

~

between a functor

and its derived functor (cf. definition of derived functor, and obtain the following diagram:

[I.5])

f~

Qf..om'(F',G')

(i)

Q Hom'(f.F',f.G')

~ ~f~ Q ~om ( ~ ' , G )

Rf. R__ Hom'(QF',QG ")

(3) l~Hom R Hom'(Qf~F',QfwG')

(~)

: %

f~

~

R_ H o ~ ' ( 0 f j " ,~f~QG " )

Here

~

f~ (5)

R-- Horn"(Rf~QF',Rf~QG') -

.

is deduced from the well-known natural map f~ HOmx(F,G )

for any two sheaves of

> Homy(f~F,f~G)

~X-mOdules

F,G.

Now we use the last

statement of [I.5.1] to deduce that certain of the ~'s are isomorphisms.

lO5

(i)

Since

G"

is made of injective sheaves, Hom'(F',G')

made of flasque sheaves [G, II.7.3.2] are involved since are

F" 6 K-

and

(note only finite products

G" 6 K+).

But flasque sheaves

f.-acyclic, so they can be used to calculate

the map

(2)

~f.

is

~f., and hence

denoted by (i) above is an isomorphism.

G" is injective (or more precisely "M satisfies the

hypotheses o f [I.5.1] for Hom'(F',')")

and so

~Hom"

is an

i somorphi sm.

(3)

We can say nothing here.

(4)

G" is injective,

(5)

F" is made of fe-acyclic sheaves, so ~fw here is an

so

~f~

is an isomorphism.

isomorphism. Therefore,

since (1), (2), (4), and (5) are isomorphisms,

there is a unique ~: Rf. R Hom'(QF',QG')

making the diagram commutative.

> R Horn"(Rf QF',

f.QG')

This is the desired morphism of

functors.

Remark. the method.

We have given the above proof in some detail to show Faced with a similar situation in the sequel, we will

say simply "we may assume F" is made of f~-acyclic objects, and

lo6

G

.

on inJective objects", and then we will drop all Q~s, and In other words~

write "=" for any ~ which is an isomorphism. we use the convention that we may erase the applied to an argument for which

~

R

before a functor

is an isomorphism.

So if F"

is a complex of f~-acyclic objects, we will write ~f.(F') = f~F'. Proposition 5.6 (Pro~ection formula).

Let

f: X

> Y

be a quasi-compact morphism of noetherian preschemes, of finite Krull d i m e n s i o n .

Then there is a natural functorial isomorphism

"

=Rf~.(F') ~y G ~

for

F" E D-(X) and

Proof.

>

=Rf~.(F" ~X = Lf'x'G')

G ~ E Dqc(Y).

Note first that both sides are defined.

the morphismj we may assume that sheaves, and that

G*

we get the m o r p h i s m b y

is a complex of

G"

is a complex of

f~-acyclic

~y-flat sheaves.

Then

composing the usual projection formula

for sheaves with a suitable

f~(F') |

F"

To define

~:

> f~(F'|

)

using the conventions of the remark above.

> Rf~(F'|

lo7

To show it is an isomorphism, we must show for each

i

that

the map of sheaves

> Hi( f.(F x

HI(Rf,(F" )=@yG')

is an isomorphism.

But this question is local on Y.

Furthermore,

the functors involved are all compatible with localization on Y, so we may assume

Y

affine.

Now, going back to the original

morphism, note that both sides are way-out left functors in Note also that every quasi-coherent free

~y-module,

Way-Out Functors

since

is affine.

[I.?.l (ii) and

to the case where a free

Y

~y-module.

G"

~y-module is a quotient of a Hence, using the Lemma on

(iv), dual

statement], we reduce

is reduced to a single sheaf

But now

G'.

G, which is

X, Y, and f are quasi-compact,

so

everything commutes with infinite direct sums, and we reduce to the case

G = ~y.

(The noetherian hypothesis ensures that

Raf~

commutes with direct sums, cf.

Then we have

Rfw(F') on each side and we are done.

Remark. the method.

[T.3.6.2] and [G,II w

A g a i n we have given a proof in some detail to show In the sequel, we may leave many of these details

to the reader.

lo8

Corollary 5.7. assume that

f

Let

f,X,Y

be as in the Proposition,

has finite Tor-dimension.

finite Tor-dimension

Let

and

F" E Db(x)

(see Definition A03 above).

have

Then

also has finite Tor-dimension. Proof.

We use condition (ii) of Proposition 4.2.

has finite Tor-dimension,

G"

is way-out right. G.

i

Dqc(Y )

> Lf~G"

Since i

the functor on

Since

F"

also has finite Tor-dimension,

> F" ~X =Lf~G"

is way-out right, and so G" is way-out right.

|

> -R-f~(F~ =~X Lf~G')

By the Proposition 3 this implies that the

functor G" is way-out right (for

> Rf~(F ~ ) | G ~

G" E Dqc (Y))"

Rf~(F ~ ) has finite Tor-dimension Proposition 4.2).

But this implies that

(using the Remark following

f

109

Problem. that and

f

Let

f,X,Y

be as in the Proposition,

has finite Tor-dimension.

~f~(F'~ ~f~G')

are both defined for

5.8.

Let

f: X

F" 6 D(Y)

noetherian,

> Y

Then there is a natural

f* R HOm'y(F',G')

for

Are they

it is easy to prove that it is an

using the lemma on way-out

Proposition preschemes.

and

The problem is to define a morphism between them,

since once we have a morphism, isomorphism,

Rf~(F')|

F" E Db(X)fTd ,

using the result of the Corollary.

G" 6 Dqc(Y) , isomorphic?

Then the functors

and assume

and

functors.

be a flat morphism of

functorial homomorphism

> R Hom'x(f~F',f~G" )

G" 6 D+(Y).

and F" 6 D~(Y),

If furthermore Y i s

it is an isomorphism.

locally

(We write

fe instead of Lf e since it is an exact functor.) Proof. of injective

To define the map, we may assume sheaves,

is a complex

and then use the natural map of sheaves

feHOmy (F, G )

To show the isomorphism,

> Hornx(f~F, f~G).

we may assume that

then we reduce to the case functors.

G"

F = 6y

Y

is affine, and

by the lemma on way-out

iiO

proposition 5.9. preschemes.

Let

~>Y

be a morphism of

Then there is a natural functorial isomorphism

Lf'(F') ~X %f'(e') :

for

f: X

~

> %f~(F'~e')

F',G" 6 D-(Y). Proof.

Left to reader.

n

Proposition 5.10.

Let

f: X

> Y

be a morphism of

noetherian preschemes of finite Krull dimensiQn.

Then

there is a natural functorial homomorphism

p:

for

F" ------->Rfe LfeF"

F" E D-(Y), which gives rise by Proposition 5.5 to a

natural functorial homomorphism 7: Rf~ R HOnt'.(Lf~F',G" )

for

F" 6 D-(Y)

and

> R Ho

(F ,Rf.G')

G" E D+(X).

If furthermore F" E Dc(Y), then

T

is an isomorphism.

111

Proof. flat

To define p, we may assume

F"

~y-modules, compose the natural map

To check that

7

Rf.G"

F" --->f.f*F"

is an isomorphism, we may assume

then reduce to the case simply

is a complex of

F" = ~y . Then

Y

with ~f.. affine, and

Lf*F'= = ~X' and we have

on each side.

Corollary 5.11.

Under the hypotheses of the Proposition,

we have

HomD(x)(Lf*F',G') in other words, to

D-(X)

and

Lf*

" > HomD(y)(F', Rf.G')

and

D+(X)

,

Rf.= are adjoint functors from

to

D+(Y)

C

Dc(Y )

respectively. '

Proof.

Apply

H~

to both sides of the isomorphism

7

of

the Proposition, and use Propositions 5.2, 5.3, and [I.6.4]. Proposition 5.12.

Let

f: X ---~Y

be a morphism of finite

type of noetherian preschemes of finite Krull dimension. Let u: Y' be a flat morphism, let X' = XXyY'

and let J

v X

v,g

be the projections, as shown.

I

-

>X

Ig f

Then there is a natural functorial y. isomorphism U . R f.F . for

F" q Dqc(X ).

N

> Rg . v ' F "

U

->Y

>Y

112

Proof. complex of u~f~F ~

To define the morphismp w e m a y assume f~-acyclic sheaves,

> g~v~F . followed by

F"

is a

and use the natural map ~g .

Both sides are w a y - o u t

in

b o t h directions,

so to prove the isomorphism we reduce by the

lemma on w a y - o u t

functors to the case of a single quasi-coherent

sheaf

F

on

Y.

Then we must show that for each i~

Hi(u~af~F)

N

> Hi(Rg~v~F)

This is [EGA III 1.4.15]o cohomology,

natural

(Recall that the proof uses ~ech

hence the q u a s i - c o h e r e n c e

P r o p o s i t i o n 5.13:

.

Let

X

hypothesis. )

be a prescheme.

T h e n there are

functorial isomorphisms F" | G" ---~-~ G" | F"

and

F ~ | (G*| H')

for all F',G ~ Proof.

and H ~ E D-(X).

Left to reader.

P r o p o s i t i o n 5.1~. natural

N > (F~174G') | H"

Let

X

be a prescheme.

Then there is a

functorial h o m o m o r p h i s m

R Hom'(F',O')

H"

>R Hom'(F',

H')

113

for

F" E D(X), G" E D+(X), and

X is locally noetherian, and Proof.

H ~ 6 Db(X)fTd 9

If furthermore

F" E D~(X), then it is an isomorphism.

Left to reader.

Proposition 5.15.

Let

X

be a locally noetherian prescheme,

and assume that every coherent sheaf on X is a quotient of a locally free sheaf of finite rank (iffr).

Then there is a natural

functorial isomorphism

R Ho m'(F', R Hom'(G',H')) ~ for

F"

and

Proof.

H" E D+(X).

G" E Dc(X) , and

To define the morphism, w e use the result of the

exercise in w

above.

W e take resolutions of

iffr's, and use them to calculate flat, and so if F'|

R Hom'(F'__@G', H')

F',G"

R Hom'.

are complexes of

F"

and

G"

by

Note that any iffr is Iffr' s, then

= F'~G', which is also a complex of iffrJs.

For the

isomorphism, we use the lemma on way-out functors and reduce to the case

F" = ~X"

Proposition 5.16.

Let

X

be a prescheme,

and let

bounded complex of locally free sheaves of finite rank. L-v

__

L"

be a

Let

Then there are natural functorial isomorphisms

114

R Hom'(F',G ~

~ L~

N >R

Hom'(F~174

.)

~ > R Horn" (F~174"w, G')

for all

F" E D-(X)

Proof.

G ~ E D+(X)~

Easy once one notes that the corresponding

hold for sheaves, Iffr~ then

and

G~L

and if

G

is an inJective sheaf, and

is inJective.

formulae L

a

115

w

Compatibilities In situations

among the relations

involving

three or more derived

there may be different ways of composing isomorphisms of functors.

Let

morphisms

functors,

the homomorphisms and

of section 5 to obtain a homomorphism

or isomorphism

One would like to k n o w that the result is independent

of any choices. i.

of w

We give three examples.

f: X

>Y,

g: Y

> Z, and

h: Z

Then by Proposition

of preschemes.

> W

5.1 there are

i somorphi sms

R+(h~g~f~)

~

> a+h. R+(g~f.)

t We would 2.

> _R+h~ a +g~ a+f~

like to know that this diagram is commutative. Let

f: X - - > Y

of noetherian preschemes

and

g: Y

> Z

be morphisms

of finite Krull dimension.

Then there are functorial homomorphisms

he three

116

>Rg. =Rf* R= Hom~F',G')

R__(g.f.)R HOmx(F" ,G" )

1 ~g. ~ Hom'y(~f.F',~f~G')

1

> a ~om;,(ag.afJ',ag~ af.G')

R= Homz(R(g.f.)F',R(g.f~S" ) = =

for

F" E D-(X)

and

G ~ 6 D+(X)~

Here the horizontal arrows are

deduced from Proposition 5.1, and the vertical arrows from Proposition

5.5, and we have tacitly included in the left-hand

vertical arrow a double use of the natural isomorphism g.f. ~ (gf).. We would like to know that this diagram is commutative.

3~

Let

X

be a prescheme.

Then there are functorial

i somor phi sms

F ~

=@(G~176

(F~ for

F',G~

N

~ | (H~174~ ~

(G~

I ~)

(F" | (G'|

i"

I'> F " ~ (

> ((F'__eG')__e~') ~ I"

and I ~ E D-(X), by Proposition

like to know that this diagram is commutative.

5.13.

We would

117

In the first example, R e m a r k after

[I.5.4].

the c o m m u t a t i v i t y

and i s o m o r p h i s m s

uniquely

d e t e r m i n e d by the c o r r e s p o n d i n g

sheaves,

and by the c o n d i t i o n

derived

functor.

the a n a l o g o u s

~

mind.

of the

then follow from

are only three of m a n y more similar come i m m e d i a t e l y

However,

to the r e a d e r ' s could prove

I w o u l d be sure to need some

to w r i t e

down.

these d i a g r a m s and c h e c k i n g

almost mechanical, I w r i t e them. existed,

for

and a l r e a d y the list of ones I can think of off-

hand is too c u m b e r s o m e inventing

to are

that they commute w i t h the

I could make a big list, and in p r i n c i p l e

later,

we

for sheaves, w h i c h w e m a y a s s u m e known.

which will

each one on the list. more

n a t u r a l maps

w h i c h occur in the d e f i n i t i o n

N o w these examples compatibilities

referred

The r e q u i r e d c o m m u t a t i v i t i e s

results

from the

In the second and third examples,

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

relevant morphisms

follows

since the chore of

their c o m m u t a t i v i t y

the reader w o u l d not w a n t

It w o u l d be c o m f o r t i n g

or to have a m e t a - t h e o r e m

one w o u l d dream up is commutative. possibilities

And

nor

to k n o w that such a list

saying that any such d i a g r a m However,

seem of an order of c o m p l e x i t y

treat in these notes.

to read them,

is

b o t h of these too great to

ll8

Unfortunately,

I will have to use many of these compatibilities

in an essential w a y in w h a t is to follow.

Perhaps

for each theorem

in the sequel one could make a list of exactly which compatibilities are needed, stage.

and verify them, but even that is too clumsy at this

So I must ask the reader's

truth of the theorems truth also.

stated,

indulgence.

for a rigorous

every commutative

proof,

that any reader will have the patience In particular,

I believe

one could imagine between

in the

and I hope to convince him of their

But I have not verified

w h i c h is necessary

I believe

and I do not suppose

to do so either.

that all reasonable

the "natural"

diagram

compatibilities

homomorphisms

and

isomorphisms

of section 5 are true, and so in the sequel I will

write

"=" instead of naming the isomorphism

simply

course,

the relevant hypotheses

are satisfied).

chapters we will deal with more homomorphisms where

the compatibilities

and may even be false

(e.g., where

we will need,

For a completely "natural"

isomorphisms

However,

of

in later

and isomorphisms,

one can imagine are not always obvious, there is a question

Hence I will name those isomorphisms compatibilities

(provided,

of sign).

and list (in principle)

the

and k e e p track of them.

satisfactory

treatment

of this question

and their compatibilities,

of

we must await

119

future developments.

Mac Lane

[12 pp.

1~,15]

problem in the context of the associativity

refers to the

of the tensor product,

where he says that he does not know even a finite number of diagrams

such as the one in example 3 above, which together imply

that all such diagrams involving associativities Perhaps the language of fibred categories or the techniques of [Giraud,

[SGA

are commutative.

60-61,

expos~ VI]

thesis] will supply what is needed.

120

InJective

sheaves on a locally noetherian

In this section we give the structure objects

in the category Mod(X)

noetherian

prescheme X.

quasi-coherent injective

of all

We show in particular

an object of A.

Let

[ 5 ], which in turn were inspired

[13] in the case of noetherian rings.

A

be an abelian category,

An inJective

hull of

I of A, together with a monomorphism a non-zero

subobject

Theorem category

7.i [5

of I, then , Ch.

with generators

object has an injective Examples. has generators

functor

of categories.

of Matlis

Definition.

in a quasi-coherent

> D + (x) qc

We use results of Gabriel from results

on a locally

that every

and hence that the natural

D+(Qco(X)) is an equivalence

of the injective

~X-mOdules

~X-mOdule can be embedded

~x-module,

prescheme.

M

be

is an inJective

object

> I, such that if

N

M M

and let

is

M N N 40.

II, Thm.

2].

If

and exact direct

A

is an abelian

limits,

then every

hull.

The category of modules and exact direct

limits,

over a commutative hence has inJective

ring hulls.

121

The category Mod(X) of ~X-mOdules on an arbitrary prescheme X has generators and exact direct limits, hence has injective hulls. Definition A

[ 5 , Ch. II, w

An object of an abelian category

is noetherian if every ascending chain of subobjects is

stationary.

An abelian category

A

is locally noetherian if it

has exact direct limits, and has a family of generators consisting of noetherian objects of Examples.

A.

The category of modules over a noetherian ring is

locally noetherian

the ring itself is a noetherian generator

for the category. The category

Qco(X) of quasi-coherent sheaves on a noetherian

prescheme X is locally noetherian, with the coherent sheaves forming a family of noetherian generators. Gabriel

This example was studied by

[ 5 , Ch. VI].

Proposition ?.2 [5 , Ch. IV, Prop. 6].

Any direct sum of

injective objects in a locally noetherian category is injective. Theorem 7.~ [ 5 , Ch. IV, Thm. 2]. noetherian category.

Let

A

be a locally

Then every injective object I of A is

isomorphic to a direct sum

~ I. j 6J ]

of indecomposable injectives

122

Ij.

Furthermore,

if

~ Ik k~K

then there is a bijection

is a second such decomposition,

h: J

> K

such that

Ij

I h (j)

for each j. Proposition 7.4 [13].

Let

A

be a noetherian ring.

Then

the indecomposable inJective A-modules are precisely the injective hulls of

I(~) of

k(~) over A, where

~

ranges over the prime ideals

A. W e also have information about the structure of one of these

injective

hulls

[Matlis, ibid.]:

Proposition 7.5.

Let

prime ideal of A, and let field

k(~) of

A I

be a noetherian ring,

let

~

be a

be an injective hull of the residue

Then one can write

I

as a direct limit of

the submodules 0 c E1 c E2 c m

where ring

Ei =

~

, and where,

I)

for each

is an A r t i n module over the local

i,

123

Now we apply these results to locally noetherian preschemes, giving first a special case. Definition. M

~x,x-module.

and

i.(M) and

Let

on X, where M

X

be a prescheme,

Then we define i: Spec ~x,X

is the sheaf on Spec

Remark.

a point of

X, and

to be the sheaf

is the natural inclusion,

associated to M.

We will be particularly interested in the case where

Supp M is just the closed point F(U,ix(M)) = M

ix(M )

> X

~x,X

x

if

x

x 6 U, and

O

of Spec ~x,X"

In that case

otherwise, i.e., it is a simple i I

sheaf on the closed subset

{x]-

of

Xi.

Moreover,

for any

~X-mOdule F, we have

HOm x( , ix(M)) -- Omspec where

F

is the stalk of

Hom

F

at

x.

X

be a locally noetherian prescheme,

X

let

Proposition 7.6.

Let

x

X, and let

I

of x.

Then

be a point of

over the local ring inJective

~X-mOdule.

~x

,X

be an injective hull of k(x) ix(I )

is a (quasi-coherent)

124

Proof. point of

By Proposition 7-5 above, Therefore,

Spec ~x,X"

and

~: F

map

F

> I, which extends to a map

x

~x,x-module.

of

ix(I )

to

Lemma 7.7. G ~X

F ~ G

is a homomorphism

> ix(I )

injective G

if

are two

X

G

x

> I

ix(I )

finite number of sections

x 6 U., generate the stalk 1 X

I

is an

this gives a map ~X-mOdule.

and let

(not necessarily quasi-coherent). U. ~ X, and a 1

sij 6 r(Ui,G ) for each i, such that at Sij' for those

x 6 X, the sections

Since

since

is an inJective

Then there are a finite number of open subsets

Proof.

gives a

be a noetherian prescheme,

be a sheaf of ideals

each point

~X-mOdules,

of sheaves,

But by the Remark above,

extending ~, so

Let

I has support at the closed

G

x

as an

i

such that

~x,x-module 9

can be covered by a finite number of open

affines, we may as well assume

X

is affine,

equal to Spec A

for

a suitable noetherian ring A. Let x 6 X. form of

For each open affine neighborhood U of x of the

Xf = Spec Af for f 6 A, consider the ideal A, where A

=

rCx, x)

> rCu, x)

0t(U) = p-IF(u,G)

125

is the natural

restriction.

gives a larger

ideal.

a neighborhood

U

C l e a r l y a smaller

Hence by the a.c.c,

of x giving

in

a maximal

F(U,~'X) are localizations

such open n e i g h b o r h o o d

U ~ U

of

x

A, there

~(Ux)

X

since the rings

neighborhood is such

01.x .

=

of A, we have

Hence, for every

that

X

(11

r(U,G)

Now if

x'

| r(U, x)

=

r(Ux,G)|

--

is a s p e c i a l i z a t i o n

of

x

in

)" U

i e., x' X ~

then every n e i g h b o r h o o d the stalk

GX

of

x'

contains

can be g e n e r a t e d

I

b e i n g an ideal in a n o e t h e r i a n

by sections

ring,

(Recall that open sets of the form of X.) (~)

Thus we have

established

For each point

neighborhood

U

X ~

So by

x.

(1) we see that

s i E F(Ux,G),

has a finite number Xf

6 Ix]- n U

"

form a base

the following

which,

of generators.

for the t o p o l o g y

result:

x 6 X, one can find an open affine

of x, and a finite

number

of sections

X

s. 6 F(Ux,. G),

such that

x'

for every

1

generate

the stalk

GX

6 [x]- N U , X

the

s. 1

I"

N o w we prove the statement w e can find an open subset m a x i m a l w i t h the p r o p e r t y

U

of the Lemma. of

that the

X

(possibly

By quasi-compacity, empty) w h i c h

lemma is true for

GIU.

is Suppose

126

that

U + X.

component of

Let

x

be the generic point of an irreducible

X-U, and choose a neighborhood

U

with the

X

property (*) above, and also so small that it does not meet any other irreducible component of U U U

lies either in

X

is also true for

U

X-U.

Then every point of

or in Ix]- N U

GIuuu x. ~

X"

so that the lemma

We conclude that

U = X, which

proves the Lemma.

Theorem 7.8.

Let

X

be a locally noetherian prescheme.

Then the category Mod(X) of all

~X-mOdules is a locally

noetherian category. Proof.

Mod(X) clearly has exact direct limits.

provide it with a family of noetherian generators. sheaves

~U' where

is the sheaf

~X on

U

W e must I claim the

is a noetherian open affine of X, and U

and zero outside, will do.

form a family of generators.

To see that

They clearly

~U is noetherian,

G 1 ~ G2 ~ 9

be an increasing sequence of subsheaves.

assume

is noetherian.

X = U

Let

G = ~ G k.

there are a finite number of open sets Sij E F(Ui,G) each

U.

1

which generate the stalk of

si-'3 we can cover

U.l

~U

let

We may

Then by the Lemma,

and sections G

at each point.

with a finite number of open sets

For

127

Uij L

such that each

Gk(ij~)

over

sij I uij~

Uij ~.

comes from a section of a suitable

Doing this for all the

k = max(k(ij~), we find that all the of

G k.

Hence

Let

X

be a locally noetherian prescheme.

Then any direct sum of injective

Corollary 7.10. Then any injective

Let

X

Definition.

~X-mOdule can be written uniquely as a direct

Let

let

x E X, and let

Let

I

be a locally noetherian prescheme,

X x'

~X-mOdules.

be a specialization of

x, i.e., x' 6 Ix]-.

be an injective hull of k(x) over the local ring

We define

J(x,x')

to be the restriction of the sheaf

the closed subset

Ix'I-

of

X = x', we write simply

Let

X

J(x)

specializing to

for

J(x,x) = i (I). x

of

X.

to

Note that

~X-mOdule.

~X-mOdules are precisely the

defined above, for x'

ix(I )

be a locally noetherian prescheme.

Then the indecomposable injective J(x,x')

~x,X"

Ix]-, as in [G, II.Thm. 2.9.1].

is an indecomposable injective

Theorem 7.ii.

sheaves

~X-mOdules is injective.

be a locally noetherian prescheme.

sum of indecomposable injective

J(x,x')

come from sections

G k = G, and so our sequence is stationary.

Corollary 7.9.

If

sij

sij , and taking

every pair of points

x

128

7.12.

Lemma let

I

of

X

be a locally noetherian prescheme, and

be an inJective

at each point ~x

Let

x 6 X

~.-module. 2~

Then the stalk

I

of

I

X

is an injective module over the local ring

x.

Proof.

•

Since

show that whenever finite type, and

x

is a noetherian ring, it is sufficient to

M ~ N ~:

M

is an inclusion of > I

is a map, then

~x-modules of ~

extends to

N.

X

W e can find coherent sheaves at

M _c N

on

X, with stalks

x [EGA.I.9.4.8], and we can find a map of

suitable neighborhood

U

of finite presentation

[EGA

N o w since

I

to

of x, extending ~, since

0.5.2.6].

I

x

extends

Proof of theorem. We define

that the stalk

I

Let

N

in a

This gives a map of NU

% to

to I. I,

~. I

be an indecomposable inJective

Supp(I) to be the set of points

is non-zero.

x

and

is a sheaf

M

is inJective, this extends to a map of

whose stalk at

~X-mOdule.

M

M

Let

x' 6 Supp(I)

x E X

be

such

a maximal

point (i.e., one w h i c h is not a specialization of any other point in Supp(I). )

O'x' - m o d u l e . of

"rXl*

Then by the lemma, the stalk Let

I

Ix.

is an inJective

be an indecomposable injective direct summand O

Then by Proposition 7.4, I ~ is of the form

I(~) for some

129

prime ideal ~ Then

of

~x'"

Let

~

correspond to the point

ix(I(~) ) = J(x), using the notation above.

there is a natural inclusion of Since the stalk ~Z,x'

J(X)x , = I

> Ix., and hence can find a map

in a suitable neighborhood image of Supp(I),

~

U

has support in Z'

and by construction,

Furthermore,

YZ in J(x), where Z = [x]-.

is mapped into

0

x 6 X.

of

we have a map X i'

~Z

x'.

since

I

extending it

Z' = {x'}-.

Let x'

> I

Then the

was chosen maximal in

is inJective at

x'.

Now applying

Lemma 7.7 to the kernel of ~, we see that ~ factors through the restriction #: ~Z

~Z'

of

> I

~Z,Z''

~Z to the closed subset Z', to give a map

(defined on U), and that by shrinking U a bit, we

may assume that

~

is injective on U.

In other words, we have an

injection of sheaves on X,

%U: where

~ Z , Z ',U

Z,Z',U

>I

is the restriction

Now it is easily seen that ~Z,Z' ,U' so

%U

of

J(x,x')

~

Z,Z'

to the open set U.

is an injective hull of

extends to give a map J(x,x' )

necessarily an inclusion is a direct summand,

since

~U

is.

Now

> I, which is

J(x,x' ) being inJective,

so must be equal to I since I was indecomposable. q.e.d.

13o Corollary 7.13.

Let

T h e n every inJective injectives

J(x,x')

Corollary

be a locally

~X-mOdule defined

7.14.

inJective A-module,

X

Let

noetherian

is u n i q u e l y

a direct

prescheme. sum of

above. A

be a n o e t h e r i a n

and let

X = Spec A.

ring,

I

let

be an

is an inJective

Then

~X-mOdule. Proof.

Follows

and Corollary

from T h e o r e m

Propositions

7.4 and

7.6,

let

be an

7.9.

corollary 7.15. Let A-module

7.3,

of finite type,

i

be a noetherian

A

and let

= Ext

N

ring,

be any A-module.

M Then

(M,N) ~

X for all i, w h e r e Proof.

Since Hom

for any

N.

X = Spec A. M

is of finite

(M',N') = Hom

type,

(M,N) "~

Take an inJective

resolution

I" of N.

Then

~"

N

inJective

resolution

is an exact

functor,

of

N

we have

by the previous the result.

Corollary.

Since

is an

131

Lemma

7.16.

~X-mOdule.

Then

is an open cover injective

Let F

is injective

[U ]

If

F

set is injective.

G'

be a prescheme,

of

X

extended

in Mod(X)

such that

is injective, Indeed,

> F I u , we deduce

and let

F

be an

if and only if there

for each ~, F IU

is

Mod (u).

in the category

Proof.

G'

X

then

given

a map

by zero outside

restricted

G' ~ G

on

>F,

where

G ,X U.

F

This

U

extends

to any open

and a map G ,x

is the sheaf

to a map

Gx

~> F,

G ----~FIu.

hence

On the other hand, enough,

by Zorn's

lemma,

family of generators of a subsheaf sheaves small,

G'

~U (which

to show that

of the category,

F

conditions

is injective,

for every

is

~X on U and O outside),

to

F,

~

7.17.

is local, Let

X

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

sheaf

G

it is in a

and for every map ~: G'

G

of i n j e c t i v i t y

Proposition

F

of

form a family of g e n e r a t o r s

question

and let

to test w h e t h e r

extends

to G.

of Mod(X),

> F

Since the

for U a r b i t r a r i l y we

see that the

as required.

be a locally ~X-mOdule.

noetherian Then

prescheme,

the following

132

(i)

F

is an injective

(ii)

F

is isomorphic

for various

x 6 X, the stalk F

(iv)

For all coherent

(v)

There is an open cover

is an inJective

Proof. isomorphic

F

at

is an

x

sheaves G on X, ExtI(G,F)

~ U -module

(i) ~

(ii)

J(x,x' ). But

x = x', i.e., J(x,x')

(iii) ~

(iv)---> (v)

well-known

from

FIU

7.13 that

F

F

is

is quasi-coherent,

J(x,x' ) is quasi-coherent

if

= J(x).

On a noetherian where

[EGA 1.9.4.8],

affine G = ~

Ext commutes with

U = Spec A of X and

7.14,

The result

and the

ring, N is injective

for all A-modules

by the lemma.

F = ~.

Corollary

fact that on a noetherian

(i)

such that

and all the stalks are zero.

ExtAI(M,N) = O

(v) ~

Since

Since G is coherent,

Ext I (G,F)I U = ExtAI(M,N) then follows

X

Clear.

(iv)

passage to stalks,

of

= O.

for all ~.

to a direct sum of

(ii)--~2 (iii)

[Us]

We know by Corollary

J(x,x' ) must be also.

only if

of

x

•x-module.

injective

only if

J(x)

to a direct sum of sheaves

x 6 X.

(iii) For every

each

~X-mOdule.

M

if and

of finite type.

and

133

Theorem

7.18.

Let

be a locally noetherian prescheme.

X

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

~ X - m O d u l e F can be embedded in a

quasi-coherent,

~X-mOdule I.

Proof.

inJective

Indeed, we w i l l show that the inJective hull I of a

quasi-coherent

sheaf

F

is quasi-coherent.

J(x,x'),

a neighborhood U of x' also a section of

I

since

F(U).

s

By shrinking s

is Just

any power of so

I

U

x ~ x' 9 no section of ~Z''

Thus

J(x,x')

U D Z'

w h e r e Z' = Ix'}-.

sheaf F9 it must be

~Z'

of

J(x,x')

= J(x)

w h i c h is

if necessary, we may

is a section of a q u a s i - c o h e r e n t

for if

For

is an injective hull of F, there is

a n n i h i l a t e d by some power of the ideal x = x',

be w r i t t e n

7.13.

and a section s 6 J(xjx')(U)

assume that the support of Since

I

J(x,x'), b y C o r o l l a r y

as a direct sum of sheaves each such

Let

Z'.

This implies

is a n n i h i l a t e d by

is q u a s i - c o h e r e n t 2 and

is quasi-coherent. Corollary

7.19.

Then the natural

Let

X

be a locally n o e t h e r i a n prescheme.

functor

D+(Qco(X))

> D + (x) qc

is an equivalence of categories. Proof.

(Cf.

w

for notations).

Follows

from [ I. 4.8 ] .

134

AS an application, we give the following result on complexes of finite inJective dimension. Proposition 7.20. let A = Mod(X), and let

Let

X

be a locally noetherian prescheme,

F" E Ob K + (A). qc

Then the equivalent

conditions (i),(ii), and (iii) of [I.7.6] are also equivalent to the following: (i)q c

F"

admits a quasi-isomorphism

bounded complex of quasi-coherent injective (ii)

The functor

F"

> I"

into a

~X-mOdules.

a Horn9 ( " ,F " ) from D(A)"

to

D(A)

is

way-out left. (iii)

There is an integer

such that Exti(G,F" ) = O

n O

for all G E Mod(X) and all i > n . "

(iii)

O

There is an integer

such that Exti(G,F" ) = O

n

C

O

for all G E Coh(X) and all i > n . O

Proof.

(i)qc~-~ (i)--~> (ii) ---~,(iii) ~

easy as before.

It remains only to prove (iii)c ~

is similar to the proof of ( i i i ) ~ Proposition

(iii)c

7.17(iv)

and Theorem

are all (i)qc.

(i) in loc. cit. using

7.18

above.

This

135

Example. the category

If

X

is a locally noetherian prescheme,

Qco(X)

of quasi-coherent

not be locally noetherian.

sheaves on

X

may

Thus we do not know the structure of

injectives in that category,

and we do not know whether every

inJective object of

is inJective in Mod(X).

Qco(X)

Here is the example. be a line in Xl, and let

Let xI

be the projective plane,

be a closed point of E 1 .

defined

Xn,En,Xn,

Xn, let

En+ 1 be the exceptional curve, and let

point of

En+ 1 .

define

X1

Then

X

let

X -x n n

L . I (X - x n) n--1

Xn

blown up at the point Xn+ 1

be a closed

Qco(X)

,

to the open subset

is an integral,

However,

to be

Having

Define

X =

where we glue

Xn+ 1

let E 1

Xn+l-En+l

of

X n + l - X n + I.

locally noetherian scheme.

is not a locally noetherian category.

be a non-zero sheaf of ideals of

~X' and let

F

be a

noetherian generator of the category which admits a map into not factoring through

~

noetherian non-zero ideal

.

Then the image of G

of

~X"

F

must be a

Indeed,

136

For each

k = 1,2,...,

let

Yk

be a closed subset of X,

n=k

Then

Y1 > Y2 >

Then

G ~ 1 < G~ 2 < 9 .-

G

"''"

Let

~ k be the sheaf of ideals of

Yk"

which is a contradiction to the statement

is noetherian. W e conclude that the category

Qco(X) does not have a family

of noetherian generators.

Remark.

We do not know if the analogue of Corollary 7.19

is true for unbounded complexes,

i.e., whether the natural

functor

I)(Qco(x))

-

~- Dqc(X)

is an equivalence of categories. to be true when

X

However, we conjecture it

is a regular noetherian scheme of finite

Krull dimension, because in that case the category has finite injective dimension.

Mod(X)

137

CHAPTER

w

III.

DUALITY

FOR PROJECTIVE

MORPHISMS

Differentials. In this section we recall

differentials will

Definition.

eventually,

If

A

we define

differentials

over

1

~B/A'

the best

II]. of rings,

DerA(B,M ) to be the A - m o d u l e We define

B

These results

for the moment

~ B is a m o r p h i s m

of B into M over A. of

but

[SGA 60-61, expos~

seems to be

a B-module,

facts on relative

w h i c h we w i l l need in the sequel.

surely be in [EGA]

reference

some

the module

A, to be the B-module

and

M

of d e r i v a t i o n s

of relative

one-

representing

the functor M In other words, that

> DerA(B,M ) . there is a d e r i v a t i o n

for any B-module

M, the natural

d: B

> ~lB/A

given,

such

map

1 H~

( ~B/A ,M )

> Der A (B ,M )

is an isomorphism. If

1

/y,

f: X - - > Y is a morphism

the sheaf of relative

by considering ponding

modules

open affines

1

~B/A"

of preschemes,

one-differentials

of

we define X

in X and Y, and glueing

over

Y,

the corres-

138

Definition.

[EGA IV 6.8.1]

A morphism

f: X--->Y of

preschemes is smooth if it is flat, locally of finite presentation, noetherian,

and for every y 6 Y, the fibre f-l(y) is locally

and geometrisally regular

(i.e.,

"absolutely non-

singular"). Examples.

i.

A n open immersion is smooth.

2.

A composition of smooth morphisms is smooth.

3.

Smooth morphisms are stable under base extensions.

4.

A prescheme

X

over a field

k

is smooth Y

smooth morphism of preschemes over another prescheme S.

be a Then

/y is locally free (of rank n = relative dimension of X over Y), and there is an exact sequence 0

> f.

Definition.

1

1

Aj Is

~i

x/Y

>0

A closed subscheme Y of a prescheme X is

locally a complete intersection if every point neighborhood by an

U

has a

such that in U, the ideal Jy of Y is generated

~X-sequence,

such that

y 6 Y

i.e., a collection of sections

s I is a non-zero divisor in

Sl,...,s r

~X" and for each

139

i = 2,...,r,

s. 1

Proposition noetherian

Y

in

Let

X

be a locally

smooth over a locally n o e t h e r i a n subprescheme

of X.

prescheme

Then the following

are equivalent: Y

(ii)

~/S -

divisor

[SGA 6o-6111.#.10]

be a closed

(i)

o

1.2.

prescheme,

S, and let conditions

is a non-zero

is smooth over S

> j2j/

is locally

and the sequence

free,

., 1

> ~/

_ > 9

S

is exact, w h e r e

J

>o

is the sheaf of ideals

of Y, and i: Y

> X

the immersion. Furthermore, section

in that case Y is locally

a complete

inter-

in X.

Definition.

Let

X

be a prescheme,

(*)

O ---->E'

.... > E ----> E"

be an exact

sequence

of locally

s, respectively.

-> O

free sheaves

of ranks r, r+s,

and

Then we define an i s o m o r p h i s m Ar+SE

as follows.

and let

Choose

> ArE , | ASE ,,

a basis

el,...,e r form a basis

e l , . . . , e r + s of E locally

of E', and the images

e

such that

J of e r + l , . . . , e r + s

14o

form a basis of E".

Then map > ( e l A . .9 A e ) r

el A ... Ae r +s

Remark 9

| (er+IA...Ae r+s I 9

We have made a choice here, and it is e m p h a t i c a l l y

not true that all imaginable compatibilities ~(w) hold 9 then

For example,

~(~) and Lemma 1.3.

a prescheme X. O

~(~)v

of these isomorphisms

if (~) is the dual exact sequence to (w),

commute only to w i t h i n a factor of (-i) rs.

Let O ~ E 1 ~ E 2 ~ E

be locally free sheaves on

Then the four exact sequences >

--~

E1

E2

> E2/E 1

> O

>

E/E 1

> O

-----> E/E 2

J> O

0

~

E1

>

E

O

~

E2

>

E

O

-----~E2/E 1

> E/E 1

give rise to a commutative

>

E/E 2

> O

diagram of isomorphisms ~ among

their highest exterior powers. Proof.

Left to reader.

Definition. relative dimension Proposition

a) n9

Let

f: X

> Y

Then we define

be a smooth morphism of WX/Y = A n ~ /

y9

Note by

i.i that wvl Y ~ / is a locally free sheaf of rank one.

141

b)

Let

a complete

f: X

intersection

by n equations), we define

>Y

be a closed

immersion which

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

and let

J

locally

be the sheaf of ideals

WX/Y = (An(j/J2)) v , where

j/j2 is locally

n (i.e.,

free of rank n on X,

v

denotes

so that

is locally defined

of X.

dual.

Then

Note that

WX/Y is a locally

free sheaf of rank one on Y. Remarks. immersion,

f: X

> Y is smooth and a closed so the two

coincide.

If

f: X

intersection, then

Note if

then it is locally an isomorphism,

definitions 2.

i.

and if

pr~(WX/y)

differentials

~> Y

is either Y'

>Y

= WX./y , .

and ideals

smooth,

or a locally

is a base change,

This

follows

of subschemes

complete

and X' = X ~

Y',

from the fact that are c o m p a t i b l e

w i t h base

extension. Lemma

1.4.

Let

X

f> Y

g> Z

with

g

smooth.

Then

F

is locally a complete

g, hence

F: X

> X xz Y

F.

Note that PI: W F

1.2 to

v

and WX/XXzY = f wy/Z.

W = X ~

.> X is smooth by base

is a local complete

of preschemes,

be the graph morphism.

intersection,

We apply Proposition

Proof.

section

Let

be m o r p h i s m s

intersection,

and

Y, X, and the extension

from

142

=

j/j2 ~

F._I

~/X

extension,

.

V

Therefore

WX/W = F WW/x.

WW/x = P2~y/Z, *

and

f = P2F,

But again by base so

as

WX/w = f * W y ~ Z

required. Definition

1.5.

Let

X

f > Y --~-~Z

locally noetherian preschemes,

be m o r p h i s m s

and suppose that

of

f,g, and gf is

each either smooth or a local complete intersection.

Then we

define an isomorphism

f,g: ~x/z

~ > f*~Y/z | ~x/Y"

There are four cases to consider. a)

f,g, and gf are all smooth.

of the exact sequence of P r o p o s i t i o n

b) J

Then we take

to be

i~

f,g, and gf are all local complete intersections.

is the ideal of

Y

in

Z, and

K

is the ideal of

X

If in

X,

then we have an exact sequence on X,

o --+ f.(j/j2) We take

~

> (K/J)/(K/J) 2 --~

of this exact sequence,

inverse i s o m o r p h i s m to be c)

> K/K 2

then dualize,

O.

and take the

~.

f is a local complete intersection,

w i t h g and gf smooth.

We take ~ of the exact sequence of P r o p o s i t i o n WX/y , and take the inverse to get ~.

1.2, tensor w i t h

i~3

d)

f

and

gf

local complete intersections,

Then by the lemma above,

F

and

g

smooth ~

9 3-

Y

is a local P2

complete intersection,

X XzY

and we can use

b) above applied to F and P2 to obtain

~r,p2" ~x/Y ~> r*~x•174 ~~215 ~" '

Now by base extension, V

~xjx,zY = f ~/z"

Proposition

~•

Le~

X

f > Y

f,g,h,gf,hg,hgf

complete intersection.

>Z

and taking the inverse, we obtain

g > Z --~-~W

morphisms of locally noetherian preschemes, of the morphisms

gf

= PlWx/z , and by the lemma,

Transposing

1.6.

X

Z

be three

and suppose that each

is either smooth or a local

Then the isomorphisms

give a commutative

diagram ~h,g~f,hg = ~f,g~gf,h

Proof. *Remark. context

"

Use Lemma 1.3. The reader will realize later that the proper

for the notion of

Gorenstein morphism,

WX/Y just studied is that of a

and we will leave him to elaborate on the

144

following indications.

A morphism

f: X

> Y

of locally

noetherian preschemes is called Gorenstein if it is locally of I

finite type, has finite Tor-dimension, in

D+(X)

sheaf

to an invertible sheaf.

and if

f" (~y) is isomorphic

Then we call that invertible

WX/y, and prove that i

f'(F')

for all

=

|

WX/y

F" 6 D + (Y). qc

Smooth morphisms,

and locally complete intersection

morphisms are Gorenstein,

and in those two cases the sheaf

is the one we have already defined. and

g: Y

> Z

are Gor~nstein,

Furthermore,

so is gf,

if f: X

and there is an

isomorphism

For a composition of three Gorenstein morphisms, commutative diagram as in the Proposition.~

there is a

WX/Y > Y

145

w

f~

for a smooth morphism

Definition. preschemes.

Let

f: X

f. >Y

of

Then we define a functor f~:

by

D(Y)

> D(X)

f~(G') = f*(G') | WX/y[n],

where

[n] means

is flat,

so

"shift n places

to the left".

f* = ~f* is defined on all of

an invertible

sheaf on

Proposition u: Y'

> Y

2.1.

X' = X ~ natural

Y'.

D(Y), of

and

WX/Y

f is

D (X)fTd , so

[II w

>Y

L e t " f: X

be a morphism

finite Tor-dimension,

Observe that

X, hence is an element

that the tensor product is defined

let

be a smooth morphism

be a smooth morphism,

of X'

and let

V

~X

/-

Then there is a

isomorphism y.

U

~y

/

=

of functors Proof. include of

from D(Y) to D(X'). This

follows

from

[II 5.4] and

the case of finite Tor-dimension,

WX/Y with base extension

[~l above].

[II 5.9] modified

to

and the compatibility

1#6

Remark.

Following the conventions

of [II w

we write

"=" instead of naming the isomorphism and keeping track of it. However,

in the following Proposition we do not write

because the isomorphism 81 above.

"=",

depends on a choice of sign made in

In general, we will write

"=" below when there can

be no doubt about the isomorphism being compatible with all previous ones, and we will name those isomorphisms where there may be a question of sign, or of choice of coordinates, Proposition 2.2. smooth morphisms.

Let

f: X

> Y and

g: Y

> Z

etc. be two

Then there is an isomorphism

of functors from D(Z) to D(X). of three smooth morphisms

Furthermore,

the isomorphisms

for a composition

~ give a commutative

diagram. Proof.

We define

and the isomorphisms then follows

[II 5.~] and [II 5.9].

from Proposition

Proposition Then f~

~f,g using the ~ of Definition

2.3.

Let

Proof.

f: X

> Y

be a smooth morphism.

and, if X and Y are locally

it takes Dc(Y ) to Dc(X ). Obvious.

The compatibility

1.6.

takes Dqc(Y ) to Dqc(X),

noetherian,

1.5 above,

147 Proposition 2.4.

Let

f: X

>Y

be a smooth morphism.

Then a)

There is a functorial isomorphism

f*(F'~') provided either

~ > f~(F') | f*(e'~

F',G" E D-(Y), or one of F',G" is in

Db(Y)fTd , and the other is in D(Y)0 b)

There is a functorial homomorphism

f~(R Hom'(F',G'))

for

F" E D(Y)

and

G" E D+(Y).

is locally noetherian, and Proof.

Left to reader.

and [II 5.16].)

> R Som'(f*F', f~e" ) It is an isomorphism if

Y

F" E D:(Y).

(use [II 5.8],[II 5.9],[II 5.13],

148

w

R e c a l l of the Explicit Calculations. In this section we recall the calculations

of projective

of the cohomology

space, as done in [EGA III w

First we must

define the Cech r e s o l u t i o n of a sheaf, and we follow Let

X

be a prescheme,

let

[G, II w

ql = (Ui) be a family of open

sets of X, and let

F

be an

~X-mOdule.

Cech complex of F,

C'(~,F),

as follows.

Then we define the

For each p > O, and for each (p+l)-tuple of indices io < "'" < ip

let

Uio

...,ip = U.1o n...n u ip 9

cP(~,F) by giving its sections on an open set

=

io cP+I(~,F)

~(

=

-i )j pj

~i o ' ' ' ' ' i^ j' .." , I p + 1

is the appropriate restriction map on sections of F.

Finally, we define an augmentation

to the product of its restrictions

by sending a section

~.~

~

F(vnui )" Proposition 3.1. [G,

of Xo

II.5.2.1]

Then the augmentation

to the Cech complex

r

C'(%I,F) of

Suppose that It is a covering

gives a quasi-isomorphism

of

F

F (i.e., it is a "resolution"

of F, in the old language). Proposition 3.2. of preschemes, such that

let

flu .

Let

f: X

>Y

be a separated morphism

It = (Ui) be a family of open subsets of

is an affine morphism

for each i, and let

X F

1

be a quasi-coherent f~-acyclic.

~X-mOdule.

Then the sheaves cP('~,F) are

150

Proof.

Since a product of

we need only show that if

U

flu

is an affine morphism,

then

i.(F) is f.-acyclic.

is affine, other hand,

affine morphisms

F i

i: U

Now since i

X

is separated,

FIU

and

[II.5.1],

flu

On the

is acyclic

by [EGA III 1.3.2]. functors

such that

is the inclusion,

is an affine morphism.

fi,

spectral sequence of derived

> X

f

is quasi-coherent, and

sheaves is f.-acyclic,

is an open subset of and if

it follows that since

f.-acyclic

for the

Hence by the

i,(FIu)

is

f.-acyclic. Corollary 3.3. preschemes,

let

%1=

Let (Ui)

f: X

> Y

be a separated morphism of

be an open cover of

flu . is an affine morphism for each i, and let 1 coherent sheaf on X. Then the natural maps

f.(c'm.F))

are isomorphisms

>

in

D(Y).

Follows

[I.5.1] and [I.5.3B].

such that F

be a quasi-

=

(Here

~

the definition of the derived functor, Proof.

X

is the canonical map in cf.

[I.5].)

from the two previous results and from

151

Now we w i l l apply these results Y

be a prescheme,

projective the

T.

X = ~y

space over Y, i.e.,

are indeterminates.

place where and

and let

flu"

T.I ~ O.

Then

to p r o j e c t i v e

space.

Let

be the n - d i m e n s i o n a l where

X = Proj. ~ y [ T o , 9 9 .,T n] For each i, let

U. = X T , the 1 i

~t = (Ui) is a finite open cover of X,

is an affine m o r p h i s m

for each i,

where

f: X

> Y

1

is the projection. On

U

Indeed,

we fix a set of i n h o m o g e n e o u s

o

i

t i = Ti/To,

Let

w = ~X/Y

be the relative

(It is w e l l known

=

coordinates

l,..o~n.

n-differential

forms on X over Y.

that one can find an i s o m o r p h i s m

but we w i l l not use this nature.)

U i ~ n y, affine n-space.

isomorphism,

because

w ~

~x(-n-1),

of its n o n - i n t r i n s i c

Then 7 = dt I A ... A dt n

is a g e n e r a t i n g and g e n e r a t e d to a global

by

section

of

WlUo,

dtl'''''dtn"

Since

section

T 6 r(x,w(n+l)) w h i c h we will also call

7.

since

~o

/y

w(n+l)

is free of rank n, ---- ~X'

9 extends

152

M u l t i p l i c a t i o n by T ...T gives a map from w to o n w h i c h is an isomorphism on U

o,...,n'

w(n+l),

so we can consider the

section wIT

o

.--T

n

o,.-.,n

This section is an n-cocycle using C o r o l l a r y 3.3 above,

) .

in the complex

f~(C'(~,w)),

and so

defines an element

E F(Y,Rnf, cw) ) .

Theorem 3.4. X = ~ y , let

f

[EGA III 2.1.12] be the projection,

relative n-differentials.

Then

Let

Y

and let

Rnf~(w)

be a prescheme, w = WX/Y

let

be the

is an invertible

sheaf

N

on Y, and

7

is a generating

section,

hence it defines an

i somor phi sm 7:

Rnf~(w)

~

>

~y

m

b y sending

7

to i.

Furthermore,

Rif,(~x(m))

for

= O = Rn-ife(wC-m))

O < i < n, m E Z, and for i = O, m < O, and the cup-product f~(~x(m))

x Rnf~(w(-m))

> Rnf~(w)

is a perfect pairing of locally free sheaves for all m > O.

153

Remarks.

i.

Note that the isomorphism

7 we have constructed

above is compatible with arbitrary base extension, in the construction 2.

is flat over Y.

It is natural to ask whether

the isomorphism

stable under automorphisms

of the projective

see later

that indeed it is.

(Corollary

since everything

i0.2)

space,

7

is

and we will

154

w

The trace map In this

section we define

Trpf:

for

G"

and

f: ~

results using

for projective

R f ~ f ~ ( G ")

E D + (Y) w h e r e qc '

Y

space.

the trace

~ > G"

is a locally n o e t h e r i a n

> Y is the projection.

of [II.7]

The definition

on locally n o e t h e r i a n

these results,

isomorphism

we can define

preschemes.

prescheme, uses the Without

the trace map only for

b G"

E D

qc

(Y).

Lemma let X = ~ ,

4. I.

Let

Y

and let

quasi-coherent

be a locally n o e t h e r i a n > Y

f: X F

sheaf

on

prescheme,

b e the projection.

X

is a quotient

T h e n every

of a sheaf of the

form L = @ f~(Gi)(-mi)'" where

the

G. are q u a s i - c o h e r e n t 1

sheaves

on Y, and the m. > O 1

are integers. Proof. subsheaves, each

Since

F

is the direct

we m a y ~ a s s u m e

that

m > O we have a natural

F

noetherian

from Serre's open

itself

is coherent.

For

map

f~f~(F(m) ) and we k n o w

limit of its coherent

theorem

> FCm), [EGA III 2.2.1]

subset V ~ Y, the r e s t r i c t i o n

that for each

of this map to

155

f-l(v) is surJective

for large enough m.

@

Hence the map

>F

m>0 is surJective on all of X, so we are done. Lemma 4.2. previous lemma, Proof.

For any sheaf L on X of the form of the Rif~(L) = O for i ~ n.

It is sufficient

to show that for G quasi-coherent

on Y, and m > O, Rif~(fW(G)(-m))

= O for i ~ n.

local on Y, so we may assume Y quasi-compact, formula

[II.5.6]

(note f W ( G ) ( - m ) =

Rf~Cf~CG)C-m))

But

Rif~(~x(-m))

N >

= O for i 4 n

(Theorem 5.4), and

The question is

and use the projection

f~(G)| ~x(-m)):

Rf~C~xC_m))

~ G.

by the explicit calculations

Rnfw(~x(-m))

is locally free on Y, so there

is only one non-zero sheaf on the right, and we are done. Proposition let

X = ~

4.3.

and let

Let f: X

Y

be a locally noetherian prescheme,

> Y be the projection.

is a functorial isomorphism Trpf: for

G" E D + (Y). qc

Rf~f~(G" )

N> G"

Then there

156 Proof.

Since

to constructing

X

and

and

D(Qco(X))

:

f~ :

[II.?.lg]

is an equivalence taking derived Note that

Trpf:

dimension

on

says that

with this isomorphism. D(Qco(X))

since

f~

is

Qco(X).)

an isomorphism ~ > G"

G" E D(Qco(Y)).

We apply B = Qco(Y), dimension

[I.7.4]

to the categories A = Qco(X)

and to the functor F = f~.

n on Qco(X).

sheaves of the form P satisfies L(Rnfw)

Rf~f~(G" )

> D + (Y) qc

and [1.5.6] which

is compatible

In fact, we will construct

for all

> D(Qco(X)).

is defined on all of

of finite cohomological

where

J> D(Qco(Y))

D(Qco(Y))

of categories,

we reduce

for G" 6 D+(Qco(Y)),

which says that D+(Qco(Y))

functors

Rf~

are locally noetherian,

a similar isomorphism Rfw

(We use here

Y

L

Let

exists,

P ~ Ob Qco(X)

of Lemma 4.1.

the hypotheses

Now

f~

and

has cohomological

be the collection

Then by the two lemmae,

of loc. cit. and we conclude

and there is an isomorphism

that

of

157

~:

Rf.

~ >

of functors from D(Qco(X)) We apply

to

L(Rnf.)[-n]

to D(Qco(Y)).

f~G" for G" 6 K(Qco(Y)),

which gives an

i somorphi sm _R_f.f~(G ") -- ~f.(f*(G') ~(Rnf.)(f*(G')|

Now each sheaf is

f*(GP)|

Rnf.-acyclic,

| w[n]) .

is in P (since w ~

~x(-n-l)~

hence

so the expression on the right is just

Rnf.(f*(G ") | w). But for each p, the projection Rnf.(f*(GP)|

formula gives us an isomorphism

~ > Rnf.(w) | G p

and composing with the isomorphism 7 of Theorem 3.4, this becomes G p.

Composing all these isomorphisms we have the required

isomorphism Trpf:

Remarks.

i.

Rf.f~G " =

> G" .

Remember that the isomorphism

just defined depends on the isomorphism

~

Trpf

we have

of Theorem 3.4, and

so depends apparently on the projective coordinates.

158 To

If one does not wish to use [II.7.19] and

one can define

Trpf

for

G" 6 D~c(Y)

proj ectio n formula [II. 5.6 ].

[I.7.4],

simply by using the

The following proposition shows

that the two methods of constructing the trace map agree when both are defined.

Proposition 4.4. proposition, F',G"

let

Let

f,X,Y

F',G" E D~c(Y) ,

has finite Tor-dimension.

be as in the previous and assume that one of Then the following

diagram is commutative:

~" =~ ~f.dG"

> R_f.(~*F" ~ f~a" )

Trpf

F'~G"

\~

Rf.f~(F" @ G')

where the upper horizontal arrow is the projection formula [II 5.6] and the right-hand vertical arrow is the isomorphism of Proposition 2.4a.

159

Proof. Y affine.

The question is local on Y, so we may assume Then w e can take a resolution of

sums of copies of

~y ~

F"

by direct

Thus we may work entirely with

quasi-coherent sheaves, and will prove the statement for F~

E D'(Qco(Y)).

Then the result follows easily from the

definition of the morphisms involved,

since if

Cartan-Eilenberg resolution of

f~G', then

Cartan-Eilenberg resolution of

f~F" | f~G" .

C'"

f~F'~C"

is a is a

160

w

The duality theorem for projective space. The duality theorem for projective space now follows

easily from what has gone before.

At the same time it is a

model of how the duality is defined in terms of the functor and the isomorphism functorial theory of and

Trf.

When we have a satisfactory

f: (which is

f~

in the smooth case)

Trf, we will prove the most general duality theorem by

reducing to this case (Chapter VII). Let let

X

Y 9 ~,

be a noetherian prescheme of finite Krull dimension, and let

f: X

> Y

be the projection.

We define

the duality morphism

e f:

for of

=Rf* R= H o ~ ( F ' ,

F" 6 D-(X) [II.5.5]

and

~G')

>R= H o ~ ( R f . F ' ,

G" 6 D + (Y) qc

G')

by composing the morphism

with the trace morphism in the second variable

(Proposition 4.3). Applying the functor

RF(Y,. )

to both sides, and using

[II.5.2] and [II.5.3] we obtain a global duality morphism

@f:

R Ho~(F',

~G')

> R Ho~(Rf.F',

G') ,

161

and taking

the cohomology

i.

Ext~(F',f'G')

Of.

Theorem

5.1.

Krull dimension,

Let

Clearly

Then

~X-mOdule

of

for v a r i o u s

sheaf

is a quotient

X

L = for certain The

~x(-n-l)

D+

qc

so we may assume

that

sheaf on Y is a q u o t i e n t

hence,

as in Lemma of a direct

4.1,

every

sum of

We can take m large,

and using

we see that any q u a s i - c o h e r e n t of a sheaf of the form

m. > O. z

in question

so by the Lemma on W a y - o u t case

m.

G"

~_f,ef,

9 w(-mi)

integers

functors

~y,

is a quotient

w ~

on

of finite

to show that ef is an

is local on Y,

any isomorphism F

and

qc

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

sum of copies

~x(-m)

.

prescheme

D-

it is sufficient

Y is affine.

of

be a noetherian

for all F"

The question

copies

----

Then the d u a l i t y m o r p h i s m s

isomorphism.

quasi-coherent

X

and let

1

of a direct

> Exti(Rf~F " , G ' )

Y

and @f are isomorphisms Proof.

of this, we get m o r p h i s m s

are w a y - o u t

Functors

F" = L of the form above,

right

[I.?.l],

in both variables,

we reduce

and G" = G, a single

to the

injective

162

quasi-coherent

sheaf.

Furthermore,

~ Hom"

transforms direct

sums in the first variable to direct products, the case

F" = w(-m) for m > O.

so we reduce to

Thus we have to prove that the

map

e_f:

Rfe R H o ~ ( w ( - m ) ,

fW(G)|

>R Ho~(Rf~(w(-m)),

is an isomorphism.

G)

By [II.5.16] and the projection

formula,

the complex on the left becomes Rf~(~x(m)) | G[n] = f~(~x(m)) | G[n]

since

m > O and

Rifw(~x(m))

the other hand, Rif~(w(-m))

= O for i > O (Theorem 3.4).

On

= O for i 4 n, so the complex on the

right becomes Homy(Rnf~(w(-m)),

0"y) | G[n],

using again [II.5.16] and the fact that

Rnf~(w(-m))

is a locally

free sheaf of finite rank on Y. Now

ef

is the map deduced from the cup-product

f.(0"x(m)) • Rnf.(w(-m))

> Rnf.(w)

and so it is an isomorphism by Theorem 3.4.

,

q.e.d.

163

C o r o l l a r y 5.2. let that

I

Let

be an A-module, I

is inJective,

A

be a noetherian ring,

and let or that

F" 6 D q c ( X ) .

let X = ~

,

A s s u m e either

Hi(X,F ") is projective

for all i.

Then there is a canonical isomorphism

Ho%(

Hi

(x, ~'), I ) ~ ~xt

n-i

(F', ~ |

I).

0-x

Remark.

When

A

is a field,

I = A, and

F"

is a complex

consisting of a single sheaf, one recovers the duality theorem of Serre for projective

space over a field.

164

~6.

Duality for a finite morphism. f: X

Throughout this section we will let morphism of locally noetherian preschemes. functor

f~

and a morphism of functors

with the same formal properties as the

> Y

be a finite

We will define a

Trff:

Rf.f ~ ----> i,

f~ and Trpf of ~2,~ above.

Then we prove a duality theorem similar to the one of w

This

duality theorem is much more elementary than the preceding one, but it is important to set it in the right functorial context. The reader will notice that the locally noetherian hypothesis is not needed for the definition of the functorial properties,

f~, but it is needed for

and the trace map.

This suggests that

our definition is not the "right" one in the non-noetherian case. On the other hand, we show by an example that the quasi-coherent hypotheses on the sheaves are indeed necessary for a duality theorem. Let

f: X

preschemes. (X,~x)

Let

> Y f

be a finite morphism of locally noetherian be the morphism of ringed spaces

> (Y,f.~x), and let

sheaves of

f.~x-modules on Y.

Mod(f.~x) be the category of Then

and we will consider the functors

f

is a flat morphism,

165

Then

~

~.:

Mod(X)

> Mod(f.0"X)

"{*:

Mod(f.~'X )

is exact, since

> Mod(X). ~

is flat, and the two functors are

adjoint [EGA O, ~.4], i.e., there is a natural map

of functors from Mod(fw~x)

Horn0,X(u

, F)

>

is an isomorphism for

Definition.

H~

"f.x.F )

F 6 Mod(X) and G E Mod(f.0"X).

Let

f: X

noetherian preschemes. f~:

into itself, such that the resulting map

> Y be a finite morphism of locally

Then we define

D+(Y)

> D+(X)

by

f~ (Note that

-- ~* _R Hom~y(f.0"x,.)

.

R Hom~y(f~yx,. ) is considered as a functor from D+(Y)

to D+(Mod(f~x)),

and that

~

is exact.)

166

If

f

has finite Tor-dimension

[If w

then

f~x

has

finite Tor-dimension in the category Mod(Y),

since

fw = f~ o ( |

is coherent, so

On the other hand,

f~x

locally it has a finite resolution by locally free sheaves of finite rank.

We conclude that the functor Ho___mm~( f ~ x , ' )

has

finite cohomological dimension, and so in that case we can define

f~ : D(Y)

>D(x)

by the same formula as above.

Proposition

6.1.

Let

f: X

locally noetherian preschemes Then

f~

Dqc(Y)

takes D + (Y) qc

(rasp. with finite Tot-dimension).

> D + (X) qc

> Dqo(X) and Do(Y)

Proof.

> Y be a finite morphism of

and

D$(Y)

> D~(X)

(rasp.

> Pc(X)).

Follows from [I.~.3],[II.3.2],

and the fact that

takes quasi-coherent sheaves to quasi-coherent sheaves, and coherent sheaves to coherent sheaves. Proposition 6.2.

Let

X

f > Y

g > Z

be two finite

morphisms of locally noetherian preschemes (rasp. with finite Tor-dimension).

Then there is a natural morphism

(gf)~

> f~g~

f

167 of functors from Furthermore, (resp.

D

qc

D+(Z) to D+(X)

(resp. D(Z) to D(X)).

this map is an isomorphism for all

G" 6 D + (Z) qc

(Z) 1.

Proof.

For

G 6 Mod(Z) there is a natural isomorphism

~ (g'f) *H~ O"Z((g f )"x'O"X"G )----->

X

YZ(g*O"Y, G ) ) ,

whence by [I.5.4] the morphism of functors (gf)~

> f~g~ .

To show it is an isomorphism for

G" 6 D + (Z) (resp. D (Z)) qc qc

we use [I.Z.l] and [II.?.lS] to reduce to the case where a single q~asi-coherent

injective

is a quasi-coherent injective rings

A

~Z-mOdule.

~y-module

is

Then g--~HOm~z(gW~y,G )

(since for a morphism of

> B, if I is an injective A-module,

an injective B-module),

G"

then HomA(B,I ) is

so we reduce to the isomorphism of sheaves

mentioned above, by [II.?.14] and [II.7.16]. Proposition 6.3.

Let

f: X

locally noetherian preschemes finite Tor-dimension),

> Y

be a finite morphism of

(resp. with

and let u: Y' --->Y

X'

v

~g

If

be a flat morphism with Y' locally noetherian. Y' ~ Let

X' = X ~

projections.

Y', and let

v,g

>X

Y

be the

Then there is a natural functorial isomorphism

168

v~f ~ (G') for

G" 6 D+(Y) Proof.

N

> g~u~(G .)

(resp. G" 6 D(Y)).

Use [II.5.8].

Corollary 6.4.

Details left to reader.

With the hypotheses of the Proposition,

assume furthermore that u (and hence also v) is a smooth morphism. Then there is a natural functorial isomorphism ~

for

G" 6 D+(Y) (resp. G" 6 D(Y)).

Moreover, under composition

of two such Cartesian diagrams, this isomorphism is compatible with the isomorphisms of Propositions 2.2 and 6.2. Proof. fact that

Follows immediately from the Proposition and the Wy,/y

is compatible with arbitrary base extension.

Proposition 6.5.

Let

f: X ---->Y be a finite morphism of

locally noetherian preschemes (resp. with finite Tor-dimension). Then there is a functorial morphism Trff:

for

> G"

G" 6 D + (Y) qc Proof.

(resp. G" E Dqc(Y)).

Consider the natural map, for G 6 Mod(Y)

169

7:

Horn~y( f ~ x , G )

>~

HOm~y(f~x,G).

This gives rise to a functorial morphism [I.5.4] RT:=

for

G" E D+(Y) (resp. D(Y)).

G" E D+qc(Y)

since

f

I claim R7

(resp. Dqc(Y)).

the case where YX-mOdule.

> af~f~ (G')

R= HOmey( f~x,G" )

is an isomorphism for

Indeed, using [I.7.3] we reduce to

G" = G is a single quasi-coherent injective

In that case

7

is an isomorphism [EGA II.i.4.3]

is an affine morphism, and

f~(G)

is injective (as we

saw above) so we are done. Now composing ( ~ ) - i with the natural map

a Hom (f. x,G')

>G"

derived from the map

om y(f. x,G)

> G,

"evaluation at one", gives Trff.

Proposition 6.6.

i)

Let

X

f> Y

of two finite morphisms as in 6.2 above. commutative diagram

g > Z

be a composition

Then there is a

17o

Trfgf R(gf).(gf) ~

>i Trf g Trff

Rg.R f. f~ g~

+ (Z) of functors on D qc

2) above.

Let

(resp.

Dqc(Z

)) "

u: Y' ---->Y be a flat base extension, as in 6.3

Then there is a commutative diagram u* Rf.f ~

Trff

>

U~

I~

Tr fg

Rg.v. f ~

of functors on

N~

Dqc + (Y)

> Rg.g~u .

(resp. D

qc

(Y)).

(The left vertical arrow

is [II.5.12].) Proo f.

Left to reader.

Theorem 6.7 (Duality).

Let

f: X---->Y

be a finite morphism

of noetherian preschemes of finite Krull dimension.

Then the duality

morphism

f: Rf._ R= ~o~(F', f~G')

> R= HOmy'(RfJ'__,G')

defined by composing [II.5.5] with Trff, is an isomorphism for F" 6 Dqc(X) and

G" 6 D+qc(Y)"

171

Proof.

Making

the usual

following w e l l - k n o w n of rings,

reductions,

statement:

let M be a B-module

let A

we arrive

at the

> B be a h o m o m o r p h i s m

and let N be an A-module.

Then

the natural map HomB(M,

HomA(B,N))

-------->HOmA(M,N)

is an isomorphism.

Example.

One cannot

quasi-coherent of integral

noetherian

non-singular closed,

expect a duality

sheaves,

even

schemes.

Let

theorem

for a finite Y

~tale morphism

be a

__..,

curve over a field k, a l g e b r a i c a l l y

and let

X

be a double

for non-

covering

of

y i

~

Y. T

Let

y 6 Y

be a closed

point,

and let

Xl,X 2 be

Y the two points

lying over y.

be the function the sheaf

K(Y),

fields

consisting

of two copies

injective

at x 2.

~X-mOdules.

and K(Y)

at the point y.

indecomposable

One sees easily

one c o n c e n t r a t e d

K(X)

of X and Y, respectively.

concentrated

(non-quasi-coherent) [II.Z.ll].

Let

that

of K(X),

injective f~(G)

Let Then

G G

is a

[y-module

is the sheaf on

one c o n c e n t r a t e d

It is the direct

be

X

at Xl, and

sum of two indecomposable

172

Now let

F = ~X"

Then we have

f, HOm~x(F,f~G ) = 2K(X)

HOm~y(f,F,G)

= K(X) concentrated

Thus the duality morphism ~f Remark preschemes

6.8.

Let

on the results

at y

at y.

cannot be an isomorphism.

f: X

(not necessarily

a locally complete

concentrated

> Y be a closed immersion locally noetherian)

intersection

in Y [w

of

(rasp. with X

Then we can improve

of this section as follows.

We can define

: D+(Y) (rasp0

f~ :

,> D+(X)

D(Y)

> D(X)

by the same formula as above, complete

intersection,

cohomological

)

noting that if

then the functor

X

is a local

HOm~v(f~x,-)

has finite

dimension.

As in Proposition

6.2,

(gf)

there is a natural map

>

which is defined and is an isomorphism One need only note that so the reduction

~

and

ge

to the quasi-coherent

on D+(Z)

(rasp.

D(Z)).

are the identity maps, case is unnecessary.

173

The trace map of Proposition Trff: is defined

Rfwf~(G " =

for G" 6 D+(Y)

)

6.4, > G" ,

(rasp. D(Y))

since the morphism

7

of the proof is the identity. The compatibilities more general case.

of Proposition

(Here one needs to note that the quasi-

coherence assumption in [II.5.12] closed immersion,

6.6 carry over to this

because then

is unnecessary

f~

if

f

is a

is an exact functor on

Mod(X). ) Finally, and

the duality of Theorem 6.6 is valid

G" 6 D+(Y).

injective

Indeed, we may assume that

~y-modules.

f~ is exact,

Then

f~G"

G"

for F" 6 D(X) is a complex of

is also a complex of injectives;

so we have to show that

f. Ho (F" is an isomorphism.

> Ho (fj" ,G') It is true for each

hence is true for the complexes. on Way-out Functors this time. )

FP,G q

separately,

and

(Note we do not use the Lemma

174

Proposition 6.9.

Let

f: X

>Y

of locally noetherian preschemes. a)

be a finite morphism

Then

There is a functorial isomorphism f~(F') ~ __Lf~(G")

for

F"

b)

6

D+(Y), G" 6 Db(Y)fT d

There is a functorial isomorphism

R Hom'(LfeF ~ f~G') for

F" E Dc(Y ) and C)

G~

E

o

~

> f~(R Hom'(F',G'))

G" 6 D + (Y). qc

There is a commutative diagram (for F" 6 Dbc(Y) and

Dbqc(Y)fTd) [II 5.6]

F" | Rf~f~G"

I

> Rf~(Lf~F'=@f~G ")

Tr ff

F'=~"

I a) Trff


~p_l(!)

is defined by dp(eil A . . . A

For any sheaf

)j

ei ) = P

^

fjeil A...A ei. A...A e i . P 3

Z (-i

F 6 Mod(X) we define

K'(_f;F) - ~Om~X(K.(_f),F) Then a section

is determined by giving its values e. . 11, - 9 9 ,ip

=

1

and the boundary operator ( du ).11''' "' ip+ 9 1 We denote the cohomology Recall K.(f)

[EGA III.l.1]

is a resolution

means of the the maps

e i ) 6 r(x,e)

A...A

~(e i

of

augmentation

,

p

is given by = ) (-I

of

f - ~ il . ... .i j, ] ' ' .

'i p+l "

K'(f;F) by Hi(~;F).

that for (fl,''',fn) ~X/J e:

an A-sequence,

by locally free Ko(f) = ~X ~

~X-mOdules by ~X/J"

Hence

178 K'(_f;F)

> ~ H~ are isomorphisms,

and we deduce isomorphisms

~i.. ~.xS~(~x/j,F ) ~ > Hi(_f;F) for

i = O,...,n.

W e define now a map

~f: Extn( ~X/J,F ) ~ by composing sending

%n

w i t h the map of Hn(f;F)

~ 6 Kn(~;F)

isomorphism.

F/Jlm

to

l , 9 '' , n

M o r e generally,

6 F(X,F).

> F/JF defined by Then

mf

is an

one shows using the Koszul complex

[EGA III 1.1] that there are isomorphisms

(l) for all i = O,...,n. noting that

T o r o ( ~ X / J , F ) = F | ~ X / J = F/JF.

Lemma 7.1. f = (fl,...,fn) generating Let

F

W e have made explicit the case i = n,

Let and

X = Spec A be an affine scheme~

let

g = (gl'''''gn) be two A - s e q u e n c e s

the same ideal J, and let

be a sheaf of ~X-mOdules.

gi = ~ ciJfj

with

cij 6 A .

Then there is a commuative

179

diagram

/

Extn(~x/J,F)

Proof. A c

of

F/~ Idet(cij)

One has only to note that there is an isomorphism

K.(g)

into

K.(f)

which is given in the p

P

A

(cij).

In particular its action on the n

th

th

degree by

degree is det(cij).

The result follows immediately. Proposition 7.2. i: Y ---~X

(Fundamental Local Isomorphism).

be a closed immersion of preschemes, where

Let Y

is

locally a complete intersection in X of codimension n, and let F

be a sheaf on X. ~:

Then there is a natural functorial isomorphism

EXt~x (~y,F)

N

(cf. ~1 for definition of

> F |

Wy/x).

wY/x

Furthermore,

if

F

i*-acyclic, then

EXt~x(~Y. F)

o

for

j ~ n .

is

18o n

Proof.

Let

J

be the ideal of

Y

in

X.

Since

~

j/j2

is locally free of rank one on Y, we have n wY/x = HOmey( ~ j/j2

F |

F/JF)

and this latter is locally isomorphic to Thus we can define an isomorphism that

~

~f.

F/JF (non-canonically).

locally by the condition

followed by evaluation at fl A...A fn (where

f = (fl,...,fn) be

~

3

is an

~X-sequence generating J locally) n When one changes basis of J, ~ j/j2 changes according

to the determinant of the t r a n s f o r m a t i o n . Lemma we see that the definition of

~

Therefore

by

the

is independent of the

basis chosen, and hence the local definitions glue together to give a global To say

9

F

.

is

i*-acyclic is to say that

Torj (O'y,F) = 0

for

.

By the isomorphisms (i) above we see that this is equivalent to the condition on the Ext's Corollary 7.3. proposition.

Let

i:

of the Proposition. Y

> X

and

my/x

be as in the

Then there is a natural functorial isomorphism,

for all F" 6 D(X) l

, i~(F') ----> Li (F') | Wy/x[-n]

.

18l

Proof. ~Y/X

(Note that we write | on the right,

is locally free on

functor.) sheaf

Y

and so tensoring by it is an exact

In the first place,

F, which is

if

i*-acyclic,

n Ext~ (

single sheaf

not @, because

F"

is reduced to a single

then on the left we have the

F) in degree n, by the Proposition,

and

2 %

on the right we have

F |

to it by the isomorphism In the second place, dimension,

~y |

WX/y[-n ]

which is isomorphic

~ of the Proposition. i* is a functor of finite cohomological

because its derived functors are the

Tor~X(~y. , 9 ), and

~'y locally has a flat resolution of length n, namely the Koszul complex mentioned above. (left)-resolution

Therefore

by I9 * -acyclic

every F" 6 D(X) admits a

~X_mOdules"

We are thus in a position to apply [I.Z.4]. B = Sod(Y)

let

F

9 *

be the functor

(i 9 ,.) O"X

be the i*-acyclic i*(') ~y Wy/x

9

i Hom

'

~X-mOdules.

Then

by the Proposition,

G-acyclic also by the Proposition.

G = RnF

which is just what we want.

*

and let

P

is isomorphic

to

y

)

and every element of P is Hence there is a functorial

i somor phi sm RF

Let A = Sod(X),

T,G I-n]

182

Proposition 7.~.

a)

If

Z

> y

i > X

are two closed

immersions which are locally complete intersections of codimensions m,n

respectively,

and if

F" 6 D(X), then there is a commutative

diagram iJ

(ij) ~ (~')

> --L(ij)W(F') @Z ~Z/X [-m-n]

j~i~ (~')

]

j~ (~i*(F 9 )%~y/x[-n])

where

~

> =Lj*(Li*(F" )|

)eZWZ/y[-m]

is the isomorphism of Proposition 6.2, and

obtained by composing the isomorphism with the isomorphisms of [II 5.4], b)

If

i: Y

> X

8

is

~i,j of Definition

1.5

[II 5.9], and [II 5.13].

is a locally Y'

J

>X'

complete intersection of codimension n, and if

f: X'

then letting

> X y,

is a flat morphism,

be the fibred product, Y

we have a commutative diagram for

F" e D(x)

>X

183

g*i~ (F-)

j~ f*(F')

1

>

g*[=Li*(F') | ~/x[-n]]

> ~.C(f*(F')) |

%r,/x,E-n]

where again u and 8 are composed of the usual identifications.

184 !

~8.

The

functor

f'

for embeddable

morphisms.

In this section we use the fundamental to relate

the functars

f~

and

local

isomorphism

f~

defined above,

and to define

which

can be factored

into a finite

!

a functor morphism section

f"

for m o r p h i s m s

followed by a smooth morphism. is only provisional,

results we w i l l

obtain

The main result

but it is a model

in Chapter VII after

of this

for the stronger

developing

the local

techniques. Lemma

8.1.

Let

f: X

noetherian

preschemes,

Then there

is a functorial

>Y

and let

be a smooth m o r p h i s m i: Y

> X

of locally

be a section

of

f.

isomorphism

i,f" G" for all

G"

6 D(Y).

Proof.

We

local complete

first note by P r o p o s i t i o n intersection

morphism.

1.2 that

Hence

for any

i

is a G"

6

D(Y)

w e have i~ f@G" = i~ (f'G" | WX/y[n]) by definition isomorphism

of f , w h i c h

D. of C o r o l l a r y 1

is isomorphic 7.3 to

by the fundamental

local

185 Li*(f*G" | WX/y[n ] ) | Wy/x[-n ].

Using [II.5.9] and [II.5.4] this becomes

G" | i*Wx/Y ~ wy/x , which finally by the isomorphism isomorphic to

G'.

~i f

of Definition 1.5 is

We compose all these isomorphisms to obtain

~i f " Proposition 8.2. and

g: Y --> Z

(Residue Isomorphism)

Let

f: X

be two morphisms of locally noetherian preschemes,

with f finite, g smooth, and gf finite.

Then there is a functorial

isomorphism Sf,g: defined on

product

(gf)~

~ > f~g~

D + (Z). qc

Proof.

We consider the fibred

X •

p l / ~ X•

Y' with projections

Pl and P2' and let i be the graph X

morphism of f.

Then

Pl

is smooth

by base extension from g, so

i

and Z

Pl

satisfy the hypotheses of the

Lemma.

Thus

> Y

186

(gf) by

~i,p I of the lemma.

This in turn is isomorphic to

i ~ P~ g~

by Corollary 6.4, which is isomorphic finally to

by Proposition 6.2. Remarks.

This isomorphism, in the case where

f

is a

closed immersion, was first discovered by Grothendieck using a much more complicated procedure.

The present proof is due to

Cartier, as interpreted by Mumford.

This isomorphism will be

used in defining the trace map for residual complexes in Chapter VI, an important preliminary to the residue theorem. Corollary 8.3.

Let

f: X

> Y

be a morphism of locally

noetherian preschemes which is both finite and smooth. there is an isomorphism

: defined on Proof.

f~

Dqc (Y)" Let f be the identity in the proposition.

Then

187 Remark.

We will leave to the reader the verification that

this map is the same as the one deduced from the classical trace map

f. ~X ----> ~y. Proposition 8.4.

Let

f: X

> Y

and

g: Y --> Z

be two

morphisms of locally noetherian preschemes, with f finite, g smooth, and gf smooth.

*f,g: defined on

Then there is a functorial isomorphism

(gf)~ ~

f~g~

D + (Z). qc

Proof.

Considering XXzY

and using the notation of the

proof of Proposition 8.2, we have

where the isomorphisms are those of Lemma 8.1, Proposition 2.2 (twice), and Proposition 8.2, respectively. Corollary 8.5.

With the same hypotheses as the Proposition,

there is a natural map Trf: Proof.

f.

WX/Z

> wy/Z.

Apply the isomorphism of the Proposition to

and use the trace map of Proposition 6.5.

~Z'

188

Remark. irreducible,

In case and

Z

is the spectrum of a field, X , Y

K(X)/K(Y)

a separable extension,

map coincides w i t h the classical one the obvious

functorial properties:

and flat base extension.

this trace

[ 3 , Ch. VI w

It has

compatibility with c o m p o s i t i o n

It is a non-trivial map, and deserves

to be studied more closely. Proposition 8.6.

a)

The isomorphisms

%f,g

8.2 and 8.4 are compatible with the isomorphisms

of Propositions of Propositions

2.1 and 6.3 under a flat base extension.

b) If X f>x each pair

(f,g),

h

g >z

(f,hg),

of one of Propositions

> W

(gf,h),

2.2,

are three morphisms,

(g,h) satisfies the hypotheses

6.2, 8.2,

or 8.4, then there is a

commutative diagram of the corresponding

c)

and if

isomorphisms.

if s Q X

- _> P g

f

~ y

J

>z

are morphisms w i t h Q = PXzY , f smooth, there is a commutative

diagram

j,k, and i finite,

then

189

(ji)

1.

i ~ j~

k~g~j ~

~

k

~ ~ f~

using the isomorphisms of Propositions 6.2, 6.4, and 8.2. Proof.

Left to the patient reader.

Definition. morphism

Let

f: X

S

be a fixed prescheme.

We say a

> Y in the category of preschemes over S is

embeddable (or S-embeddable), if there exists a smooth prescheme P over S and a finite morphism i: X f = P2i.

> Py = P• Y

such that

Unless otherwise specified, embeddable will usually

mean over Spec ~. Examples.

A projective morphism

f: X

> Y where Y is quasi-

compact and admits an ample sheaf is embeddable (for any S). Indeed, f can be factored through some ~

[EGA II 5.5.4 (ii)].

Any finite morphism is embeddable, by taking P = S.

Any morphism

of finite type of affine schemes is embeddable in some affine space.

Note that any composition of embeddable morphisms is

embeddable (:) and that embeddable morphisms are stable under base extension.

19o !

Theorem 8.? (f" for embeddable morphisms).

We fix a base

prescheme S, and consider the category Lno(S) of locally noetherian !

preschemes over S. morphisms

Then there exists a theory of f" for embeddable

in Lno(S) consisting

of the data i) to 5) below,

to the conditions VAR i - VAR 6.

Furthermore

unique in the sense that if 1')-5')

subject

this theory is

is another set of such data

satisfying VAR 1 - VAR 6, then there is an isomorphism of the functors

i) and i') compatible with the isomorphisms

2)-5) and

2')-5'). i)

For every embeddable morphism

f: X

> Y in

Lno(S),

a functor

f:" D+qc(Y) ~ 2)

D~c(X)

For every composition

morphisms,

X

f >

I

I

(gf)"

> Z of embeddable

f: 9

I

> f'g" .

For every finite morphism df:

4)

g

an isomorphism of functors Cf,g:

3)

Y

f, an isomorphism

> f~

For every smooth embeddable morphism el.9

f"'

>

f~

.

f, an isomorphism

191

5)

For every embeddable

mcrphism

u: Y'

every flat base extension

(where

v

g

and

VAR i).

> Y, an isomorphism

>g'u

v'f"

u,f:

> Y, and for

!

!

b

f: X

are the two projections

of

X' = X ~ Y ' ) .

cf,id = Cid, f = I, and there is a commutative

diagram of four

c's for a composition

of three embeddable

morphisms.

VAR 2).

For a composition

compatibility

of

via

d g.

df

and

VAR 3). Proposition

2.2, ef and

in Corollary

of

Cf,g

6.2

of Proposition

of smooth morphisms,

square of embeddable

6.4, compatibility

using

morphisms

of that isomorphism with

c

as

v~f

via df, dg, eu and ev. For a composition

the hypotheses with

VAR 6). embeddable

f,g,

e . g

For a Cartesian

VAR 5). satisfying

with the isomorphism

f,g

Ditto for a composition

VAR 4).

and Cg,u

c

of two finite morphisms

@f,g

of Proposition

compatibility

2.1 or 6.3.

of

morphisms

f,g

8.2 or 8.4, compatibility

via the appropriate

For a flat base extension

morphism,

of Proposition

of two embeddable

d's and e's.

of a finite or smooth b

u2f

with the isomorphism

192

Proof. more

We will give only a sketch, since a similar but

difficult

theorem

is

proved

in

some detail

in

Chapter

VI.

!

To define f" one chooses an f embedding

i: X

> Py, and

!

defines

f" = i~p~.

The product of two embeddings is again one, !

so

one

shows

that

f"

is

independent

using Propositions 8.2 and 8.6b.

J: Y

the

To define

one notes that given embeddings of and

of

f

and

embedding

chosen

by

Cf,g for a composition, g, say

i: X

>Py

> QZ" then

(j xs P)i:

x

is an embedding of

> (P Xs Q)z gf, and one can define

Cf,g

using the

isomorphisms of Corollary 6.4.

Of course

of the embeddings chosen . . . .

One defines df and ef using

Propositions 8.2 and 8.4, and Checking the properties

b

u,f

using

is independent

Proposition 8.6a.

VAR 1 - VAR 6 requires many

commutative diagrams, but no imagination. tedious but straightforward.

Cf,g

The uniqueness is

By the way, the reader will note

that 5) and VAR 6 are not needed for the uniqueness statement. Remarks.

One of the main goals of these notes is to obtain !

a theory of

f"

such as the one given in this theorem,

for

arbitrary morphisms of finite type of locally noetherian preschemes.

193

The obvious difficulty is that the derived category is not a local object.

That is to say, if

X

is a prescheme,

J.

then the presheaf on X.

> Dqc(U ) is not a sheaf of categories

One can give a cover of

complexes. in

U

FI: 6 Dqc(Ui)

D + (U..) qc 13

X

by open subsets Ui, and

and isomorphisms

~ij: Ft."IUij

which are compatible in Dqc(Uijk)

there does not exist a complex

F" 6 D + (X) qc

> F~IUij

but where

whose restriction

to U. is F'. 1 l

Even worse, given two complexes F',G" 6 D + (X), and qc

isomorphisms

~i:

D qc(Uij), +

~:

F"

the

~i

F" IU"

> G " IU"

such that ~iiu.. = ~j IU..

in

may not glue into a global isomorphism

>G'.

Thus although every morphism of finite type is locally !

embeddable, we cannot glue the local functors

f"

into a

global one. To overcome this difficulty, ~ notion of residual complex.

study in Chapter VI the

These are actual complexes, and !

hence can be glued.

We develop a formalism of

f"

for residual

complexes similar to the one given here, expanding from the two easy cases of finite and smooth morphisms.

Then after proving !

the duality theorem we can recover a theory of

f"

for arbitrary

194

complexes~ but only under the additional hypotheses that our schemes be noetherian of finite Krull dimension, and admit a residual complex (e.g., anything of finite type over a regular scheme of finite Krull dimension),

and that our complexes have

coherent cohomology. Proposition 8.8.

Let

f: X

> Y be an embeddable morphism

of locally noetherian preschemes.

6)

Then

There is a functorial isomorphism !

!

f'(F') =~ =~f*(G') for

F" 6 D + (Y)

qc

7)

and

~ ;

G ~ 6 Db

f'(F'_,G')

c (y) fTd"

There is a functorial isomorphism |

R= Hom'(__Lf*(F'),

for

F" E Dc(Y)

and

!

f'(G'))

G" 6 D + (Y). qc

Proof. Left to reader.

(Factor

followed by a smooth morphism, Remark.

~ > f'(R Ho_am'(F',G'))

f

into a finite morphism

and use Propositions 2.4 and 6.9.)

We w o u l d like to have an isomorphism such as 6)

above when f is flat, F ~ q D~c(Y)fTd , and

G" 6 D~c(Y).

Both

sides make sense in that case, but we do not know how to define a map between them, and hence cannot construct the isomorphism. However,

if Y admits a dualizing complex, w e can get a result of

this kind for complexes with coherent cohomology [V 8o6].~

195

w

The residue This

symbol.

section w i l l not be used in the sequel,

omitted at a first reading. Res[

w ] t I ' -.. ' t n

of residue. regular

In it we define

w h i c h is a g e n e r a l i z a t i o n

For

X

a non-singular

differential

at P, of the d i f f e r e n t i a l results

t

Let

f: X

> Y

tl,...,t

subscheme

Z

of

finite and h e n c e

X

w/t.

[EGA IV w

,t n

]

X

Let

relative

is

w E F(X, Wx/Y) to

Y.

Under

r(Y,a'y)

Z

T h e n by the r e s i d u e

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

dimension

symbol

Let

g = fi.

a

such that the c l o s e d

can be defined as follows. X, and let

w

to the reader.

over Y.

form on

the r e s i d u e

ReSx/Y[tl, 9

in

k,

defined by the ideal I = (tl'''''tn)

flat

notion

Since w e w i l l not use these

be a smooth m o r p h i s m of r e l a t i v e

be a global n - d i f f e r e n t i a l these c o n d i t i o n s

of the classical

a function w i t h an

q r(x,~._) be functions

n

symbol

is just the o r d i n a r y residue,

later, we leave their proofs

Let n.

form

the r e s i d u e

curve over a field

form on X, and

i s o l a t e d zero at a point P, Res[~]

and so may be

i: Z

> X

8.2 we have an i s o m o r p h i s m

be the i n c l u s i o n isomorphism

of

$i,f

196 > i~ f"(~y) = i t (.x/y[n]) ~

> i*(~x/Y) | ~z/x = Hom 0" ( A n I / I 2 , i * W x / y ) , -Z

where the second arrow is given by the fundamental isomorphism ~. of Corollary 1 imply that

Z

7.3.

(Note that our hypotheses

is locally a complete intersection

Z is a local complete intersection and is defined by [ZS, vol. tl,---,t n

is an

rank 1 on Z, so

in Z ~

P~(tl'''''tn )' hence

II, App. 6, Thm. 2]. ~X-sequence.) ~IA...A~

n

we obtain via the isomorphisms

in X.

Indeed,

X, by Proposition

this is an

1.2, uence

•Zx•-Seq

But now by faithfully flat descent, Now

AnI/I 2

is a basis for it.

element into the global section

local

~

of

i*Wx/Y

is locally free of By sending this obtained

from

w,

above, a global section of g~(~y).

Applying g. we get a global section of

But since

Z

can erase the of

is flat over Y,

R.

g*@'Z

A p p l y i n g our g l o b a l

is locally free, and so we

section

g*~z' we obtain a global section of

to the unit

section

~y, which is by definition

197

ReSX/y [

] 9

=

tl, 9 . 9 ,t n

The residue symbol has the following properties

(we assume every

time we write a residue symbol that the conditions

for its

existence are satisfied) : (RO).

It is

(RI).

Let

Res[

In particular,

~y-linear

in

=.

si = Z cijtj"

w ] tl, 9 9 9 ,t n

=

Then

Res[det(cij )wl Sl,-..,SnJ

the symbol is alternating in

(~2).

Localization.

(R3).

Restriction 9

.

tl,..-,t n. f

It is stable under etale localization

on Y. Let

X'

X

be a

complete intersection ~n X, also smooth over

Y, defined by functions

in F(X, ~X) .

Let

F(X, ~X) , where t~, 9 -. "t'n

tl,...,t n X

S I,

'''

,S p

Y

be in

is of relative dimension

be their restrictions

to

X'.

Let

n+p, and let n w 6 F(X,~/y).

Then

Resx'IYLt, ... t'-I = ReSxlYLtl,...,tn l'

" n

,Sl,-.-,Sp

198 (R~). morphisms,

Transitivity.

Let

X

f > Y

g > Z

be two smooth

of relative dimensions n,p respectively.

Let

ti,...,t'n ~ r(X'#x)" ~' e r(X,~x/Y); Sl,'",sp ~ r(v,~ W 6 F(Y,~y/z) , and let with

f.

I

|

be the compositions of

el, 9 " 9 , Sp

Then w

Res

~XW ' , n, Sl,

Z t , xj[i

(R5).

s. 1

,

Base change.

9 )S

'p3 =

Resy/z[ w'ResX/Y t l , ' ' ' , t el, 9 9 ,Sp

I

n]

9

Formation of the residue symbol commutes

with base change. (R6). be in

Trace formula (Normalization). F(X,~x).

In particular,

for

tl, 9 9 9 ,t n

and

Then

_~dtlA. Res L

Let

Adt n_

Trz/y(~Iz)

tl,.il,t n J

~ = i, one has

aes[dtlA...Adt n ]

=

rank ( Z / Y ) - ~ .

tl, 9 9 ,t n (RT).

Intersection

formula.

For any collection of integers

kl,''',k n > O, not all equal to one, ReS[dk IA"

Adtn=

n

O

.

199

(RS).

Duality.

Res[

If

W E

w

]

?(ItiWx/y),

then

= O

tl, 9 9 9 jt n fw and conversely if

] =0

Res[

for all

tl,''',t n

f E r(X,~x),

then w 6 F(ItiWx/y).

(R9). and

Exterior differentiation.

n-1 w 6 F(X, Z / y )

and for

dw

Ros[k tlZ .-. 3

(RIO).

]

kl,...,k n > O

=

k i Res

n

Residue Formula.

morphism where

X',X

tl,.--,t

n ~ r(X,#x),

we have

dt. A w

tkn 9

For

ki+l t I ,--.,t i ,''-,t n

Let

g: X'

are both smooth over

~> X

be a finite

Y.

Let

tl,''',t n 6 F(X,~x).

Let

I

w' 6 F(X',WX./y)

and let

be their compositions with

Resx./Y[tl, 9 9 9 ,t' ] n where

Tr

g

g.

Then

Resx/Y[t~r?(. W')

is the map of Corollary 8.5.

9 3 t n]

9

,

|

tl, 9 9 9 ,t n

2O0

w

Trace for projective morphisms. In this section we show that in the situation of the

Residue isomorphism space morphism,

(Proposition 8.2) if

then our trace morphisms

g Trf

is a projective for finite

morphisms and Trp for projective space morphisms are compatible. This allows us to expand from these two cases to arrive at a theory of the trace map for any projectively embeddable morphism. This result,

like the one of aS, is only provisional, because we

want eventually a theory of trace for an arbitrary proper morphism. This will come in Chapter VII. Proposition iO.I. let

X = ~,

a section of

Let

Y

be a locally noetherian prescheme,

let

f

be the projection,

f.

Then for every

and let

s: Y

> X

be

G" 6 D + (Y) the composition qc '

of maps G9

#s,f

is the identity.

> Rf, Rs, s ~ f~G .

Trf s > R f ~

f~G"

Trpf > G"

(The maps are those of Lemma 8.1, Proposition 6.5,

and Proposition 4.3, respectively.) Proof 9

i)

We note that both

$s,f

and Trpf

are

calculated by using a Cartan-Eilenberg resolution of f~(G') = f*(G') | A/W"IY[n]"

We can use the same resolution for

201

each, and thus reduce to the case where coherent and

sheaf on Y.

Rnf~-acyclic,

Then

G

f~(G) | WX/Y

is a single quasin Ext ( s ~ y , ' ) - a c y c l i c ,

is

so we have to show that the composition

n

> Rnf~(f G|

> G

is the identity. 2)

Noting that the functors above are all right exact in

G, and commute with direct sums, and noting that the question is local on Y, we may assume that the case ~y

G = ~y.

Y

is affine, and thus reduce to

Thus we must show that the composition

~ > f~ E X t ~ x ( S ~ y , mX/y)

8 > Rnf~(WX/y)

7 > yy

is the identity.

3)

In other words,

obtained a map of D(s) 6 F(Y,~y), everything over

~y

from the section s of ~ into itself,

in the composition

~n.

of morphisms

Since

in 2) above is flat

is stable under arbitrary base change.

Now our given section A:

i.e., a section

and our problem is to show ~(s) = i.

Y, this construction

section

, we have

> ~n x ~n

s

can be obtained from the diagonal of projective

space over Spec

into its product with itself by the base extension

p2s: Y

> IP n.

202 Thus Now

D(S) = (P2S)*8(A) D(A)

and w e reduce to showing that B(A) = I.

is an integer,

what integer,

since

F(~

To find out

it is sufficient to make the base extension at

some closed point,

n

= T n = O of ~P , consider

say T 1 . . . .

Y = Spec Z, s = the section of ]P~ and show that 4)

n) = 2Z.

given by T 1 . . . .

= Tn = O,

8(s) = 1 in that case.

We show more g e n e r a l l y that for any prescheme Y, if

s is the section T 1 . . . .

= T n = O of ~

, then 5(s) = i.

is a formidable exercise in explicit calculations,

This

of w h i c h we

w i l l give a mere outline. R e c a l l i n g the notation of w the

7

we calculate

of T h e o r e m 3.4) by means of the cover

the section

7 = dtlA...Adt n

of

7

(which is

~i = (Ui) of X,

WX/yI U , and the n - c o c y c l e O

7/To...Tn of the Cech complex

f.(C'(l[;WX/y).

To calculate u, we use the notation of w complex

K" (t;Wx/y) w h e r e

O'0'''

O

the fundamental the isomorphism map

-'J/J2

O

and the Koszul

tl,''',t n are the local coordinates The map

u

is obtained by composing

local isomorphism D.l of Proposition 7.2 w i t h ~s,f

of Definition

.~ 1 > I A/~-/S of Proposition

1.5.

R e c a l l i n g that the

1.2 is defined by sending

203

t E J to dt, we see that u(1) is the cocycle

K'(_t;.x/y) given by elA...Ae n

;

>

9 .

Finally, we calculate

8 by means of the morphism of complexes

K'(tl,'..,tn;W )

> C'(Is

defined by sending a p-cochain e

(where

i1

A... Aeip

!

> f" .-. i dtlA'''Adtn ] x I, , p

fi l,...,ip E F(Uo,YX))

to the p-cochain

TP-n-lf. 7 o ll,''',i p E r(u

T. ... T. lI P

and

0 E F(

Ujo" Jl' .. "' jp,W)

,m) , ~

when all

Then 8 applied to the cocycle elA.--Ae n

.

'

> dt 1 A...Adt n

Ji40}

204 gives

9 /'r 9 9 F

~

9j

--

" T

0

n

6 F(U 0 , . . . , n,~)

as required. Corollary 10.2.

The isomorphism

7

of Theorem 3.4 is

compatible with an automorphism of the projective space, i.e., it is "independent of the choice of homogeneous coordinates". Proof.

Indeed,

SS, f

and

Trf s

choice of coordinates, hence Trpf

Remark. of rank

n+l

in w

E

on a prescheme Y, we can define an isomorphism

Rnf~(w)

X = ~(E), and

projection

does not either.

This shows that for any locally free sheaf

7: where

do not depend on a

f: X

>Y

- -

>

w = WX/Y. as in w

~y

We can define

Trpf

for the

and get a duality theorem as

for this morphism. Proposition 10.3.

Let

u: X

of locally noetherian preschemes,

>Y

be a finite morphism

let V

f

be the projection of projective

n-space over Y, and fill in a Cartesian diagram as shown.

Then there is a

in ~ X

commutative diagram of morphisms of functors on

U

D + (Y), qc

>Y

2o5

Trpg

'I~

Trf

~ f ~ 1 9

where

e

is composed of [II.5.1] and Corollary 6.4.

Proof.

Left to reader.

One follows through the

definitions of the maps concerned.

The only tricky point

is to note that if C'" is a Cartan-Eilenberg resolution of f~(G')9 where Y, then of

G"

is a complex of quasi-coherent sheaves on

v)(C "') is not necessarily a Cartan-Eilenberg resolution

v~f~(G').

However 9 one can find a Cartan-Eilenberg

resolution D'" of this latter which dominates it (i.e., there is a map of double complexes D'" ---->v~(C")), which is good enough for the proof. Proposition 10.4.

Let

X

and

Y

locally noetherian preschemes 9 and let be a finite morphism of X into P ~ . g: ~

be f

Let

> Y be the projection 9 and assume

that gf is finite.

Then there is a commutative

diagram of morphisms of functors on

D + (Y), qc

Y

206

Trf

l

gf

>1

Trpg Trff > ~g,g~

where the vertical arrow on the left is [II.5.1] composed with the residue isomorphism #f,g of Proposition 8.2. Proof.

Considering ~

= X ~

~

as in the proof of

Proposition 8.2, the result follows from Propositions

6.6,

i0.i,

and 10.3. Definition.

A morphism

projectively embeddable if p: ~

f: X f

> Y of preschemes is

can be factored

f = pg

where

is the projection of a suitable projective space

> Y

over Y, and

g: X

Example.

If

> ~y

is a finite morphism.

f: X ---->Y is a projective morphism, where

is quasi-compact and has an ample sheaf, then

f

Y

is projectively

embeddable [EGA II.5.5.4 (ii)]. Theorem 10.5 (Trace for proJectively embeddable morphisms). We consider the category (Lno) of locally noetherian preschemes. There is a unique theory of trace for projectively embeddable morphisms

f: X

> Y in (Lno), consisting of a morphism of

207 !

functors

> 1 on D + (Y) for each such m o r p h i s m qc

Trf: Rf~f"

f,

subject to the conditions TRA 1 - TRA 4 below. TRA i).

For a composition

embeddable morphisms,

X

f ~> Y -g

there is a commutative ,

>

I

Trf

!

Rg~Rf~f' g"

via

df

with

TRA 3).

!

> Rg~g"

f: X

For a finite morphism

> ~

Trf is compatible

Trff. For the projection

space, Trf is compatible, TRA 4).

I Trg

Cf,g !

diagram

Tr

gf

2)

> Z of projectively

f: ~ y

of projective

via el, w i t h Trpf.

For a p r o j e c t i v e l y

and a flat base extension

>Y

embeddable morphism

u: Y'

f: X

> Y

> Y, there is a commutative

diagram , u ~ Rf~f"

Trf -

>

Tr

[II.5.12] b !

=Rg~v ~ f" where

v

and

g

---

u~

U

'

f

are the two projections

g

! 9

--> =Rg~g u of

X' = X Xy Y'

.

208 !

Proof. b

u,f

Of course

refer back

to T h e o r e m

To construct in the definition defines Trppo

Trf

the notations

Trf,

f" , Cf,g,

8.7.

one chooses

of p r o j e c t i v e l y

embeddable

to be the composition

to Proposition

are all straightforward, Remarks. a theory of morphisms

f.

similar

above,

and and

of the factorization,

one

Cg,p,

can be dominated The properties

10.4.

by a third, TRA

1 - TRA

but tedious.

The second main object Trf

morphism

pgas

dg, ep, Trfg,

of

that any two factorizations

and thus reduces

f=

a factorization

To show that it is independent

observes

df, ef, and

of these notes

is to obtain

to the above one for all proper

It is not simply a question

of localization,

as

!

for the theory

of

f', b e c a u s e

a proper morphism

is not locally

projective. Therefore

we resort

the c o n s t r u c t i o n the projective morphism

of the general

case,

complexes

relative

prove the r e s i d u e

different

trace map.

and work purely

to define a trace map

of residual we w i l l

to an entirely

We

technique

forget

for

entirely

from the trace of a finite

(which is a map of graded to a m o r p h i s m

theorem w h i c h

sheaves:)

of finite type.

says that

for a proper

Then

4

209

morphism,

the trace map is a morphism of complexes.

Finally,

after proving the duality theorem, we lift ourselves by our bootstraps,

and obtain the general trace map (but under the

restrictive hypotheses that cur schemes be noetherian of finite Krull dimension admitting a dualizing complex, and that our complexes have coherent cohomology).

210

w

Duality for projective morphisms. I

Combining the results of 888 and iO we have a notion of f" and Trf for projectively embeddable morphisms,

and we are in a

position to prove the following duality theorem. Theorem ll.1 (Duality for projectively embeddable morphisms). Let

f: X

> Y be a projectively embeddable morphism of noetherian

preschemes of finite Krull dimension. I

@f: Rf.R Hom~(F',f'G') defined by composing

Proof. P: using

> R Hom~(Rf.F" G')

[II.5.5] with

an isomorphism for all We factor

Then the duality morphism

Trf

in the second place, is

F" 6 Dqc(X)

and

G"

f

with

g

into

pg

qc (Y).

o+

finite and

---~ Y the projection of a suitable projective space.

Then,

TRA 1 of Theorem 10.5, and [II 86, ex. 2] we see that

@f = @ e c . P g g,P

Thus it is sufficient to show that

@

and P

are isomorphisms.

@

g

This follows from Theorems 5.1 and 6.7, using

the compatibilities of Theorems 8.7 and 10.5. Remarks. isomorphisms.

i.

As in w

the variants

ef

and

i @f

are also

211

2~

This result, like the ones of w167 and iO, is provisional.

We will prove a more general duality theorem for proper morphisms in Chapter VII. 5.

Taking global sections, and

H~

on each side, we have

(using [I.6.4]) !

HOmD(X)(F',f'G') For

F" E

c(X)

and

>

G" 6

HOmD(y)(Rf~F',G') . c(Y)

this says that

f"

(and the

!

map

Trf: Rf~f"

> i) is a right adjoint of the functor

Therefore the pair

(f',Trf) is uniquely determined on

~f~.

D c(y)

It is conceivable, however, that there are non-isomorphic !

functors

f"

on

D + (Y) qc

Corollary 11.2. morphisms a) is

f: X

each of which gives a duality theorem.

We consider smooth, p r o j e c ~ vely embeddable

>Y

of locally noetherian preschemes.

For each such morphism

f

of relative dimension n, there

>

~'y.

a map n

7f:

R f.r

Then

212

b)

For each pair

f: X

> Y and

g: Y

of such

> Z

morphisms, of relative dimensions n and m, respectively,

there

is a commutative diagram Rn+m(gf)~(WX/Z )

~f,g

> Rmg~Rnf~(f~Wy/Z | ~X/y)

|

7gf

Z (Note that

~g


Y, > Y f~x

~f

is the map

~

of Theorem 3.4.

a finite smooth morphism, > ~Y

is the ordinary classical trace map.

213

f)

For F quasi-coherent

on X and G an inJective quasi-

coherent on Y, the duality map

i ef:

Ext ~X (F, f*G | ~X/y)

>Hom~yt"Rn-i ~.(F), G) -

defined via 7f, is an isomorphism. g)

The map

7f

f

is an isomorphism if and only if

is

surJective and has geometrically connected fibres. Proof.

We obtain the map

G = ~y . Property

7f

by applying

b) follows immediately from

Trf

to

TRA I.

To prove c), the question is local on Y, so we may assume Y is affine.

Then

closed immersion, morphism.

f and

can be factored N

p: ~ y

[EGA II 5.5.4].

> Y

f = pi, where

Oxj, --- '"

is a

is a projective space

We can calculate

the fundamental local isomorphism

i

7f

by considering

(Proposition 7.2) which gives

o,

Then there is a natural map

Rnf.(wx/y) which followed by

7p

> ~Np.( P/Y) , gives

7f 9

Everything in sight is flat

over Y, and hence commutes with arbitrary base extension.

214

d)

follows

e)

we w i l l leave to the reader as an exercise.

f)

is a special case of the Theorem,

g)

follows from [EGA III 4.3.1], using f).

Remarks.

from

i.

TRA 3.

and

Later we w i l l prove this theorem

for an

a r b i t r a r y smooth proper m o r p h i s m of locally noetherian preschemes. 2.

In case Y = S p e c k

projective

scheme /k (i.~,

variety"),

and

F

w i t h k a field, G = k, X is a smooth "absolutely non-singular

a coherent

sheaf on X, the duality formula f)

above reads

x (F, x/Y) where

v

projective

--

means the dual k - v e c t o r

>

Hn-i(X;F) v

space.

215

CHAPTER

IW.

LOCAL COHOMOLOGY.

This chapter consists

for a g r e a t p a r t in d e f i n i t i o n s ,

w h i c h g e n e r a l i z e those of the L o c a l C o h o m o l o g y lecture notes [LC].

Notable new material

filtered topological

~i.

is the s p e c t r a l s e q u e n c e of a

space, and the C o u s i n c o m p l e x of a sheaf.

L o c a l c o h o m o l o g y g r o u p s , sheaves, Throughout

space, and

F

and complexes.

this section, X w i l l be an a r b i t r a r y t o p o l o g i c a l a s h e a f of a b e l i a n g r o u p s on X.

T h e r e are three

w a y s in w h i c h one can v a r y the b a s i c d e f i n i t i o n of F w i t h

s u p p o r t s in a c l o s e d

Z b y a family of supports; c o h o m o l o g y if Z' ~ Z everything definitions

subset Z of X:

of the c o h o m o l o g y one can r e p l a c e

one can define a r e l a t i v e

are two c l o s e d subsets;

local b y sheafifying.

and one can m a k e

T h e r e f o r e w e w i l l p r e s e n t the

in the form of a theme and v a r i a t i o n s ,

all p o s s i b l e c o m b i n a t i o n s

local

of the g e n e r a l i z a t i o n s

to a l l o w for

suggested

above. We groups,

state our r e s u l t s m o s t l y in terms of the c o h o m o l o g y and leave to the r e a d e r

terms of the d e r i v e d category.

the a p p r o p r i a t e When

X

statements

is an a r b i t r a r y

in

216

topological space, w e work in the derived category D(Ab(X)) of the category of abelian sheaves on X.

If

X

we work in the derived category D(X) = D(Mod(X)) of

~X-mOdules.

is a prescheme, of the category

All the derived functors considered in this

chapter are compatible in the two cases with the natural functor

D+(X) since any injective

D+(Ab(X)) , ~X-mOdule is flasque, and flasque sheaves

are acyclic for the functors considered

(Motif C below).

Theme. Let

Z

be a closed subset of X.

Define Fz(X,F) to be the

group consisting of those global sections of lies in of

Z.

Define

~(X,F)

F

whose support

to be the i th right derived functor

?Z (which is a left-exact functor), and define ~F z

the right derived functor on the derived category

to be

D+(Ab(X)).

These are called the local cohomology groups of F with supports in Z.

They have the following properties: Motif A. O

If >F'

>F

>F"

>O

is a short exact sequence of abelian sheaves on X, then there

217 is a long exact

sequence

of groups

> rz(x,F' ) --+ rz(x,F)

O

Hz1(X,F ,) --+ Proof.

This

o

B.

H~ ) , =

There

is

a

....

which long

it is.

exact

sequence

> r(x-z,F)

>

HzI(X,F)

> ~(x,F)

> ?(x-z,F)

> Hze(X,~")

For any sheaf

> rz(X,F) F

is flasque

resolution

F

one has an exact

> r(x,F)

Thus

of F, we have an exact > r z(x,I" )

gives rise

> r(x,i" )

to a triangle

> ....

sequence

,~r(x-z,~) ,

(in particular

write a zero on the right.

which

>

> r(x,~)

O

O

> ~zl ( x , ~ ' " )

> rz(X,F)

Proof.

and if

>

is the same as to say that F z is left exact,

and so is equal to Motif

(x,F)

> rz(x,~ )

if

if F is injective) I"

is an injective

sequence

of complexes

> r(x-z,i" )

in D+(Ab),

> o

one can

218

RV(X-Z,F) b

f

=Rrz(x,~)

-

9

> ~r(x,F)

Taking cohomology gives the result.

Mos

C.

Proof.

For This

F

Remark.

F

= O for i > O, and F(X,F)

= O.

> r(x-z,F) is

flasque.

In fact, F is flasque

subspace Z of X,

Variation

and i > O, ~ ( X , F )

follows from M o t i f B, and the fact that

Ri(x,F) = ~i(x-z,F) surj ective for

flasque,

~(X,F)

= O

for every closed

[LC, l.lO].

1.

Definition.

A family of supports on a topological

space

X is a set ~ of closed subsets of X such that (a) Z'

if Z 6 ~, and Z' is a closed subset of Z, then

6 ~, and (b)

if ZI,Z 2 E ~, then Z 1 U Z 2 6 ~.

Now let ~ be a family of supports,

and define F (X,F) to be

the group of global sections of F w h o s e support is in ~. R= F ~ (X,F) and Hi(X,F)

to~'be the right derived functors.

Define

219

Motif A.

Repeat as above, with

Z

replaced by ~.

Motif D.

Proof.

It is true for i = O, and direct limits commute

with cohomology of complexes. Motif C. Proof.

For

F

flasque, and i > O, H~(X,F) = O.

Follows from Motif D, and Motif C above.

Variation 2. Let Z' c Z be closed subsets of X.

Define FZ/Z, (X,F)

to be FZ(X,F)/FZ,(X,F).

Define RFZ/Z, and ~ / Z '

right derived functors.

Note that in general FZ/Z, is not

left exact, so that FZ/Z, 4 Motif A.

to be the

o "z/z,"

Repeat with H ~ in place of F, and Z/Z' in

place of Z. Motif B.

o

There is a long exact sequence

> q.(x,F)

~ >rz(X,F)

"zl, (X.F) ~ 2 Hz. (X,F) Moti f C.

Repeat.

o

~.~/z,(X,F)

1 .zm(X.F) ---~ "Z/Z > ..

(X,F)

> >

220

Variation 3. Let

Z

be a closed subset of X.

Define F_z(F) to be the

sheaf whose sections over an open set U are the elements of FZNu(U,FIu ).

Define

=RFz(F ) and Hz(F)

to be the right derived

functors, which are now complexes of sheaves, resp. Moti f A.

Repeat with underlines.

Motif B.

There is a four-term exact sequence

0

where

> F

J : X-Z

> X

sheaves on X.

----, O

> &(FIx_z ) --~(F)

is the inclusion t and there are isomorphisms

for i > O, Rij,(FIx_z )

Motif C. Proof. ~(F)

N >~+I(F).

For F flasque and i > O, ~ ( F ) One sees easily that for any

is the sheaf associated

F

= O. and for any

i,

to the preshea f

Now since the restriction of a flasque sheaf to an open subset is flasque,

the result follows from Motif C above.

221

Motif E.

There is a spectral sequence

H (x,F) or equivalently,

,

in terms of the derived categories,

RF_ = RF.RF . =Z = ~-Z Proof.

Referring to [I.5.4], we need only show that

~Z takes injectives into F-acyclic objects. injective is flasque,

Indeed, any

and ~Z of a flasque is flasque, and

any flasque is F-acyclic.

Variation 4. Combining variations 1 and 2, let # _c ~ be two families of supports.

Define __~/$(X'F) and its derived functors R F /$

and H i

Motif A.

Repeat.

Motif B.

Repeat.

Motif C.

Repeat.

Motif D.

H~/,(x,F)

li3 R i Z6~ Z'E~ Z'CZ

Z/Z '

(x,~)

222

Variation

5.

Combining

variations

Definition.

1 and 3 leads us to a

A sheaf of families

space X is a sheaf of sets ~, ~(U)

is a family of supports

restriction

map _~(U)

of supports

such that

for every open

on U, and such that

> ~(V)

on a topological set U,

for V c U the

is given by Z 6 ~(U)

goes to

z n v 1.

Remarks.

If ~ is a family of supports

on X, we can

N

define a sheaf of families sheaf a s s o c i a t e d

u

2.

If

].

is a sheaf of families

define a family of supports 3. inverses compact

Nos

to each other.

is the m a x i m a l maximal

the operations

For example,

of a locally

on X, we can

F and N are not

if ~ is the family of

compact H a u s d o r f f

sheaf of families

family of supports.

of supports

F(_~) on X by F(~) = _~(X).

that in general,

subsets

~ on X by taking the

to the p r e s h e a f

> {znulz

_~

of supports

of supports,

space,

then

and F(~) is the

But if X is not compact,

~ # F(~).

223

4.

Conversely,

let f: X

type of locally noetherian

> Y be a m o r p h i s m

preschemes.

and let ~ be the sheaf of families ~(U)

= the set of (relatively)

for every y 6 f(U), Z N X X

y

.

Then

specific

in general example,

y

Let

p

of supports

closed

of finite

be an integer, on X given by

subsets Z of U such that

is of c o d i m e n s i o n > p in the fibre

F(~) N is different

let Y = Spec k[x,y],

from ~.

(For a

let X = ~

and let

p = 1. ~ Then

if Z ~ X is obtained

Z is locally

in ~ except over the origin of Y, but Z ~ F(~).)

5-

For another

example

of a family of supports,

be a subset of the topological (e.g.,

space X,

stable under

Z = Ix E X of c o d i m e n s i o n > p] for some p.

codimension largest

by b l o w i n g up the origin

of a point

integer

x in a t o p o l o g i c a l

n, or +co,

is a proper

specialization,

Then the set of subsets

of X w h e r e

i.e.,

Z

specialization

By the

space X, we mean the

such that there

X o , X l , X 2 , . . . , x n = x of points

let

of Y,

exists a sequence

for each i, x.l

> Xi+l

x i + 1 6 {x~i] and x.1 4 x i + 1 ).

of finite unions

of Z is a family of supports

~.

associated

sheaf of families

of supports

noetherian

(as a topological

space)

of closures

of points

One can also consider ~.

If

X

the

is locally

and if every closed

irreducible

ee4

subset of X has a unique generic point, correspondence between

then there is a i-I

subsets of X, stable under specialization,

and families of supports ~, such that ~ = F(~).

In these cases

we will w r i t e F Z for F ,

etc"

~

for H ~, i

F_Z f~

~,

Now let ~ be a sheaf of families of supports on X.

Define

F (F) to be the sheaf w h o s e sections on an open set U are ~F(u)(U'FIu ).

Define

R F (F) and -~HI(F) to be the right derived

functors. M o t i f A.

Repeat.

M o t i f C.

Repeat.

M o t i f D.

If _~ is a sheaf of families of supports of global

nature

(i.e.,

such that _~ = F(_~) ), then

x)--Hz(F) z

Hi(F) =

li

"

M o t i f E.

There is a spectral sequence

Epq = HP(x,_~(F))_____ > En = HF(~ ) n (X,F) or, in terms of the derived categories,

,

)

--

r.ffr.

225

Variation

6.

Combining variations of X.

2 and 3, let Z' _c Z be closed subsets

Define _FZ/Z. , RFZ/Z. , and HZ/Z'"

Repeat Motifs A, B, C,

and E. Variation

7.

Combining v a r i a t i o n s

i, 2, and 3, let _% c _~

of families of supports on X.

be two sheaves

Define F/S,__ RF/_#,__ and ~ / ~ .

Repeat Motifs A, B, C, D, and E.

V a r i a t i o n 8. In this case we define a purely punctual invariant.

Let

x

be a point of X and define F (F) to be the subgroup of the stalk x F

x

consisting of elements

s w h i c h have a representative

suitable n e i g h b o r h o o d U of x, w h o s e support is Ix] N U. the right derived

functors RFx(F ) and Hi(F). = x

s in a Define

Note that

H~(F) = H z ( F ) x , w h e r e Z = Ix], and the subscript x denotes the stalk.

Repeat Motifs A and C.

M o t i f F.

A s s u m e that X is a locally noetherian topological

space and that every closed irreducible

subset of X has a unique

generic point.

Let Z' c Z be two subsets of X,

specialization,

and such that every x 6 Z-Z'

stable under

is maximal in

Z

226

(i.e.,

if x E Z and x ---> x' is a n o n - t r i v i a l

then x'

6 Z').

Let

F"

6 D+(Ab(X))

Then there is a canonical

specialization,

be a complex

functorial

of sheaves.

isomorphism

xC. where

for any group G,

and O elsewhere. of the sheaves of V a r i a t i o n Proof. sufficient a single

i (G) is the constant x

(By abuse

of families

sheaf G on

of notation we write Z,Z' of supports

instead

they define as in Remark

5 above. ) Since both

sides are derived

to establish

a canonical

functors,

functorial

it w i l l be

isomorphism

sheaf F 6 Ab(X)

fz/z,(F)

~ >

11

x6Z-Z '

ix(q(~)).

For an open set U we define a map

rznu(u,~l U) by sending

{x],

a section

>

I I )nu

x~(~/~'

r

x

(F)

into its germ at each stalk.

finitely many are non-zero,

because

section

[z~i] U "'" U [ ~ ]

s is a finite union

the support

Only

of any

with

z.1 6 Z

for

5

227

(since X is locally noetherian:) maximal

in Z.

A section

in FZ,Nu(U,FIu ). above. germ

goes

each x 6 Z-Z'

we see that it is surjective,

s E F (F) comes x U, w i t h

from a section support

is

to zero if and only if it is

Thus we have defined an inclusion

Finally,

neighborhood

s

and b e c a u s e

s

of

F

of sheaves

because

every

in a suitable

Ix] N U.

Coda. H a v i n g made all these g e n e r a l i z a t i o n s local cohomology, Motif G. Space.

(The Spectral

Sequence

=

be a filtration F"

O

~o

of

D

~o

X

be a complex

Then there

back w h e r e we started: of a Filtered

i

D

i.e., there

Topological

..-

by sheaves of a b e l i a n

is a spectral

of families sheaves

is b i r e g u l a r n

is a diagram

and

sequence

[EGA Oiiiw

= ~ for some n.

of supports,

on X b o u n d e d below.

EPq = H P + q -'l (F') ----> E n = H n(F')

which

of

Let X

let

we are p r a c t i c a l l y

of the notion

Or,

if the

,

filtration

is finite,

in terms of the derived

of triangles

category,

(shown in the convergent

case)

228

Rin_

Rr o

l/mn(F" )

/''x Proof.

I"

I"

>... > ~_ I(F')

Take an injective resolution I' of F'.

is filtered

=

>RI'=o(F')=F'.

This is just the spectral sequence of a filtered

complex [M, Ch. XV]. Then

/"x

> Rr=_n_l(F')

O=RF=_n(F')

I(F)

_r o ( i ' )

a n d the q u o t i e n t s

~ _r

are

l(X')

"'" = F

_F i~ i+l (I')"

--

S

(I') = O , n

229

w

Depth and the Cousin Complex. Throughout

topological

this section,

space in which every closed irreducible

a unique generic

Let

F"

(bounded below)

supports.

subset has

point.

Definition. groups

X will denote a locally noetherian

be a complex of sheaves of abelian

on X and ~ a sheaf of families of

Then the ~-depth of F" is the largest

integer

n

(or +co) such that Hi(F ") = O for all i < n. Remark.

If

X

is a locally noetherian

family of subsets of a closed coherent

sheaf,

then this definition

the usual definition Proposition noetherian

2.1.

Let

the following (i)

Let

is maximal F

F

a

are equivalent:

H0

>_z/z(F)

of (the locally specialization,

(with respect

be an abelian

The natural maps

are isomorphisms.

X, and

the

of depth coincides w i t h

space) X, stable under

conditions


l, and is flasque.

F has supports in Z, and Z'-depth > 2.

Proof.

(i)----> (ii) follows immediately from Motif F of

Variation 8 above. (ii)---~. (iii) in Z

Condition (ii) implies that F has supports

and that F_z,(F ) = O, i.e., Z'-depth F ~ i.

F is flasque

because it is a direct sum of sheaves ix(M), each of which is a constant sheaf on an irreducible space, hence flasque. (iii)--->. (iv) i > O, by Motif C. F has Z'-depth

(iv)--~ supports in Z.

o

>fz (F)

Since F is flasque, ~ , ( F )

= O for all

But HzO,(F) = 0 s i n c e F has Z'-depth > I, so

+co >_ 2. (i)

_Fz(F )

,>F is an isomorphism since

F

Now by Motif B there is an exact sequence >-.-

9

has

231

Since Z'-depth middle

F > 2, the two outside

If

F

satisfies

Lemma

2.2.

F

a unique

(up to unique

be a sheaf on X w i t h

u: F

and cokernel Proof.

in Z.

first.

proposition, Then there

is

Z'-isomorphism

~>G lies on the Z/Z'-skeleton.

we m e a n a h o m o m o r p h i s m

have

supports

(By

of sheaves w h o s e k e r n e l

in Z'. )

(i) To show the existence

G = ~/z,(F),

of X.

but prove a lemma

supports

isomorphism)

into a sheaf G w h i c h

Z '-isomorphism

Then

complex,

conditions

Let Z' c Z be as in the previous

and let

F

the e q u i v a l e n t

we say F lies on the Z / Z ' - s k e l e t o n

W e now come to the C o u s i n

of

so the

is an isomorphism.

Definition. above~

terms are zero,

of u, we take

and let u be the n a t u r a l map of F = F_z(F ) into G.

from the exact

Z'-isomorphism,

sequence

of M o t i f B we see that ~ is a

and from M o t i f F we see that G lies on the

Z/Z'-skeleton. (2) ~: F functor

To see that u is unique up to isomorphism, > G' be another O H~/Z,,

such Z '-isomorphism.

(which takes Z ' - i s o m o r p h i s m s

we get a c o m m u t a t i v e

diagram

let

Applying

the

into isomorphisms)

z3z

8

F

G

where

7 = _~/Z,(8) (3)

8: G

>G

~G'

is an isomorphism.

To show that the isomorphism in 7 is unique,

let

> G' be any other homomorphism w h i c h gives a commutative

diagram F

G'

Then applying the functor

~7/Z'" we obtain a commutative

diagram id

G

>~

GI

Proposition 2.3.

Let X = Z O _D Z 1 D -'' be a filtration

of X by subsets Z p stable under specialization, for each p, each

x q zP-z p+I

an abelian sheaf on X.

is maximal in Z p.

Then there is a unique

isomorphism of complexes)

and such that

augmented complex

Let

F

be

(up to unique

233

F

0

~ C~

>C 1

>

"..

with the following properties: (a)

For each p > O, C p lies on the zP/zP+l-skeleton.

(b)

For each p > O, HP(c" ) has supports in Z p+2.

(c)

The map F ---->HO(c" ) has kernel with supports in Z I,

and cokernel with supports in Z 2. Furthermore, Proof.

C" depends functorially on F. We prove by induction on p, that there exists a

unique (up to unique isomorphism of complexes)

augmented complex

F

O

> C~

> C1

> ...

> Cp

with all the properties above, except that instead of saying HP(c" ) has supports in Z p+2, we say that

cP/Im C p-I has supports

in Z P+I (or, if p = O, instead of saying that the cokernel of F

> HO(c" ) has supports in

supports in zl).

Z 2, we say that

cO/Im F has

Case p = O.

This is precisely the statement of the Lemma,

with Z = Z O and Z' = Z I. Induction Step.

Suppose the statement proven for p, and

take a complex C" as above, defined in degrees < p. Lemma to cP/Im C p-I with Z = Z p+l and Z' = Z p+2 be the sheaf G thus obtained. cP/Im C p-1 ~ is a zP+2-isomorphism,

properties. let

C '"

< p+l.

and let C p+l

Now since C P+l

its kernel, HP(c" ) and its cokernel,

cP+l/Im C p, have supports in Z p+2. the zP+i/zp+2-skeleton,

Apply the

Furthermore,

C p+l lies on

so we have a complex with the required

To show that it is unique up to unique isomorphism,

be another such augmented complex,

defined in degrees

Leaving off C 'p+I, we have a complex satisfying the

conditions of the induction hypothesis,

therefore there is a

unique isomorphism of augmented complexes C" in degrees < p.

> C'"

defined

Thus there is a unique isomorphism

cP/Im C p-I

> c'P/Im C 'p-l.

Now again by the Lemma, this extends to a unique isomorphism of C p+I

> C 'p+l.

235

Definition.

The complex of the above proposition

is

called the Cousin complex of F (with respect to the filtration

Z'). Examples.

(1) If

F

has supports

in Z I, then its Cousin

complex is the zero complex. (2)

If

F

c ~ = F /_FZ I(F),

is flasque,

then its Cousin complex is given by

C p = O for p > O.

Lemma 2.~.

U n d e r the hypotheses

of Proposition 2.3,

_Hzp/zp+I(F ) = 0 for ail i > p. Proof.

Using M o t i f F, we have only to show that for all

x 6 zP-z p+I, H~(F) = O.

But this group can be calculated as the

.th 1 derived functor of Fx on the topological of all

generizations

to X).

Indeed,

of

x

(i.e., points x' which

it can be calculated by flasque

and the r e s t r i c t i o n x ~ Z p+I

,

(Motif C),

X

dimension

Let

P

"

X = Z O D Z 1 D ..- and

Proposition 2.3, and assume furthermore (i.e., ~ Z n = ~).

~

or[LC 1.12]

"

Proposition 2.5.

Now

f

l

Hi(F) = O for i > p [G, II,4 15.2],

separated

sheaves

the space X(x ~ has combinatorial %

and so

specialize

of a flasque sheaf to X(x ) is flasque. %

since

space X(x ) consisting

F

be as in

that the filtration is

Then the natural map

,

236 ~: F

> HO

I(F) makes the complex

"z'Vz"

0o

0

of

> H~ I(F) dl -z~/z ~

I0

> H1 (F) dl _zl/z2

E~OD terms of the spectral

>_z~/zH2o~(F)

> --.

sequence of M o t i f G into a Cousin

complex for F. Proof. (i.e.,

We must check

first that

~

gives an a u g m e n t a t i o n

d~O.~ = O), and then that the properties

of Proposition 2.3 hold. each point separately.

These properties Thus

(b), and (c)

can be checked at

for an x 6 X, we can replace X by

the space X(x) of generizations dimension

(a),

of x, w h i c h has finite combinatorial

(since the filtration Z" is separated).

we may assume that the filtration Z" is finite. the reader, we will check only property

In other words, To avoid b o r i n g

(b) w h i c h is the hardest.

Given p > O, we wish to show that H p of the complex above, w h i c h is nothing but E2PO of the spectral sequence of M o t i f G, has supports that

in

Z p+2.

We show, by descending

E pO has supports in Z P+r for each r > I. r

enough,

induction on r, For r large

E pO = E pO = O, since the abutment of the spectral r

E n, is O for n ~ O.

sequence,

237

Now let r > 1 be given.

Then E p-r'r-1 = O, since r

E pq = O for q > O b y the Lemma. r sequence

T h e r e f o r e we have an exact dP O

O

~ EPO r+l

EPO r

r

By the induction hypothesis,

E =-~n r+l

>

p+r,-r+l = HP+l and E 1 _Zp+r/zp+r+I(F), E p~ has supports in r in

Z p+r

> EP+r,-r+l. r has supports in has supports in

and in particular,

Z p+r+l" Z p+r.

Thus

E pO has supports

Z p+2 .

Now we ask when the Cousin complex is a r e s o l u t i o n of F. Proposition 2.6.

Under the hypotheses

of Proposition 2.5,

the following conditions are equivalent: 4

(i)

= 0

for all i,p w i t h i ~ p.

(ii) zP-depth F > O

for all p.

(iii)

for all i,p

(iv) Proof.

p/zp+l(F) = o

with

i ~ p.

The Cousin complex of F is a (flasque) r e s o l u t i o n of F. (i)--~->(ii) by definition of depth.

They are zero for i > p by Lemma 2.4. exact sequence

(ii)----~(iii).

For i < p we use the

238

Hi

H_p(F) (iii)----~

(iv).

i+l ,

> _zp/zp+l(F)

> p+l F)

The condition is pointwise,

we can assume the filtration Z" is finite. sequence of M o t i f G degenerates: means that the complex is the Cousin complex) (iv)----~ (i).

Let

so as before

Then the spectral

E~ q = O for q ~ O.

That

E~ O (which by the previous proposition is a r e s o l u t i o n of F. C"

be the Cousin complex.

Since it

is a flasque r e s o l u t i o n of F, we may use it to calculate cohomology:

H i p = H i ( -F~ p(C ")) . --Z

zP/zP+l-skeleton

Since

for all p, F p(C')

Cp

lies on the

is the truncated complex

f.J

(ci)i>_p.

C l e a r l y it has cohomology only in degree p, since

the original C" was exact. Definition.

If the equivalent

conditions

are satisfied, we say that F is C o h e n - M a c a u l a y

of the p r o p o s i t i o n (with respect to

the filtration Z'). Remark.

If X is a locally noetherian prescheme,

and Z p

the set of points of c o d i m e n s i o n > p (i.e., points x 6 X w i t h dim

~x

then

this notion coincides with the usual definition.

,X

> p)' and F a coherent --

sheaf on X with

support X, Indeed,

239

using condition

(iii) and Motif F, F is Cohen-Macaulay

only if for each x q X, d e p t h ~ F

if and

> dim ~x' which is the usual

-x x-definition. Example. scheme, let

let

Z"

F = ~__.

[ZS vol.

Let

X

be a non-singular

locally noetherian

be the filtration by codimension as above,

Then

F

is Cohen-Macaulay

and

(usual sense)

II, App. 6], so the Cousin complex gives a flasque

resolution of

~X"

Furthermore,

complex is isomorphic isomorphic

to

to

the p

th

component of this

H/ z P p

zp+I(~X) which by Motif F is

codim x = p Now for x of codimension p, we know [LC.4.13] an injective hull over the local ring k(x). of

that

~x of its residue

field

Thus our Cousin complex is in fact an injective resolution

~X' and its component

in degree p is isomorphic

sum of sheaves J(x) (see definition in [II w

to a direct

where

x

is a

point of codimension p. This is an example of the notion of residual complex, which will be studied in more detail in Chapter VI.

24o

w

Generalization

to Complexes.

In this section we generalize section to complexes. Cohen-Macaulay relations

the results

In particular,

complexes,

Gorenstein

of the previous

we w i l l discuss complexes,

and their

to Cousin complexes.

Throughout topological

this section,

X

will be a locally noetherian

space in which every irreducible

a unique generic

point.

closed subset has

We will denote by D+(X)

the derived category of the category of abelian or D+(Mod(X))

if

X

is a locally noetherian

either D+(Ab(X)), sheaves on X,

prescheme.

The

results are valid in both cases. We will consider always

filtrations

suppose the following

(l)

Z" = (ZP)p6~

conditions

on X, and will

satisfied:

Each Z p is a subset of X, stable under

specialization,

and Z p-1

(2) proper

for some

(4)

Z p+I

Each x 6 zP-z p+l is maximal in Z p (i.e., x is not a

specialization

(3)

Zp

of any other x' E zP).

The filtration

is strictly exhaustive,

i.e.,

p 6~. The filtration

is separated,

i.e.,

N Z p = ~.

X = Zp

Definition.

A Cousin complex on X, with respect to the

filtration Z', is a complex of sheaves G', such that for each p, G p lies on the zP/zP+l-skeleton of X. Proposition 2.1.)

(Cf. definition following

Note that a Cousin complex is necessarily

flasque, and bounded below.

We denote by

Coz(Z';X) the category

of Cousin complexes and morphisms of complexes.

It is an

additive category. Example.

The Cousin complex of a sheaf F (as in

Proposition 2.3 above) is a Cousin complex. Definition.

Let

F" 6 D+(X).

Then we denote by E(F')

the complex

p _(F') -zp/zp+.,-

HP+I

> _zp+i/zp+2(F )

of ElPO terms of the spectral sequence of Motif G. by Motif F that the p

th

. .

We observe

term EP(F" ) lies on the zP/zP+l-skeleton

of X, and so E(F') is a Cousin complex.

(Note that even if

F" 6 Db(x), the complex E(F') need not be bounded above. ) Proposition 3.1.

Let

conditions are equivalent:

F" 6 Db(X).

Then the following

2~2

(i)

a)

H i (F') = 0 for i < p, and ~P

b)

the map

> ~i(F')

H i (F') -z p

is surjective for i = p and bijective for i > p.

(ii) ~p/zp+l(F" ) = o for i + p. (iii) There is an isomorphism where

o+(x)

Q

~: F"

> QE(F') in D+(X),

is the functor sending a complex to its image in

[i w Furthermore, the isomorphism in (iii) can be chosen so that

the isomorphisms

Hi(~):

the isomorphisms

E.: 1

Hi(F ")

Hi(E(F'))

> Hi(E(F')) are inverse to > Hi(F ")

determined by the

degenerate spectral sequence of Motif G. Remark.

One should beware, however, that the isomorphism

is not in general unique, and so is not functorial. Proof of Proposition.

(i)-----> (ii).

This follows immediately

from the exact sequences ~_~p/zp+l(F") "H ~_~zp+I(F') >_.zp

(ii)---> (i). as in w a)

Condition

(i) can be checked pointwise,

so

above, we may assume that the filtration Z" is finite. For i < p, the exact sequence of Motif B shows us that

9

H i

(F')

>-zp is bijective.

Therefore by iteration,

9

H~p+r (F") is bijective

Hi >_zp(

for any r > O.

F" ) But for r large enough,

Z p+r = ~,

so Part b) is proved similarly. (i) + (ii)-----> (iii). such that

F" E D-(X), we can find an i

Since

Hi(F ") = O for i > i . --

By condition

(ii), the

o

spectral sequence of Motif G degenerates, we see that

o

Hi(E(F')) = O for i > i --

so checking pointwise,

also. o

We will construct, by descending induction on p, an isomorphism in D+(X) of Q(7>_p E(F')) of

R Fzp(F" ) with the truncation

QE(F') in degrees >_ p (see [I ~7] for notation).

Then, since the filtration is strictly exhaustive, we have

244

F" = R_Fzp(F" ) and QE(F') = Q(7>_pE(F')) which will give us the isomorphism

for p small enough,

~.

For p large enough (say p >_ io) , RFzp(F" ) has a unique non-zero cohomology group.

H~p(F') = o

for

indeed, i < p

by (i) a)

for i > p

by (i) b),

and

.i (F') -- Hi(F ") = O -z p since

p > i . --

(1)

o

Furthermore,

there is an exact sequence

o

> Hp (F')

-zp

> ~P+I_(F')

> HzPp/zp+I(F" )

>0

-zp+-

from Motif B. At the same time, for p > i , --

non-zero cohomology group.

7>p E(F')

Indeed,

H i = O for i < p since

the complex is zero in those degrees. p > i --

and

(2)

(see above) 9

has a unique

o

H i = O for i > p, since

Finally since this is a truncated complex,

o

HP(E(F')) = O, there is an exact sequence

0

> HP(~>_pE(F'))

> EP(F ")

> EP§

")

Now comparing (1) and (2), and using the definition of the complex E(F" ), and noting that

245

> E p+I(F ")

HzP+I p+l~"F " ) is injective, we see that

HP (F ") _--"HP(~>pE(F')). -Z p Hence there is an isomorphism [I.~.3]

~p: R~=_pCF')

~ > o( ~>_pP(F"))

in D+(X) . We continue the induction as follows.

In terms of the

derived category, the spectral sequence of Motif G is expressed as a diagram of triangles

~ p/zp+X(F")

> ~rp§ l(F')

-R_--Fzp-1/zp(F")

>R_~p(F') --~ R rzp_l(F') --~ ...

So suppose by induction we have an isomorphism ~p of R__~Fp(F') with Q(T>_pE(F')), for any p.

By (ii)j _R_Fzp_I/zp(F') has a

single non-zero cohomology group, so there is an isomorphism

246

of it in the derived category with the complex consisting of that single sheaf, in the right place [I.4.3], namely Q(E(F')P-I[-p+I]).

Thus we have a commutative diagram

R~ (F') ~p

> ~_rzp_l(F")

O(~(F')P-I[-p+I])

O(,>pE(F'))

i~p-i

> Q(~>p_IE(F" )

of distinguished triangles.

By the axiom (TR3) of triangulated

categories, we deduce an isomorphism on the third side of the triangle, which allows us to continue the induction9 small enough, ~p gives the required isomorphism ~.

For

p

Furthermore,

by choosing the isomorphisms # to be the obvious ones, we get the further condition on the Hi(~). (iii)---~(ii).

We can use the (flasque) complex E(F') to

calculate cohomology, by Motif C. whence the result.

Then %p/Zp+I(E(F'))~

= EP(F'),

2/-I.7 Definition.

If F" 6 Db(x) satisfies the equivalent

conditions of the Proposition, we say that F" is Cohen-Macaulay with respect to the filtration Z'.

If

F

is a single sheaf,

this is the same as the notion of the previous section (Proposition 2.6). Lemma 3.2. a)

If

Let

f,g

F',G" 6 Coz(Z';X).

Then

are two morphisms of F" into G" such that

Hi(f), Hi(g):

Hi(F.)

> Hi(G. )

are the same map for each i, then f = g. b) if

G"

If

3:

Q(F')

> Q(G')

is a morphism in D+(X), and

is a complex of injective sheaves, then

represented by an actual morphism of complexes Proof. if

f: F"

a)

By considering

f

is

f: F"

> G'.

f-g, we reduce to showing that

> G" induces the zero map on cohomology, then

the zero map itself. Then since H~

We may assume by translation that X = Z ~

") ~ F ~ fo

f

Fo

and H~

is the zero map, the map

> G~

passes to the quotient to give a map Bl(F" ) ....> G ~

is

248

But

BI(F" ) has supports in Z l, and G ~ has zl-depth > i,

i. e., --Z F 1 (G ~ ) = O, so there are no non-zero maps of into G ~ .

We conclude that

B1 (F" )

~o, and hence also fo, is the

zero map. Thus

fl

maps

yl:

B2(

ZI(F ") into O, hence gives a map

")

> G1 .

Proceeding as above we shows that

fP

is the zero map for

all p.

b)

This follows directly from [I.4.5] or [I.4.7],

and

is included only as a reminder.

Corollary 3.3.

Let

F',G"

6 Coz(Z';X).

If

f: F"

> G"

is a morphism of complexes which induces an isomorphism on the cohomology sheaves,

and if

F"

is an injective complex,

then

f is an isomorphism of complexes. Definition.

A complex

F" q Db(X) is Gorenstein with

respect to the filtration Z" if it is Cohen-Macaulay, H~(F')

is either zero or injective

denote the (additive) by

Db(XlGor (Z.) .

and if

for each i E Z, x 6 X.

subcategory of Db(x) of Gorenstein

We complexes

249 Remarks.

i.

If

X

is the spectrum of a local noetherian

ring A, and if Z" is the filtration by codimension,

and if

F" = A, then this is the usual notion of a Gorenstein ring [LC 4.14 and Exercise 2 ff]. @

2.

If

F" is Gorenstein,

then E(F') is an injective Cousin

complex, by Motif F. *Example.

We will see in Chapter V that a dualizing

complex on a prescheme in Gorenstein with respect to the corresponding filtration by codimension.

Proposition 3.4. E:

,

The functor

Db(X)Gor (Z.)

> Icz(Z',X)

is an equivalence of categories of the category of Gorenstein complexes with the (additive) category Icz(Z',X) of injective Cousin complexes.

Furthermore,

the natural functor Q is an

inverse to E. Proof.

and

We must construct isomorphisms of functors ~:

1

> QE

~:

1

> EQ

on the two categories. Gorenstein complex

To construct ~, we choose,

F" 6 Db(X), an isomorphism

for each

250

~(F'):

F"

> Q~(~')

as in Proposition 3.1 (iii) such that

~i(~(F.)) : Hi(F.)

>~i(~(~.))

is inverse to the map derived from the spectral sequence. claim the collection of morphisms

I

[~(F')] is an isomorphism of

functors ~: as required.

1

> QE

Indeed,

Gorenstein complexes

~(F')] F"

Q~tF') where

~(F'),~(G')

let

f: F"

in Db(x). -

be a morphism of

Then we have a diagram

f

~E(f)

> G"

I~(G')

> G"

~ eE

")

are isomorphisms

in the derived category.

To show this diagram is commutative, QE(f) = ~(G')f~(F') -I in D+(X).

we must show that

Now by the lemma, part b),

~(G')f~(F" )-i is represented by an actual morphism of complexes

g:

E(F')

> E(G')

.

251

Since E is a functor,

our condition on Hi(~(F'))

and Hi(~(G'))

shows that E(f) and g have the same effect on cohomology. by the lemma, part a), they are equal.

Hence

Thus ~ is functorial,

as required. For $, one need only note that if EQ(F ~

F" E Coz(Z',X),

then

= F" in an obvious way.

Remark.

This Proposition will be used in an essential way

in Chapter VI in the construction of residual complexes dualizing complexes.

from

252

CHAPTER V.

w

DUALIZING COMPLEXES AND LOCAL DUALITY

Introduction. In this chapter we discuss dualizing complexes on a

locally noetherian prescheme X.

A dualizing complex is a

complex R" E D+(X) such that the functor

D:

M"

> R Hom'(M',a')

induces an auto-duality of the category

D~(X) consisting of

those bounded complexes in D+(X) which have coherent cohomology. We will show that a large class of preschemes admits dualizing complexes, and that they are almost uniquely determined. The notion of dualizing complex will allow us to write the duality theorem in a new way.

For example,

n-space over a field k.

R" = win]

Then

sheaf of relative n-differentials

let

X

(where w = WX/k is the

[III ~i]) is a dualizing

complex for X, and k is a dualizing complex for k. canonical isomorphism any complex

be projective

We have a

Rfw(w[n]) ~ k [III 3.~], and hence,

F" E Db(x), a homomorphism

DXCF)

>_Dk(af CF')

,

for

253

where

~X

and

respectively. isomorphism,

~k

are the d u a l i z i n g

The duality i.e.,

theorem

functors

[III 5.1]

that the d u a l i z i n g

on X and on k,

says that 8 is an

functors

on X and on k

commute with ~f.. In the latter duality

theorem,

part of the chapter we discuss

as in [LC w

The proof of the duality

or

theorem

the local

[SGA 62, expos~

given here

2,4].

is different,

however. The reader w i l l complex has

on a locally noetherian

finite Krull

notion,

see that the existence

dimension.

that of a pointwise

case of infinite

Krull

For convenience dualizing

complexes

prescheme

of a dualizing

X implies

that X

Hence we w i l l give a more general dualizing

complex,

to cover

the

dimension.

we defer until

the q u e s t i o n

of existence

the end of the chapter

of

(section

IO).

254 w

Example: Let

X = Spec Z.

sheaves on groups.

Duality for abelian groups.

X

Then the category of quasi-coherent

is isomorphic to the category

(Ab) of abelian

We know that ~ / Z gives a good duality for finite

abelian groups,

i.e., that the functor

exact contravariant

Hom(., ~/~) is an

functor, which, when applied twice to a

finite abelian group, gives that group back again. the group generated

~

Similarly,

gives a good notion of duality for finitely

free abelian groups.

Combining the two, we consider

the complex

and work in the derived category

D~(Ab),

consisting of those

bounded complex in D+(Ab) which have finitely generated cohomology. generated"

(c is for "coherent", which means in our case.)

"finitely

Since the complex above is an

injective resolution of ~, hence isomorphic

to ~ in the

derived category, we arrive at the following proposition. Proposition D:

M"

1.1.

The functor

> R Hom'(M',2Z)

255

is a c o n t r a v a r i a n t

~-functor

there is a n a t u r a l

functorial

D:

Proof. if

M"

id

from D~(Ab)

N

D(Ab)

> DD

then so is

and

isomorphism

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

6 Db(Ab),

into itself,

.

of

D(M').

D(M')

is Extl(M',~),

The n a t u r a l

functorial

c

homomorphism

D is defined

it is an isomorphism, applying

in Lemma

the case M" = Z r for some r.

the proof, and

O

Since

[I.?.l],

the functors

M"

and

we reduce

to

in q u e s t i o n

to the case M" = ~, and to c o m p l e t e

one need only observe

F"

1.2.

6 D(X). 7:

Proof. erase the R. D(X),

one reduces

functors

of

that

Exti(z,Z)

= Z

if

i = O,

otherwise.

Lemma let

To show that

by taking a free r e s o l u t i o n

the lemma on w a y - o u t

are additive,

1.2 below.

i.e.,

F"

Let

X

be a prescheme,

Then

there is a n a t u r a l

let

R"

6 D+(X)

b y an i n j e c t i v e

R')

.

resolution,

As usual, w e need o n l y define ~ on o b j e c t s complexes.

and

functDrial h o m o m o r p h i s m

> R Hom'(R H o m ' ( F ' , R ' ) ,

Replacing

R"

G i v e n an index p, and a section

we can of s 6 FP(u)

256 over an open set homomorphism

U

in

X, we must define,

for each q, a

of sheaves Homq(F" ,R" )U -----> R~ +q '

w h e r e the subscript

U

Given a h o m o m o r p h i s m

f 6 Homq(F',R')u(V)

set

denotes r e s t r i c t i o n of sheaves to U.

V ~ U, we send it to the section

sv is the section s, r e s t r i c t e d to V.

defined on an open

f(Sv)

in F'.

where

One checks that this

indeed does define a m o r p h i s m of complexes, functorial

6 RP+q(v),

and that it is

257

w

Dualizing Complexes. Throughout this section, X will denote a locally noetherian

prescheme.

We will consider complexes

R" 6 D+(X) which have

coherent cohomology and finite inJective dimension, R" 6 D~(X)fi d (see [I.7.6] and [II~?.20]). quasi-compact, on a complex

i.e.,

Note that if

X

is

then the condition of finite injective dimension R" E D$(X) can be checked locally (e.g., by

[II ?.20(iii)]).

Note also that such a complex R" is quasi-

isomorphic to a bounded complex of quasi-coherent

injective

sheaves on X [II ?.20(i)qc] , and hence (using [II 3.3]) the functor D:

7"

> R

sends Db(x) into itself, and interchanges Definition.

Let

F" 6 D(X) is reflexive

R" 6 D+(X).

D+(X)c and

Dc(X ).

We say that a complex

(with respect to R') if the natural map

~: F" ---->R Ho___mm'CRHom'(F',R'), of Lemma 1.2 is an isomorphism.

R')

258 Proposition 2.1.

Let

+(x) lid"

c

Then the following

conditions are equivalent: (i)

Every

F" 6 Dc(X) is reflexive

(ii)

Every

F" 6 D~(X) is reflexive.

(with respect to R').

(iii) Every coherent sheaf on X is reflexive. (iv)

The structure

Proof.

Clearly

sheaf

(i)~

~X is reflexive. To

(ii)---> (iii)===~ (iv).

(iv)--~,(i) we note that the question is local on X. may assume that

X

is affine.

is the quotient of a free sheaf are additive, Functors

Then every coherent

which implies that Definition.

R Hom'(',R') Let

A complex R" 6 D$(X)fi d the proposition Definition.

X

Hence we sheaf on X

G~; the functors in question

so the result follows

[I.7.1 (ii) and (iv)],

prove

from the Lemma on Way-Out

taking into account

[II 7.20(ii)]

is way-out in both directions.

be a locally noetherian

prescheme.

satisfying the equivalent

conditions

of

is called a dualizing complex for X. The Krull dimension of a locally noetherian

prescheme X is the largest n (or 4-oo) for which there is a chain Zo < Z1 < "'" < Zn of closed irreducible

subsets

Z.I of X.

is also the sup of the dimensions of the local rings of X.

It

259

Example 2.2. Krull dimension.

Let

X

(A regular prescheme is a prescheme X such

that all the local rings rings.)

be a regular prescheme of finite

G

of points of X are regular local

x,X

Then the structure sheaf

~X is a dualizing complex

for X. Indeed, condition (iv) of the proposition is trivial, because EXti(~x,~x ) = ~x for i = 0, and o otherwise. 6 Dc+(X).

~X has finite injective dimension,

To show that

we use condition (iii)c of [I.7.20]. dimension of

X.

Clearly

Let

n o be the Krull

Then for every coherent sheaf F on X,

Exti(F,~x ) = O for i > no, because Ext i commutes with taking all have cohomological dimension

stalks, and the local rings X

< n q

since they are regular local rings [14 (28.2)] O'

Corollary 2.3.

Let

R"

6 Dc+(X)fid.

if and only if for every x E X, R"

Then

R"

is dualizing

is dualizing on the local

X

scheme

Spec

closed points Proof.

~x"

Furthermore,

it is sufficient to take the

x 6 X. For x 6 X, we see that the stalk R"

has finite

X

injective dimension in Dc+(Spec and [II.7.17 (iii)]. of the Proposition.

~x ) by using [II 7.20 (i)qc ]

Thus the result follows from condition (iv)

260

Proposition 2.4.

Let

f: X

locally noetherian preschemes, complex on Y.

Then

> Y be a finite morphism of

and let

R"

be a dualizing

f~R" is a dualizing complex on X

(cf. [III w Proo f.

We may assume that

quasi-coherent

R" is a bounded complex of

injective sheaves on Y.

Then

f~(R') = Homy(f~O'x,a" )'~ ,

which is a bounded complex of quasi-coherent Hence

f~(a') E Db(x)fi d.

we apply condition

To see that

injectives on X.

f~(a') is dualizing,

(iv) of the previous proposition,

and consider

the map 7: ~X

Applying morphism

> R Ho~(R

H o ~ ( ~ X , f~(R')),

~f~ to this map, and using duality for a finite [Ill 6.7] twice, we see that ~f~(~) is an isomorphism.

But we are dealing with quasi-coherent faithful,

sheaves, and

f~

is

so ~ is an isomorphism.

Remarks. morphism,

f~(R')).

then

I.

We will see later that if f is a smooth

f~

(Theorem 8.3 below).

of a dualizing complex is dualizing

261

2.

This result,

joined with the example above,

that any closed

subscheme of a regular

Krull dimension

admits a dualizing

Proposition

2.5.

A complex R"

and only if for every closed point consisting

of the residue

zero elsewhere, Proof. where

X

residue

I)

field

6 D:(X)fi d is dualizing x 6 X, the sheaf

k(x)

particular,

A-module

at the point

Using Corollary

The assumption

for M = A),

that

Using induction M

if

k(x), x

and

is reflexive. 2.3, we reduce to the case

is the spectrum of a local noetherian field k.

of finite

complex.

M

of finite type M (in

is reflexive

on X.

on the length, we show first that every

of finite length is reflexive.

M = i, we have assumed

ring A, with

is then that k is reflexive,

and we wish to show for every A-module

2)

prescheme

shows

it.

Indeed,

for length

For length M > I, one can write a

short exact sequence O

> M' ----->M

>M"

> 0

where M' and M" have length < length M. hypothesis M' and M" are reflexive,

Then by the induction

and by long exact sequences

262

(DD is a ~-functor~) 3)

and the five-lemma,

N o w we show that any m o d u l e

reflexive,

using

induction

then M has finite length,

Then

support ~ ,

M' has finite

M/M'.

Let t 6~

t

Therefore,

Hi(DD(M))

N o w these Nakayama

Hi

's

O

0

>

~

....t

M' be the submodule is the m a x i m a l

so it is s u f f i c i e n t

Let

of

ideal of A.

to c o n s i d e r ~ A s s M.

is not a z e r o - d i v i s o r

in M,

sequence > M/tM

> O

.

so b y the i n d u c t i o n h y p o t h e s i s

M/tM

for each i ~ O, we have an exact ~> Hi(DD(M))

are A - m o d u l e s

lemma,

Consider

where

>M

N o w dim M / t M < dim M, reflexive.

Let

be an e l e m e n t w h i c h

>M

If dim M = O,

we have r e d u c e d to the case ~

so that we have an exact O

of M.

and M is r e f l e x i v e by 2) above.

length,

In other words,

of finite type M is

on the d i m e n s i o n

M be a m o d u l e of finite type. elements with

M is reflexive.

Hi(DD(M))

sequence

, > O.

of finite type, =

and t 6 ~

, so by

0 for i ~ O.

the c o m m u t a t i v e d i a g r a m t M > M

I~

> HO(DD(M) .t > HO(DD(M))

is

>O

> H~

)

>0

263

with exact rows.

A diagram chase shows that

H~

= ~(M) + tHO(DD(M))

so by Nakayama's Lemma, ~ is surjective. injective,

let x 6 M.

Choose

n

x

does not become O in

M/tnM ~ H~

so

To show that

so large that

draw the same diagram as above, but with Then

,

tn

x ~ t~,

M/tnM, and

Taking M = A, we find that R" is a q.e.d.

Proposition 2.6. Let let R" 6 D~(X)fi d

X

R Hom'(-,R'). A complex

D

be the

Then

G" 6 D~(X)

has finite Tor-dimension

D(G') has finite inJective dimension.

There is a functorial isomorphism D(R Ho__~m'(F',G'))

for

be a locally noetherian prescheme,

be a dualizing complex, and let

[II ~.2] if and only if

b).

F" E D c(X)

G" 6 D+(X)fid.

and

in place of t.

dualizing complex.

a).

is

~(x) ~ O.

Thus M is reflexive.

functor

~

and

N >

F" =~ D(G')

G" 6 D+(X), or for

F" E D (X) C

and

26# c).

There is a functorial i s o m o r p h i s m

D(F" |

>R

r162162

for

F"

and

G"

Proof.

E

D-(X), c- "

Hom'(F',

or for

D(G'))

F" E D (X) C

and

G~ 6

Db(X)fTd,

The natural m a p of sheaves F | Hom(G,R)

------>Hom (Hom(F,G),R)

gives rise to a m o r p h i s m of functors on the derived category,

( -It-)

F ~ =~ R Horn ~ ( G ~ ,R" )

> R Eom'(R Hom'(F',G'),R'),

p r o v i d e d either

D+(X)

and R" E D+(X)fi d,

1)

F ~ E D'(X),

G" ~

2)

F" E D(X),

G',R" 6 D+(X), and

If under the set of conditions F" E D~(X),

then by [I~

~ Hom'(G',R') E Db(X)fTd .

1), w e assume furthermore

the m o r p h i s m

Indeed, we reduce to the case

or

that

(~) is an isomorphism.

F = ~X" w h i c h is trivial.

Taking

the inverse i s o m o r p h i s m gives b) under the first set of hypotheses. As a corollary, we see that if dimension, the Remark

then

D(G')

following

is an integer

no

G" E D+(X) has finite inJective

has finite Tor-dimension.

Indeed, by

[II 4.2] it is enough to show that there

such that

Tori(F,G~ ) = O for all i > n o

265 and all coherent sheaves Thus

F

on

X, and this follows from b).

we have established one half of a).

then as Just remarked,

Now if

G" E D+(X)fid,

D(G') E Db(X)fTd , so by 2) and

again, we obtain the isomorphism

b)

[I.7.1]

under the second set of

conditions. For statement c), let D(G'), and apply D 2 = id

on

D

to the resulting isomorphism.

then

7.20 (iii)c]) D(G ~ )

other half of

F"

and

Then since

D (X), we have the required isomorphism under the c

first set of conditions. [II

F',G" E Dc(X) , apply b) to

As a corollary we deduce (using

that if

G" E D~(X)

has finite Tor-dimension,

has finite inJective dimension.

This gives the

a), and applying D to the second half of b)

gives the second half of c)~

q.e.d~

266

w

The Uniqueness

Let

Theorem 3.1. prescheme,

of the Dualizing

and let

X

R"

Complex.

be a connected,

be a dualizing

locally noetherian

complex on X. |

R'"

6 Dc+(X)

complex

Then

be any other complex.

R

9

Let

is a dualizing

if and only if there exists an invertible

sheaf L on X,

and an integer n, such that R'"

Furthermore,

~ R"

| L[n]

.

L and n are determined

uniquely

to within

isomorphism. Proof 9

i)

Observe

that such an R'"

Indeed, we may assume that R" is a bounded coherent

injectives.

of injectives,

[If ?.16], Corollary 2)

Then

R" | L[n]

since the property

and L is locally

is a dualizing

complex of quasi-

is also a bounded

of being injective

free.

R'"

complex.

complex

is local

is dualizing by

2.3. Observe

that L and n are uniquely determined by the

identity L[n] ~ R Hom'(R',R''). 3) R'"

Now we show that existence

is a dualizing

complex 9

of L and n, assuming

that

Let D and D' be the dualizing

267 functors

corresponding L" = D'D(~x)

W e w i l l prove

Define

to R" and R''.

= R= Hom'(R',R'')

that

L"

.

is isomorphic,

in D+(X),

of the form L[n], w h e r e L is an invertible R'"

=N R "

to a complex

sheaf,

and that

| L" .

Lemma 3.2.

There

is a natural

functorial

isomorphism,

for

M" M" | L"

~

(Note that this tensor cf.

> D'D(M')

product

makes

.

sense

since L"

6 Dc(X) ,

[II ~ ] . ) Proof,

AS in Lemma

1.2 above,

one defines

a functorial

homomorphism M" ~ R H o m ' ( R ' , R ' ' )

Since

it is an isomorphism

way-out

functors,

>R

Hom'(R

for M" =

Hom'(M',R'),R'').

~X' by the lemma on

it is an isomorphism

for all M"

6 Dc(X).

|

We now define L " = DD'(~X) , and use the lemma to deduce that

L|

' = D'D(L'')

= D'DDD'(

=

o

268

Lemma 3.3. prescheme, Then

Let

X

and let L',L

L" = L[n] Proof.

$ 9

X

X

that

p+q

HP(

')

L" = ~x[n]

Hq(L ' ' )

such that

Hq(L '') # O.

O"x .

that

L" = L[ (9 ~ ' x [ - p ] ,

only in dimensions < p. L'|

" =N

~X

L" ~

Similarly,

on modules

[cf. EGA, Ch. O,

where

L1

~X is locally has c o h o m o l o g y

L'" = L I ' ~ ~x[p].

Thus

from

we d e d u c e

E-pl from which

p+q = O, and

Now since

) -- ~X

be the

be the

It follows, by a simple ~ m m a

H p(L') ~ Hq(

free, one shows that

q

p

H n ( L ' @ L '" ) ~ O,

Therefore

of finite type over a local n o e t h e r i a n ring 5 4.3],

Let

Then one sees easily

for w h i c h

'') = HP(L.)| =

for some n.

HP(L') # O, and let

is the largest integer

and HP+q(L'|

we reduce i m m e d i a t e l y to

is the spectrum of a local noetherian ring A,

such that

largest integer

L ' " ~' O"X.

sheaf L, and L'" = LV[-n].

is connected,

and we wish to show that largest integer

locally noetherian

6 Dc(X) be such that L" |

for some invertible

Since

the case where

be a connected

(,i

,i

it follows that L l

~x[-p] , and

L'" ~

) = o and L'I" are O in Dc(X),

~x[p] , as required.

so that

269 This proves the lemma, R'" = R'|

Indeed, R"

R'|

= D'D(R ~

so we have only to show that

6 D~(X),

residue

q.e.d. Theorem.

= D'(~X) = R'"

We now give an application Proposition 3.~.

so applying Lemma 3.2,

Let

field k, and let

A

of the uniqueness be a noetherian

R" E D~(A).

Then

theorem.

local ring with

R" is dualizing if

and only if there is an integer d such that Extl(k,R ") =

(Here we write D+(A) of A-modules.

f

O

for i 4 d

k

for i = d.

for the derived category of the category

The subscript c denotes complexes whose cohomology

modules are of finite type.

We carry over the definitions

results of the previous sections to this case.

(Cf.

and

[II.7.19]

which ensures that we will not get into trouble.)) Proof. j: S p e c k

First suppose that

R" is dualizing,

> Spec A be the inclusion. J~(R') = R Hom'Ck,R')

and let

Then by Proposition

2.~,

270

is a dualizing complex J~(R')

complex on k.

But k itself

on k, so by the U n i q u e n e s s ~ k[-d]

for some d.

For the converse, Proposition inJective r = dim M,

Theorem

This d will

it is clear

Indeed,

for any A - m o d u l e

Exti(M,R ") = O

that k is reflexive,

we w i l l

R" has

type,

so by

finite

show by induction

M of finite

for

above,

do.

2.5 we have only to show that

dimension.

is a dualizing

on

that

i ~ [d-r,d].

Indeed,

for dim M = O, M is of finite

length,

follows

from the case M = k by induction

and our statement

on the length

of M.

For dim M = r > O, we induct as in the proof of Proposition 2.5.

First we may assume

maximal

ideal of A.

divisor

in M, we have O

and

>S

dim M / t M < r.

hence

that

4~ ~ Ass M, where

Then uhoosing

t 6 4~

t>M

----> S / t M

Then

for i ~ [d-r+l,d]

for i ~ [d-r,d]

we have

with

4~

is the

t a non-zero-

>o

Ext i(M/ ,R ")=0,

we have

from the long exact

sequence

> Exti(M,R ")

.

of

Ext's, Exti(M,R .) It follows by N a k a y a m a ' s

t

> O

lemma that ExtI(M,R ") = O .

q.e.d.

271

Corollary 3.5. let if

R" 6 D~(A)o

R" |

A

Proof.

Let

Then

R"

is dualizing Clearly

R" |

Exti(k, R | A) A

for all i.

A

be a noetherian

local ring, and

is dualizing on

A

if and only

on t h e c o m p l e t i o n

A

of

~ E Dc+(A).

A.

Furthermore,

-- Ext (k,R') |

Thus the "only if" implication is clear.

For the

A

"if" implication, flat, so if non-zero. M |

M

note that the functor

| A

is a non-zero A-module,

then

Furthermore,

A ~ k, then

if

M ~ k.

M

is an A-module,

(Indeed, map

M

to

is faithfully

M |

A

is

such that k

via the

A

natural map

M

>M |

A.

This map becomes an isomorphism

A

upon tensoring with A, hence was an isomorphism). result thus follows from the proposition.

The

272

~.

Local C o h o m o l o g y on a Prescheme. Let

of

X

be a prescheme,

~X-mOdules.

Y a closed subset,

and

F

a sheaf

We interpret the local cohomology groups ~ ( F )

in terms of Ext's. For each n > I, let Y --

be the subscheme of

X

defined by

n

n where Iy,

the sheaf of ideals

some scheme structure on Y.

Iy

is the sheaf of ideals of

Then for each n we have a natural

map H O m ~ X ( ~ Y n , F)

> Fy(F)

,

since an element of the first group is given by a global

section

n and hence has support on Y of F w h i c h is a n n i h i l a t e d by Iy, Taking the direct limit as n varies, we deduce a map

li B H o m ( ~ y , F) n n

. > Fy(F).

Taking derived functors, we deduce a map of functors

from

D+(x) to D+(Ab) (*)

R= li~ n

H o m C ~ y , F') n

Theorem 4.1. F"

6

(cf.

If

X

> R= FyCF')

is a noetherian

.

prescheme,

D + (X), i e. the complex F" has quasi-coherent qc " " [II w

then (*) is an isomorphism.

and cohomology

273

Remarks.

i.

Since there are enough flasque ~X-mOdules,

R= Fy(F') is the same as if one had calculated it in the category D + (abelian sheaves on X). 2.

The derived category does not have direct limits, so

we cannot write

li~ R= Hom(~y ,F'), as one is tempted to. n

n

However, upon descending from the derived category this difficulty disappears, and we have the Corollary below. Corollary 4.2.

Under the hypotheses of the theorem, the

map induced on cohomology,

I is an isomorphism, Proofs.

If

for all i. X

is noetherian, and F quasi-coherent,

then

the map li~ H o m ( ~

,F)

> Fy(F)

n is an isomorphism,

since every section of F with support on Y

will be annihilated by some power of Iy. lemma on Way-Out Functors

Therefore, by the

[I.7.1], the morphism of derived

functors (*) is an isomorphism for F" 6 D + (X). qc

27~

For the Corollary,

just observe that taking cohomology of

complexes commutes with direct limits. Corollary ~.3.

If

X

is locally noetherian,

and

F" E D~c(X), then the analogous maps

R liz Hom(~ ,F') n

> __Rf~(F')

n

and

n

n

are isomorphisms. Proof.

By definition of the derived category, it is

sufficient to prove the second. since

Ext and H ,, ,

Ext a n d S .

,

-my

This follows from Corollary ~.2,

are sheaves associated to presheaves given by

275

~5.

Dualizing In this

functors

section we recall w i t h o u t

and some properties

of dualizing

ring A, with m a x i m a l Proposition residue Then

on a local n o e t h e r i a n

ideal

5.1.

4~ .

Let

I

ring.

proof the definition

functors

on a local n o e t h e r i a n

For proofs,

see

be an i n J e c t i v e

[LC,w

hull of the

field k of A, and denote b y T the functor Hom(.,I).

for every A - m o d u l e

M

M of finite

length,

the n a t u r a l

map

T T(M)

is an isomorphism. Definition. category

A contravariant

~ f of A - m o d u l e s

a dualizing

functor

of finite

for A at

~

hull I of k and an isomorphism Proposition

~f

into

and

itself

T(k)

is

length

if there

a dualizing

functor

into itself exists

of functors

A contravariant

functor T from the

T ~ Hom(',I).

additive

if

is called

an injective

functor T from

and only

if

~ k.

it

is

In that case one can take I = l i ~ T(A/4~n), / n is a canonical i s o m o r p h i s m T ~ Hom(.,I).

then there

Proposition the functor (dcc)

5.2.

additive

5.3.

Hom(-,I)

of A - m o d u l e s

If

I

is an injective

descending

A-modules

of c o - f i n i t e

type)

of finite

type over the c o m p l e t i o n

of the c a t e g o r y

chain c o n d i t i o n

and the c a t e g o r y A of A.

and

hull of k, then

gives an a n t i - i s o m o r p h i s m

with

exact,

(acc)

(we call them of m o d u l e s

276

w

Local Duality. Throughout this section we let

ring, with maximal ideal ~

A

be a noetherian local

, and residue field k, and let R"

be a dualizing complex on X = Spec A (cf. 62).

Recall

Proposition 3.4 which gives a necessary and sufficient condition for a complex

R" E Dc+(A)

Definition.

to be dualizing.

We say that the dualizing complex R" is

normalized if the integer d of Proposition 3.4 is zero. Remark.

Since the translate of a dualizing complex is

again one, we can normalize by translation. Proposition 6.1.

If

R" is normalized, then RFx(R'), where

x is the closed point of Spec A, is isomorphic in the derived category to an injective hull I of k. Proof.

The cohomology of R Fx(a" ) is

Hi(a" ) ~

li~ Exti(A/~n,R ") n

by Corollary 4.2 above.

On the other hand, we saw from the

proof of Proposition 3.4 that all M of finite length. functor

Ext~

Exta(M,R ") = O for i ~ O and

Furthermore, by Proposition 5.2, the

,R') is a dualizing functor, and corresponds to

277

the inJective hull of k given by

I = li~ n So we see that

Ext~

Hi(R~

= O for i ~ O, and H~

=R Fx(R" ) ~ I in the derived category

Now let

M

be an A-module.

") = I.

Therefore

[I.4.3].

Then there is a natural

homomorphism FxCM )

> HomCHom(M,R" ),

FxCR" ) ).

This gives rise to a morphism of functors on D+(A),

> R Hom(R Hom(M',R'),

R= ~xCM" )

R Fx(R" ))

since we can take R" to be an injective complex, R F (R') = F (R') is also injective. =

x

in which case

Indeed, whenever

J

is

x

an inJective A-module, Fy(J) is injective A-modules).

and

Y

a closed subset of Spec A, then

(cf. [II w

for the structure of injective

We now assume (without loss of generality)

is normalized,

and so, by the Proposition,

have a morphism of

functors G:

R ~x(M')

that R ~

> Hom(R HomCM',R'),

I).

278 (Following the usual conventions, we do not write R before a functor which is already exact.) Theorem 6.2 (Local Duality).

Let

A

be a local noetherian

ring, let R" be a normalized dualizing complex, corresponding injective hull of k, and let M"

let I be the

E D~(A)

the cohomology modules of M" are of finite type).

(i.e.,

Then

8

as

defined above is an isomorphism. Corollary 6.3. of finite type. :

With

A, R', I as above,

let M be a module

Then the homomorphisms H (M)

> Horn (Ext -i(M,R'),

I)

induced by @ are isomorphisms. Proof.

Using the Lemma on Way-out FunctDrs

[I.?.l] one

sees that the Corollary is equivalent to the theorem. the Corollary,

To prove

note that @i is an isomorphism for M = k, because

H~(k) = O for i ~ O, and H~(k) = k; note also that for all M of finite type, both sides are modules with support at x.

Thus we

are reduced to proving the following LEMMA 6.4. cohomological

Let @i: S i

> Ti

be a morphism of covariant

functors on the category of modules over a

noetherian ring A.

Assume

279

(i)

for every maximal ideal ~

A, @i(A/~) is an

isomorphism, and (ii) for every non-maximal prime ideal ~ c A , S i(A/~) and Ti(A/~) have support < Supp A/2. Then @i(M) is an isomorphism for every A-module M of finite type. Proof.

By assumption, ei(A/~)" --" is an isomorphism for every

maximal ideal 144 .

If

M

is any module of finite type, then

M

admits a finite filtration each of whose quotients is of the form

A/~

case

M = A/y

with ~

prime.

with T

we may assume that

Thus by the

not maximal.

5-1emma we reduce to the By noetherian induction,

@i(M') is an isomorphism for every M' of

finite type with support < Supp (A/~). Now for each f 6 M = A/~,

f ~ O, we consider the exact

sequence O

>M

f > M

and apply the functors

0

>K.

> M/fM

> O ,

S i, T i to it, splitting it as follows:

------> si (M)

f ~ Si(M)

.> , . .

I

0

. > Lf ~

Ti

) ----~T i

)

~>

9

280

and 9 .. - - >

...

si-l(M)

> Ti-I(M)

> si-l(M/~)

> Kf

> o

> Ti-l(M/~)

> T.f

> 0 .

Now by our induction hypothesis, surjective.

Since

each element of Thus

Si(M) =

of M.

so ~ is

Supp Si(M) < Supp M by our assumption (ii),

Si(M)" "

U Kf

Similarly

5 is an isomorphism,

is annihilated by some non-zero f 6 M.

as

f

ranges over the non-zero elements

Ti(M) = U Lf.

Therefore 8 is surjective.

This is true for each i, so also ~ is surjective. implies ~ is injective,

so ~ is an isomorphism.

But that Then, as above, q.e.d.

8 is also an isomorphism 9 Corollary 6.5.

For

M

of finite type, the modules

Hix(M)

are of co-finite type (see Proposition 5.3), and we have a functorial isomorphism 9

^

ExtI(M,R ")

> Hom (H

(M)~ I)

A

of

A-modules.

Proof and

use

the

Apply the functor Hom ( , I ) fact

that

for

any A-module

of

to the isomorphism @ i finite

type

A

functor applied twice gives the completion M.

[LC p.

M,

this

61].

281

Remark.

If

A

is a regular

local ring,

then

A

itself

is a dualizing complex for A (see Example 2.2), and we recover the old local duality theorem

[LC 6.3].

more g e n e r a l l y

local rings

for G o r e n s t e i n

This is also true (Theorem 9.1 below).

282

w

Application Let

X

to Dualizing Complexes.

be a locally noetherian prescheme,

be a dualizing complex on X.

and let

R"

Then for each point x 6 X, R"x is

a dualizing complex for the local ring

~x (Corollary 2.3).

Thus by Proposition 3.~ we can find an integer d(x) such that I EXt~x(k(x)'

Proposition is an immediate d(y)

Proof.

7. i.

and let

Z j: Z

for

k(x) for

i ~= d(x) i

d(x)

With the hypotheses

specialization, = d(x)

.

above, if x

t> y

then

+ i.

Since the question is local around y, ve may

assume that X = Spec Let

RX) =

O

~y.

be the reduced induced subscheme > X

be the inclusion.

on Z by Proposition 2.~, and for any

Then

structure on Ix], j~R"

is dualizing

~Z-mOdule F we have

EXt~z(F , j~R') = EXt~x(J.F , R')

by duality for j [III

6.7].

calculated on Z with

j~R', and so we reduce to the case X = Z

Thus

d(x), d(y)

a~e the same if

integral local of dimension l, with generic point x and closed point y.

283

By translating,

we may assume that

R"

is normalized,

i.e., d(y) : O.

To calculate d(x) we consider

EX p] for each p.

Then b y virtue of the previous proposition and

corollary, Z" is a finite filtration of X, and each x E zP-z p+I is maximal. Proposition 7. 3 . let

R"

Let

X

be a locally noetherian prescheme,

be a dualizing complex on X, and let

associated filtration.

Then

Z"

be the

R" is Gcrenstein with respect to

z" (cf. [ ~ w

Moreover,

JCx) of k(x) over

~x occurs exactly once as a direct summand of

the (injective)

complex

for each

ECR').

x 6 X, the inJective hull

285 Proof.

Using condition (ii) of [IV 3.1] and [IV.I.F], it

is sufficient to show that

o .~(a') =

for i 4 d(x)

J(x) for i -- d(x).

But this follows from Proposition 6.1 above.

286

w

Pointwise dualizing complexes And f~. In this section we give a generalization of the notion of

dualizing complex, which may exist on a locally noetherian prescheme of infinite Krull dimension. to show that

f~

We apply this notion

of a dualizing complex is dualizing, where

f is a smooth morphism.

Throughout this section, X will denote

a locally noetherian prescheme. Definition.

A complex R" E D:(X) has pointwise finite

in~ective dimension (pfid) if for every x q X, R" 6 D+(Spec ~x ) --

'

X

C

has finite injective dimension. Proposition 8.1. inJective dimension.

Let

R" 6 D~(X)

have pointwise finite

Then conditions (i)-(iv) of Proposition 2.1

are equivalent. Proof. The same as loc. cit. since one can check reflexivity pointwise. Definition.

A complex

R" E D~(X) with pfid

is pointwise

dualizing if it satisfies the equivalent conditions of the proposition. Remarks.

i.

Since

lid implies pfid, we see that any

dualizing complex is pointwise dualizing. 2.

If

X = Spec A where A is a local ring, then

fid sup d(x) ,

x~

has finite injective dimension [If 7.20 (ill)c].

Theorem 8.3.

Let

f: X

>Y

be a smooth morphism of

locally noetherian preschemes, and let

R" E D~(Y)

dualizing.

is pointwise dualizing

on X.

Then

f~(R') (cf. [III w

If furthermore Y is noetherian,

R" dualizing, then

be pointwise

f of finite type, and

f~(R') is also dualizing.

289 Proof.

For the first statement,

the question is local

on X and Y, so we reduce immediately to the case Y = Spec A, with A a local ring, and X of finite type, affine over Y.

Then

n X admits a closed immersion into a suitable affine space I y , using

[III 8.~] and Proposition

"pointwise dualizing"), Case i.

so

2.4 (which is valid also for

we reduce to the case X = Any.

A is a regular local ring.

since A is local, and by uniqueness after shifting if necessary,

Then R" is dualizing,

(Theorem 3.1) we may assume,

that R" ~ A .

Then f@(a') ~ Wx/y= ~ / y ,

which is locally free of rank one, and hence pointwise

dualizing

(Example 2.2, since X is regular if Y is). Case 2. Y' = Spec A'.

A is a quotient of a regular local ring A'. Then X is obtained

by the base extension

Y

from the morphism

~.

Let > Y'

> Y', and we reduce to the previous

case by means of [III.6.4],

Proposition

2.4, the uniqueness

(Theorem 3.1) on Y. General case.

A is an arbitrary noetherian

local ring.

A

Let

A

be the completion of A, and let Y' = Spec A.

the base extension

u: Y'

on Y' by Corollary 3.5.

> Y, and then But

A

u*(R')

We make

is dualizing

is a quotient of a regular local

290

ring by Cohen's structure theorem [ 14,(31.1)], so f is dualizing on X' = X ~ x E X, the completion

Y', by Case 2.

(u*(R))

Now for any point

~x of its local ring is also the A

completion of the ring of the points of

X'

~X |

A, which is the semilocal ring

lying over x, and so applying Corollary

3.5 once in each direction, and [III 2.1] we see that f@(R') is pointwise dualizing on X. Now if Y is noetherian, f of finite type, and R" dualizing, then

Y

has finite Krull dimensiQn by Corollary 7.2, hence

has finite Krull dimension, and so

f~(R') is dualizing by

Proposition 8.2.

q.e.d.

Corollary 8.4. ~

and ~

X

Under the hypotheses of the theorem, let

be the codimension functions associated with the

pointwise dualizing complexes R" and f~(R'), respectively. Then for each

x E X,

~(x) where

= ~(y)

+ tr.d. k(x)/k(y)

y = f(x). Proof.

Chasing through the reductions above (and noting

that this formula holds trivially for a finite morphism and f~(R')), we reduce to the case

Y = Spec A,

A

a regular local

egl

ring, R 9 = A, and X = ~ yn .

Then

f~(R') = WX/y, which is

locally free of rank one, so for any x 6 X we have ~ ( x ) and for any

y E Y, ~ ( y )

= dim ~y.

[LC w

= dim ~x'

Our formula is

then the usual dimension formula for a local homomorphism of local rings (cf. [EGA OIV 17.3.3]). Remark. Using Proposition 2.4, the result of the theorem can be extended to show that for any embeddable morphism f of !

noetherian preschemes [III w to dualizing complexes.

f" takes dualizing complexes

The formula of the Corollary also

extends to this case. Proposition 8.59

Let

f: X

L> y

be an embeddable

morphism of locally noetherian preschemes, be a dualizing complex on Y, and let

D

let

R" q D+(Y)

denote the functor

!

R HO~(-,R')

or

R Ho~(-,f'R')

from the context).

(the one meant will be clear

Then there is a functorial isomorphism

$

f" (F') for all

)))

F" E D+(Y).

Proof.

Apply [III 8.8, 7)] to D(F') and R'.

292

Corollary 8.6.

Under the hypotheses of the proposition,

assume furthermore that Y

is noetherian.

f

is flat, of finite type, and that

Then there is a functorial isomorphism

!

!

f'(F*@G ~

for

F* E D+(Y) Proof.

and

"

> f~*(F~

| f*(G ~

G ~ E ~Dc(Y)fTd~

Note first that Theorem 8.3 and the Remark |

following apply to show that

f" (R')

is dualizing on

The result now follows from the Proposition, and Proposition 2.6 above.

X.

using [II 5.8]

293

89.

Gorenstein preschemes. Theorem 9.1.

Let

A

be a noetherian local ring.

Then

the following conditions are equivalent:

(i)

A

is a dualizing complex for itself.

(ii) A (iii) A

has a finite inJective resolution. is Cohen-Macaulay-i Xl,...,x n

and whenever ideal

(Xl,. -. ,Xn) (iv)

is a maximal A-sequence,

k

(v) filtration

(vi)

There is an integer

d

such that

O

for i ~ d

k

for i = d

is the residue field of

A.

is Gorenstein w i t h respect to a suitable

A

Z"

of

Spec A (cf. [IV w

There is an integer Hi(A)

~

= O

for

d

such that i + d

is inJective where

x

then the

is irreducible. )

~xti(k,A) = {

where

(i.e., A is Cohen-Macaulay,

for

is the closed point of Spec A.

i=d

294 Proofs.

(i)~-~(ii) (ii) ~

[2, Theorem ~.I].

By definition of dualizing complex. (iii) ~ ( i v )

Proved by Bass

This article, incidentally,

is a good

summary of all previous occurrences of Gorenstein rings in the literature. (iv)---~, (i)

This is Proposition 3.~ above.

(i) ~

This is Proposition 7.3 above.

(v)

(v) =--~(vi) By [IV 3.~], E(A) is an inJective resolution of A, and furthermore,

for each p E Z,

direct sum of constant (inJective) y E ZP-Z p+l.

of Spec A with Hi(~(~-(A) ) ).

EP(A) is a

sheaves on subspaces

We can calculate

H~(A)

[y]as

by what we have Just seen,

But

consists of a single inJective sheaf in some degree (say d), since

x

is a minimal point of

Spec A.

First note by [LC o, where

loc. cit. that

n = dim A, so A

n = d.

6.~,

part ~] that

(The hypothesis in

is a quotient of a regular local ring can

be circumvented by passing to the completion [LC 5.9].) Therefore

A

is Cohen-Macaulay

be a maximal A-sequence,

and let

[LC 3.10].

Let

(Xl,...,Xn)

~ = (Xl,...,Xn).

Then

295

H xis annessential ( Aextension )

of

Prop. 2], and so is irreducible,

since

A-module,

and hence irreducible.

is an irreducible

[11, proof of A/~

is a simple

We conclude that

H~(A)

injective A-module with support at

and so is an inJective hull of

x,

k [II 7.~].

Now for any A-module M,

.om (k,M)~ Bom (k, ~(M)). Furthermore,

the functor

F

x

takes inJectives

into injectives,

so we have a spectral sequence of derived functors, which degenerates

in this case to give

Ext1(k,A)

= Horn

(k, ~(A)) -- t 0 k

Definition. equivalent

i~n i = n

A local noetherian ring satisfying the

conditions of the theorem is called a ~ocal)

Gorenstein ring. Remark.

For

a

local ring, regular ---~.complete

intersection ~ G o r e n s t e i n - - - - ~ . implications

are all strict

[2].

Cohen-Macaulay,

and these

296 Definition.

A prescheme is Gorenstein if all of its local

rings are Gorenstein local rings. Corollary 9.2.

A localization of a Gorenstein local ring

is Gorenstein. Proof.

Follows from condition

(i) of the Theorem,

and

Corollary 2.3 above. Proposition 9.~. morphism, where

k

Let

f: X

> Speck

is a field.

Then

X

be an embeddable

is Gorenstein if and

|

only if on

f'(k) is isomorphic,

in

D+(X), to an invertible sheaf

X. Proof.

Indeed, b y the remark at the end of the last I

section,

f'(k) is a dualizing complex on X.

Gorenstein if and only if

~X

But

X

is

is a dualizing complex on

X.

So the result follows from the uniqueness of the dualizing complex (Theorem 3.1 above). Corollary 9.4.

Let

X

be a Gormnstein prescheme of

finite type over a field k, and let Then

X |k K

k~K

be a field extension.

is also Gorenstein. !

Proof.

The question is local on X, and

w i t h flat base extension [III 8.7, 5)].

f"

is compatible

297

Corollary 9.5.

Let

K,L

be extension fields of a field

k, and assume that one of them is finitely generated. K |

Then

L is a Gorenstein ring (i.e., every localization of it is

a local Gorenstein ring). Proof.

Similar to the above.

One can also give a direct

proof, by induction on the number of generators. generalizes

[EGA IV 6.7.1.1] which says that

This

K|

is Cohen-

Macaulay. Proposition 9.6.

Let

f: X

> Y be a flat morphism of

locally noetherian preschemes, with Y Gorenstein finiteness assumption on f). only if all the fibres Proof. following:

Xy,

Then

X

(but no

is Gorenstein if and

for y E Y, are Gorenstein preschemes.

The question is local, so we reduce to the let

A ---->B

be a local homomorphism of local

rings, with A Gorenstein) Gorenstein if and only if

and B flat over A. B/~B

Then

B

is

is Gorenstein.

To prove this, we use condition

(iv) of Theorem 9.1.

By

flatness, B)

for a suitable d.

=

I

0

for

i ~ d

B/~AB

for

i = d

Thus the spectral sequence of

Ext's for the

298 change of ring from B to B/4~AB

ExtiB(kB, B)

degenerates, and we have

i-d %B(kB, =~ ExtB/

for each i, whence the result. Exercise 9.7.

Let

f: X ---->Y be a flat morphism of

finite type of locally noetherian preschemes. f: (~y)

(which is defined locally on X) is

Show that

isomorphic

to an invertible sheaf if and only if all the fibres f, for

y E Y, are Gorenstein preschemes.

in D+(X) X

Y

of

Also show that

|

f" (~y)

has a unique non-zero cohomology sheaf, which is flat

over Y, if and only if all the fibres of preschemes.

f

are Cohen-Macaulay

Such morphisms are called Gorenstein morphisms

(resp. Cohen-Macaulay morphisms). Compare [EGA IV 6.3 and 6.7] for statements analogous to Corollary 9.5 and PropositiDn 9.6 for Cohen-Macaulay.

299

w

Existence

of D u a l i z i n g

W e now draw together chapter

to make

dualizing

about

gained

discussion

in this

the e x i s t e n c e

complex on a locally n o e t h e r i a n

complexes

of a

p r e s c h e m e X.

for p o i n t w i s e

dualizing

to the reader.)

Sufficient

conditions.

finite Krull dimension, In particular,

If

If

X

is G o r e n s t e i n

~X is a d u a l i z i n g

any regular

f: X

noetherian,

1.

then

d i m e n s i o n has a d u a l i z i n g 2.

the e x p e r i e n c e

some remarks

(We leave the a n a l o g o u s

X.

Complexes.

prescheme

and of

complex

for

of finite Krull

complex.

> Y is a m o r p h i s m of finite type, w i t h Y

and if

Y

admits a d u a l i z i n g

see in the next chapter

that

X

admits

c o m p l e x R', we w i l l

a dualizing

complex

|

f'(R').

(If

above.)

Thus any n o e t h e r i a n

Gorenstein dualizing 3.

f

is e m b e d d a b l e

prescheme

then

X

prescheme

follows

from ~8

of finite type over a

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

admits

a

complex. In particular,

any p r e s c h e m e

field k admits a d u a l i z i n g

4.

this result

If

X

(31.1)].

complex.

is the spectrum of a complete local ring A,

admits a dualizing

of a regular

of finite type over a

complex.

Indeed, A is a q u o t i e n t

local ring by the C o h e n structure theorem

300

Necessary conditions. complex,

X

then

9.

If

catenary

I.

If

X

admits a dualizing

has finite Krull dimension admits a dualizing complex,

X

(in fact universally

(Corollary Z.2). then

X

is

catenary since any prescheme

of finite type over X admits a dualizing complex,

at least

locally). 3.

More precisely,

4.

If

complex,

A

~ ~ A

Proposition

|

If

IO.I.

A

~ |

k(~) is a

from the following is a local noetherian

domain,

field K, which admits a dualizing complex,

then

K is a Gorenstein ring. Proof.

R" ~A A

Let

R'|

R"

be a dualizing

complex on A.

Then

is dualizing on A by Corollary 3.5, and so by

localization,

R'|

is a prime ideal, then

This follows

ring.

with quotient

function d.

is a local noetherian ring admitting a dualizing

and if

Gorenstein

X admits a codimension

(R" |

K is dualizing on

~ |

K.

But

is dualizing on K, so by Theorem 3.1 we may assume ~ K after translating

"|174

=

"| K)|174 A ~

if necessary. ) so A |

But

K is a dualizing complex

for itself, and we are done (Theorem 9.1 (i)).

3Ol

Example. which

There are local n o e t h e r i a n

are not catenary,

2 which

and local

are not u n i v e r s a l l y

no dualizing

complexes

domains

noetherian

catenary,

domains

and which

[i~, A p p e n d i x A1,

of dimension

3

of dimension

therefore

have

Ex. 2].

A

Problems. ring,

even if

i.

W e do now k n o w if

A |

K

is a G o r e n s t e i n

A

is a local domain

of dimension

i, hence we do

not k n o w w h e t h e r

every local domain

of dimension

1 admits

dualizing 2.

complex. If

if it admits codimension complex

a

X

is n o e t h e r i a n

a dualizing function

globally.

and of finite Krull

complex

locally,

dimension,

and if it admits

d, we do not k n o w w h e t h e r

X admits

and

a

a dualizing

302

C H A P T E R VI.

w

RESIDUAL

COMPLEXES

Introduction. In this chapter we return to the problem of constructing |

a functor

f"

for a morphism of finite type, which should

reduce to

f~

for a finite morphism,

morphism.

We ran into difficulty earlier

and

f~

for a smooth

[III w

because the

derived category is not a local object -- one cannot glue together elements of the derived category given locally.

Now

we overcome that difficulty in a special case by using residual complexes.

The residual complex is a very special

complex of quasi-coherent

injective

below) which is almost unique

sheaves

(it was called

(see definition "residue complex"

in the Edinburgh Congress talk [ 9 ]). Modulo some technicalities Krull dimension, complex

arising from possibly infinite

this is how they work:

R" E D+(X)

is associated

To each dualizing

functorially a residual

C

complex

K" = E(R'), and

complex

K

in

Q(K')

(which is the image of the

is isomorphic to R" in D

(X>

Thus we

!

will define

f"

locally for a residual complex

K"

by

303 |

|

f'(K')

=

!

where

f"

is the one we know for embeddable morphisms.

Then

|

since

f'(K') is an actual complex defined locally, we can glue |

to get a global

f ' ( K " 1.

We w i l l

give

the statement

and proof

|

of the existence of important

f"

in some detail,

since this is an

step towards the general duality theorem.

Later,

after proving the duality theorem, we will pull ourselves up !

by our bootstraps

to obtain a definition of

in the derived category

f"

for objects

D~(X)~

!

Once having

f',

we will define a trace map for residual

complexes, which will be a map of graded sheaves Trf:

f.f'K ~

> K'.

We will prove in the next chapter that if

f

morphism,

(Residue Theorem).

then

Trf

is a map of complexes

is a proper

304

w

Residual Complexes. Throughout this section, X will denote a locally noetherian

prescheme. hull of

If

k(x)

x

is a point of

over the local ring

the ~uasi-coherent sheaf

I

X, and if

on

inJective

Ix]-, and

Definition.

0

is an inJective

~x" we will denote by J(x)

~X-mOdule, which is the constant elsewhere

(notation of [II w

A residual complex on

of quasi-coherent

I

inJective

X

is a complex

K"

~X-mOdules, bounded below, with

coherent cohomology sheaves, and such that there is an isomorphism

pEW. Example.

If

xEX X

is a regular prescheme,

then the Cousin

~X is a residual complex for

complex of the structure sheaf X (see example at end of [IV w Proposition then its image b) then

If

a)

i.I. Q(K')

If

6 D~(X)

R" E D~(X)

K"

is a residual complex on X,

is a pointwise dualizing complex.

is a pointwise dualizing complex on X,

E(R') is a residual complex on X.

(Here E is the notation

of [IV 93] , with respect to the filtration Z" associated with R ~

305

(cf. [V w c)

and [V w If

X

Remark ~]).)

admits a residual complex (or pointwise

dualizing complex) with bounded cohomology, then there is a functorial isomorphism ~:

R"

N >

QE(R')

for pointwise dualizing complexes. E:

Ptwdual (X)

Hence the functor > Res (X)

is an equivalence of the category of pointwise dualizing complexes of

D~(X)

and the category of residual complexes

(and morphisms of complexes). Proof.

a)

Its inverse is

Q.

The question is local, so we may assume

X = Spec A, where

A

is a noetherian local ring.

we have only to check that there is an integer

Exti(k'K') = { kO for

where

k

where

x

Since

k ~ k(Xo) ,

is the closed point of X0 we have O

(k,

= k

Hom (k, J(x)) = O

such that

ii =~ dd

is the residue field of A.

Hom

d

By [V.3.~]

for

X

~: X 0 .

3o6 The result now follows from the definition of a residual complex. b)

By the pointwise convergence

[IV.I.G] we see that E(R ~

of the spectral sequence

Hi(R ") = Hi(E(R'))

for all i, and hence

is a complex which is bounded below and has coherent

cohomology.

By [IV.I.F]

the question of whether it is a

residual complmx is local, so we may assume

X

is the

spectrum of a local ring, in which case our result is [V~ c)

This follows from [V, w

Remarks. complex

I.

Remark 5] and [IV.3.~].

In particular,

Dual (X)

admits a dualizing the functor

> aes (X)

is an equivalence of categories,

with inverse Q.

I expect that the statement c) is false without

boundedness

assumption.

E(R') = E(R'').

the

That is, there may be two non-isomorphic

pointwise dualizing complexes

R"

In particular,

theorem similar to [V.3.1] 3.

X

(and hence has finite Krull dimension), E:

2.

if

and

R'"

such that

there may not be a uniqueness

for pointwise dualizing complexes.

It follows from the proposition

that

X

admits a

residual complex if and only if it admits a pointwise dualizing complex,

so the remarks of [V w

apply.

307

Exercise.

Show that there is a uniqueness theorem

analogous to [V.3.1] for residual complexes, i.e., two residual complexes can differ only by shifting degrees and tensoring with an invertible sheaf. to show that if Hom'(K',K'')

K',K''

The touchy point is

are residual complexes, then

is a complex with coherent cohomology:

We now give two technical results which will be used in the following sections. Lemma 1.2. on X.

Let

K"

and

K''

Then to give an isomorphism

equivalent to giving, for each

be residual complexes #:

K"

> K''

is

x E X, an isomorphism

> Q(KI') in then

D+(Spec c

~x

is obtained from

Proof. described.

~x), such that whenever

Clearly

$

#y

x

> y

by localization.

gives rise to a system ($x)x6X

Conversely, given the isomorphisms

note that by c) of the Proposition above, unique isomorphism

is a specialization,

as

#x' we first

Sx comes from a

308 ~x:

K"x

> K'' x

of the actual residual complexes,

~x

and these

are

compatible with localization.

Already we deduce that the

codimension

d'

functions

are the same.

and

And since for

no non-zero maps of ~x

d

J(x)

associated

d(x) = d(y) and

into

to

and

K"

K''

there are

x ~ y

J(y), our system of isomorphisms

gives rise to, and is determined by, a collection of

isomorphisms : x

I(x)

> I'

for each x E X, where subsheaf of

I(x)

K" (resp. K'')

for an immediate

(X) (resp.

I'(x)) is the (unique)

isomorphic

specialization

x

to

> Y' ~x

J(x),

such that

and

~y are

compatible with the boundary maps of the complexes

K"

K''.

(~x)

But to give such a collection of isomorphisms

to give an isomorphism

~:

X, and let

Suppose

K" v

furthermore

given an isomorphism

is

K" ---->K'', so we are done.

Lemma 1.3 (Glueing Lemma). of

and

Let

(Uv)

be a residual complex on

be an open cover U , for each v.

that for each pair of indices

~,v

we are

309

~v of the restrictions

:

K" v

>K"

of these complexes

U

to

= U AU

such

that for each triple ~,v,A, m

on

U

K"

can be chosen uniformly

v

Suppose

~vA"

unique residual

8 V: for each

finally that the lower bound of the complexes

complex

for all v.

K"

K'[U

on

>

X, together w i t h isomorphisms

K'v

v, w h i c h are compatible with

Proof.

K" on X, which is bounded below. a complex of quasi-coherent and has coherent

cohomology.

together

local isomorphisms

pE~

so it is

into a global complex

~X-mOdules

K"

[II.7.16],

J(x)

are

subsets of X, w e can glue

to obtain a global isomorphism J X() .

x6X

is a residual

UUv.

for glueing,

Finally since the

sheaves on irreducible

I Y

is a finite morphism of locally noetherian

preschemes, we define a functor fY:

aes(Y)

> aes(X)

on the category of residual complexes by fY(K') = Ef ~ Q(K'). Note by Proposition complex; by [V.2.4] by Proposition

l.lb,

l.la

that

Q(K') is a pointwise dualizing

f~Q(K') is pointwise dualizing on X, and Ef~Q(K" ) is a residual complex on X.

312

If

f: X

> Y

and

g: Y

. > Z

are two finite morphisms

of locally noetherian preschemes, we define an isomorphism (gf)Y of functors from

"~ >

fYgY

Res(Z)

to

(I) Res(X)

By Lemma 1.2

as follows.

it will be enough to define this isomorphism in the scheme Spec(~x)

for each

localization.

x E X, in a manner compatible with

Afortiori,

it is enough to define the

isomorphism after making a base extension each point

z E Z, and thus we reduce to the case where

(and hence also dimension.

Spec(Gz) ---->Z, for

X

and

Y)

Z

are noetherian of finite Krull

In that case pointwise dualizing complexes are

dualizing, and we have a functorial isomorphism

~: 1

N > QE

on the category of dualizing complexes on Y (Proposition l.lc). N o w for a residual complex

K"

on

Z

we define our isomorphism

as follows:

(gf)YK'def.

,(gf) QK" = Zf g Q " [I~ 6.2]

-f Q g QK ~

= fYgY( '). def.

We w i l l use this same technique of reduction to the case of finite Krull dimension without explicit mention below.

It

enables us to carry over isomorphisms defined for dualizing complexes to residual complexes.

313

For a composition of three finite morphisms, the usual commutative isomorphisms

diagram

there is

(referred to as (i)) of the

(I).

For a smooth morphism

f: X

> Y of locally noetherian

preschemes we define a functor fz :

aes(Y)

> Res(X)

by f'(K')

= Ef~Q(K 9 ) .

This takes residual complexes into residual complexes by virtue of Proposition

1.1a,b, and [V.8.3].

For a composition of two smooth morphisms

f,g, we define

an isomorphism

(gf)Z

~

>

fZgZ

(II)

using the above reduction to the case of finite Krull dimension, and carrying over the isomorphism

[III 2.2].

For a composition of three smooth morphisms,

there is a

compatibility

(referred to as (ii)) of the isomorphisms

which follows

from the compatibility

of [III 2.2].

There are two other isomorphisms, between the functors

fY

and

z

g :

(II),

expressing compatibilities

the Cartesian

square, and

the residue isomorphism.

For the first, we suppose given a

Cartesian diagram as shown (i.e. ;

k

W W = X ~

Y) with

f

>Y

(and hence k)

a finite morphism, and g (and hence h) a smooth morphism.

In that case there

x

~,,

>z

is an isomorphism hzfY

--

> kYg z

(III)

obtained as above from the isomorphism of [III 6.~]. If we have another

Cartesian

diagram as shown, with finite morphism,

r

also a

p

k

>w

ql

Ih

! II

>

P

>X

f :

~ j""

Z

(kp)Yg z

lI qZryfy

>Y

then there is a

commutative diagram of isomorphisms qZ( fr)Y

Q

(iii) .III

_ > pYhzfy

III

> pYkYg z

315

Also

if we have another

diagram

doubling

square,

i.e., w i t h r smooth,

is a similar

the smooth

commutative

W

then there

diagram

These

the isomorphisms

two c o m p a t i b i l i t i e s

P

side of the 9

>Y

(iv) X

involving

P ~ ~

Q

Cartesian

(II) and

>Z

(III).

follow from

[III 6.~]. For the residue

isomorphism,

we P

suppose which closed

that we have a finite m o r p h i s m

is factored

into pi, w i t h

i

immersion,

and p smooth.

Then

f

a

X

Y there is an i s o m o r p h i s m

fy

obtained

N

as above

./

iYpZ

from that of [III 8.2].

If we have a diagram shown, w i t h immersions,

f,g and

finite, q,p

there is a c o m m u t a t i v e

such as the one

A

i,k closed

smooth,

then

diagram

\I p Z

316

(gf)y

IV

> iY(pq)Z

II

~I

> iYqZpZ

(v)

IIV

fygy

~V

> fykyp z

~

~kf)yp z

This follows from [III 8.6b] applied to the triples (i,q,p) and (f,k,p). There are also two commutative diagrams expressing compatibility between the square isomorphism (III) and the residue isomorphism (IV). For the first, suppose one has

i finite,

J,k,~

closed immersions,

Q

~

f,g

smooth.

>P

j y

and

-

, >Z

Then there is a

commutative diagram (ji) y --

1

iYjY _

IV

:

~ (~k)Yf z (vi)

IV

> kYgZ jy

which follows from [III 8 ~6c ] ~

III

> ky ~y fZ

317

Q

YI"

For the second, suppose there is a Cartesian diagram as shown (W = X ~ f,g

Z; Q = P ~

finite,

Z)

w

with

~

iv

~z

i,J closed

immersions, and

p,q,u,v,w smooth. X

Then there is a commutative

f

> Y

diagram uZfy

I

IV

>

uZiYpZ

III

jYvZp z

(vii)

I~l

gYw z

..IV

> jYqZwZ

This follows from [III 8~

II

> JY(wq) z = JY(pv) z .

modified as in [III 6o~]~

318 !

w

f"

for residual complexes. |

In this section we construct the functor residual complexes, which we call f~

f"

for

to avoid confusion.

We refer back to the previous section for the notations fy fz

the isomorphisms

(I)-(IV), and the compatibilities

We will work in the category of locally noetherian preschemes, and w e will consider only morphisms which are of finite type, and such that the dimensions of the fibres are bounded.

It will be understood in the following that

these conditions hold for all schemes and morphisms considered. W e are now in a position to state our theorem. Theorem 3.1.

There exists a theory of variance consisting

of the data a)-d) below subject to the conditions VAR 1 - VAR 5. Furthermore,

this theory is unique in the sense that given a

second collection of such data a')-d') there is a unique isomorphism of the functors a) and a' ) isomorphisms

a)

compatible with the

b)-d) and b' )-d' ).

For every morphism

: Res(Y)

f: X

~ > Y, a functor

> Res(X)

on the category of residual complexes.

319

b) and

For every pair of consecutive morphisms

g: Y

(gf)A

LN > fA g A

For every finite morphism fA

~f: d)

> Y

> Z, an isomorphism Cf,g:

c)

f: X

N

fy

>

"

f, an isomorphism

9

For every smooth morphism g, an isomorphism

VAR i).

,~ > g Z

ga

9 g:

For any f,

Cid, f = cf,id = id, and if

are three consecutive morphisms,

f,g,h

then there is a commutative

diagram (hgf) ~

~

Cf,hg

> f~(hg) 6

Cg,h

Cgf,h

(gf)Ah A

Cfpg

(In other words, the cliv~e normalis6e"

fA

and

_ > fag Ah~

Cf,g

define a "cat~gorie

in the terminology of [SGA 1960-61, VI] ).

320

VAR 2). then

Cf,g

If

f,g

are consecutive finite morphisms,

is compatible, via

#f

and

#g, with the usual

isomorphism (I) above. VAR 3). Cf,g

If

f,g

are consecutive smooth morphisms, then

is compatible, via

~f

and

~g, with the usual

isomorphism (II) above. VAR 4).

Given a Cartesian

k

W

~Y

diagram as shown, with f,k finite and

g,h

smooth, the isomorphisms

b) are compatible via

c), d) with

f

X-

the isomorphism (III) of a square above, i.e., there is a commutative diagram

(fh) A

h~f~

=

(gk) a

~h'~f

Ck,g

> hzfy

> kAg A

III

> kYgZ

.

>Z

321

VAR 5). shown, with f finite, and

P

Given a diagram as i

a closed immersion, p

X

P

smooth, we have

Y a commutative diagram C,

fA = (pi)A

Z.p ~ iAp~

,[%f fy

i.e.,

1 $i'~p IV

> iypZ

c. is compatible with the residue isomorphism. 1,p The proof of this theorem requires drawing a great many

diagrams and checking their commutativity.

We will therefore

carry out in detail only a few of these verifications, by way of example, and will leave the others to the reader, marking them with the symbol (:) which indicates that he has some work to do at that point.

The proof per se will follow after some

definitions and lemmas. Definition. U c X

Let

be an open set.

f: X

>

Y be a fixed morphism, and let

We define a chart on

following collection of data:

U

to be the

3a2

Res(Y)

I)

A functor

fa

2)

A factorization

> Res(U). i

U

flu = pi w h e r e

>P

P

i is a closed immersion into a scheme

Y smooth over Y. 3)

An isomorphism Ti,p

Definition.

:

~ > iYp z .

fa

If

a second chart on the

~,

fl U = Jq,

same

open set

q: Q

' is 7j,q

> Y, and

U, a permissible

i s o m o r p h i s m b e t w e e n the two is an isomorphism

fa of functors,

R

such that for every

c o m m u t a t i v e diagram such as the one shown, w i t h and

k a closed immersion

r,s smooth,

commutative

fa

there is a Y

diagram of isomorphisms

Ti,p > iYpZ

IV

>

IV

> kYsZq z

kYrZp z

!

t_.b

Tj,,q > jyqZ

=

~: >k

(qs)

= kY

323

Lemma 3.2.

Given two charts on an open set U, there

exists a unique permissible Proof. R = P ~

Q

isomorphism between them.

The uniqueness and

k

is clear, because one can take

the diagonal map

ixj.

Then the isomorphism

is determined by the condition of the definition. For the existence, above, and let above.

let

(R,k,r,s)

fa,fb and

(S,~,...)

be two diagrams as

We must show that the isomorphisms

by the condition above are the same. diagram

etc., be two charts as

(R•

k•163

morphisms.

uS

defined

the third

) and comparing each to this one, we

diagram,

the one shown, with immersions and

and

By considering

reduce to the case where S dominates R. have a commutative

uR

r, s, t

k,L

In other words, we

such as

closed

smooth

We must show that

the following diagram is commutative: fa

7 i , p > iYpZ IV > kYrZpZ

~

kY(pr)Z = kY(qs)Z< I I ~ s Z q Z < _ ~ _ ~ q Z ~ f b

fa ~iypz ~>#(~gpz =>#(~)z - #(qst)Z kYrZpZ

~

IV

id

=a

~YtZr z pz

id

~o

1,p > iYpZ

IV

>

Y(rt)ZpZ

The left-hand square is identically commutative.

The right-hand

one is commutative by virtue of a special case of compatibility (v) above, where

X = Y.

On the right-hand side of our long diagram we fill in two analogous commutative squares.

This leaves in the middle the

following diagram kYrZpZ !IV ~y ZrZpZ ~II ~Y(rt)ZpZ

kY(qs)Z


kY(pr)Z

=

II

t~IV > ~y (pr)Z

IIII = ~ytZ(~s)Z
~Y(prt)Z

=

q!t I Ly ( )z
f~

is independent of the restrictions of Furthermore it follows from the lemmas

that on a triple intersection

U vA, these isomorphisms are

.527

compatible:

Thus,

since we are dealing with functors on residual complexes,

we can use Lemma 1.3, and glue the functors isomorphisms

u

fv

via the

to obtain a functor Uv

f4 : ResCY)

> Res(X)

(together with isomorphisms compatible with the u

b)

~v

Construction

f: X ---->Y

and

to construct

> Z

that

X

ooze. i: X

>

~

and

f,g

.

Let

It will be sufficient

isomorphisms

of change of chart.

Thus we may assume that

f

and

g

and we may even assume

in an affine space over Y, since that

C oosoem e On s k: Y

over Z, and note that ~ y = that we have a commutative shown, with

for each v

chart by chart, and show that it is

in smooth morphisms,

is embeddable

c

be morphisms.

compatible with the permissible

are embeddable

f~

>

of the isomorphisms

locally,

For then we can glue.

fA[U

) which is the one we want.

g: Y

Cf,g

By:

>P,

smooth

Xp Y,

~

so

diagram as

i,J,k closed immersions,

/k--

z p,q,r

smooth, and the

328 upper right square Cartesian.

Define

the charts chosen, and taking

Ji: X

gf)

Cf,g (depending on n

> ~p

as a chart for

as follows:

Cf,g: (gf)a

>(Ji~(rq~

I,~

Now we must show that permissible

isomorphisms

is sufficient

iYjyqZrZ
P •

,.

Furthermore,

P.

the third chart

to the case where Q' we reduce re

Q

So we have a

diagram such as the one shown, and we must show that the two Cf,g are compatible,

i.~

the following diagram is commutative:

that

\'/'. Z

It

if k': Y----> Q

n

dominates

~gC.

Cf,g is compatible with the

is another chart for g, by considering k" = k•

7rr iYpZkYrZ
(ji)Y(rq)z

~I > iYjYqZrZ ~,

and as before we may assume that

This involves checking another analogous

diagram of isomorphisms,

which we leave to the

(!).

c) morphism

Construction f.

Let

of the isomorphism

f: X

> Y

#f

for a finite

be a finite morphism.

As before,

it will be sufficient to construct the isomorphism on charts,

provided our definition

isomorphisms i: X----> P

of charts.

~f

is compatible with permissible X

So let

i

be a chart for f, and Y

de fine fa f:

7i

z ~P > iYp

IV
P

330

If that

J: X J

> Q

dominates i, and one checks (~) using compatibility

(V) above that d)

is another chart, one may assume as usual

%f

For a smooth morphism

the isomorphism ~g

~.

is compatible.

~g.

Take

g

g: X

> Y we must construct

as its own chart, and take

There is no choice involved, hence nothing to check.

Having constructed the data a) - d), we must verify the conditions

VAR 1 - VAR 5.

VAR i). morphisms

Given three

f,g,h,

\/

we use

Y

~n

~ ~

P

a diagram such as the one shown to calculate

Cf,g,

etc., on the charts.

Z

The

~

\/ -

f

P

W

diagram of the condition becomes a large diagram whose commutative one checks (:) using (iii) and (iv) above. VAR 2).

If

f

one checks (~) that isomorphisms

%f

and

and Cf,g

g

are consecutive finite morphisms, is compatible with (I) via the

%g, by using (v) and (vi) above.

331

VAR 3). isomorphism

Trivial.

One has only to observe that the

(III) is the identity if one side of the square

is the identity. VAR 4) and VAR 5) follow (:) from the definitions

and

a few more commutative diagrams. This comp%ees the proof of existence of the theory of variance,

and we now show its uniqueness.

If', Cf,g, 6, $~, ~]

suppose that

are two sets of data

a)-d),

and

For that purpose,

Cfx, C~,g, $~, ~c~]

each satisfying the conditions

VAR 1 - VAR 5 (which we will call VAR 18 , VAR 1 • precise)

We will construct an isomorphism

compatible with the data observe that Let

8

b)-d)

and

etc., to be

8: fA

b')-d'),

> fx

and we will

is unique.

f: X ----->Y be a morphism of

i X -----------> P

finite type, and choose locally an embedding

i: X

smooth over Y. 8:

fa

> P Define

c~ i~p> ~pA

Y

into a scheme 8

by

~ A x x x ~i'~P > iYpZ< $i'~p i XpX < c.x,p

(Note that in order to be compatible with the

Ci,p,

fX . #i' ~p'

we must choose 8 this way, which proves the uniqueness of 8.)

332

To see that

5 is independent

of the 0

embedding another

chosen,

one.

let J: X

Replacing

Q

> Q by

be PxQ

as >P

X usual we reduce P, and h a ~ a

to the case Q dominates

diagram

such as the one Y

shown.

We must check that the perimeter

of the following

diagram

c ~.

a

> i~ pa

1~

id

>

..~ ~ .a

"q j~q~p~

I

Ca

That

VAR 3 ~

Thus U

v

X

id

ditto x

IT T

P,q -

w h i c h are c o m m u t a t i v e

5

f

'd"

X

x

X

in fact it is, is shown by chopping

right we use

x c.l,p

.X X p
] q Pzz

~

f~

x x ~i,~ p

A

~i'~p

l,p f8

VAR

is commutative:

VAR

i x, VAR 3 x

is w e l l - d e f i n e d

for w h i c h

local i s o m o r p h i s m s

jX(pq)X< cj, fx

it into little

for the reasons

On the

shown.

and VAR 5 x similarly locally.

this is possible,

N o w cover

X

by open

and glue together

the

333

(fj~)A where

J:

U

isomorphic

>X

via

can glue the

8

~A Jv

~

> (fj~)x

is the open immersion. with

jz v

Since

J~A

is

which is the restriction, we

once we have checked (:) that the isomorphisms

v

are compatible with restriction. 8:

fA

_

Thus

> fX

is defined. Now we must check that isomorphisms the conditions

Cf,g, #f, ~g. VAR

5

is compatible with the

This can be done locally using

1 - VAR 5, and we leave the details to the

reader (~).

q.e.d.

Proposition 3.~.

Let

conventions above) and let Let

d

f: X K"

> Y

be a residual complex on Y.

denote the codimension function on X (resp. Y) associated

with the pointwise dualizing complex for each

be a morphism (with the

f~K" (resp. K').

x E X, if y = f(x), we have

d(x) = d(y) + tr.d. k(x)/k(y).

Then

334

Proof.

The question is local, and compatible with

composition of morphisms. finite (trivial)

Thus we reduce to the case

or f smooth (which is [V 8.4]).

Corollary 3.5.

Let

the conventions above:

f: X

Then if

(resp. dualizing complex)

Proof.

> Y

Y

Y

admits a residual

The existence of a residual complex on X, given If

Y

admits a dualizing

R', then the aseociated codimension

is bounded.

associated to

(with

so does X.

one on Y, follows from the theorem. complex

be a mDrphism

f is of finite type, and the dimensions

of its fibres are bounded). complex

f

By the proposition, QfAE(R')

pointwise dualizing,

the codimension

is also bounded.

hence dualizing

function

But

d

on

function

QfAE(R')

(cf. proof of [V 8~

is

335

w

Trace

for R e s i d u a l

Complexes.

In this section we define complexes.

For a m o r p h i s m

of the previous Trf:

where

K"

section)

the c l a s s i c a l differential

> Y (with the c o n v e n t i o n s

c o m p l e x on Y.

only if

f

is proper

(see the

theorem that the sum of the r e s i d u e s

of a

on a curve is zero). the trace map for a finite morphism,

by carrying over the m a p of [III.6.5] f: X

It w i l l be a

in the next chapter, w h i c h g e n e r a l i z e s

First we define

Let

sheaves

> K"

is a r e s i d u a l

Residue Theorem

f: X

for r e s i d u a l

it is a map of graded

f.fAK"

m o r p h i s m of c o m p l e x e s

the trace map

> Y be a finite morphism.

to r e s i d u a l

complexes.

W e w i l l denote by f'

the functor

f Hom~.y( f*~'X' " ) (using the n o t a t i o n of Lemma let

K"

residual

~.i.

Let

be a r e s i d u a l

[III ~6]) f: X

so that

> Y be a finite morphism,

c o m p l e x on Y.

c o m p l e x on X, and

f~ = Rf'.

Then

f'(K')

f.f'(K ~ ) is a C o u s i n

and

is a

complex

336 on Y, w i t h r e s p e c t

to the f i l t r a t i o n Z 9 a s s o c i a t e d w i t h

Q(K') (cf. [IV ~3]). Proof. injectives

It is clear

that

f'

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

on Y to q u a s i - c o h e r e n t

is also clear that

inJectives

f'(K ~ ) has c o h e r e n t

cohomology,

have only to check that it is i s o m o r p h i c sheaves of the form J(x). to show,

for each

Indeed~

y E Y,

f'(J(y))

~

A

~

have a local h o m o m o r p h i s m k

J-

HOmA(B,I )

k. 1

over

k

of > B

X

B

the

lying over

where

B

y, we

is a finite

field of A, and

I

is

A, we m u s t show that r sum Z Ii w h e r e k l , 9 9 9 ,k r i=l

fields of B, and

B.

Y, and

on

y

over

is a direct

are the r e s i d u e of

A

x

is the r e s i d u e

an i n J e c t i v e h u l l of

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

>y

be the local ring of

If

to a sum of

J(x)

s e m i l o c a l ring of the points

A-module.

so we

that

x

Letting

on X, and it

For any B - m o d u l e

is an i n J e c t i v e hull

I. 1

of finite type M, we

have i s o m o r p h i s m s

HomB(M,J) = HomB(M, HomA ( B , I ) )

~

HomA(M,I)

.

337

Now

J

is inJective,

of Spec B. copies

and has support at the closed points

Hence it is a direct

of the inJectives

I.. 1

sum of some number

of

To find out h o w many, we

have only to calculate

H~ which and

= "~

is true since I

is dualizing.

Now

f.f'(K ~

x which

Hence each

sum,

length,

occurs

Just once.

for each

y 6 Y, of

sheaf spread out on [y]-, and which

p = d(y), w h e r e

function.

with respect

I. 1

of finite

>y

in degree

codimension

is an A - m o d u l e

is a direct

is a constant

occurs

k. 1

Hence

to the f i l t r a t i o n

d

is the a s s o c i a t e d

f.f'(K') Z~

is a C o u s i n

associated

complex

w i t h Q(K" )~ q.e.d.

N o w we are in a p o s i t i o n the finite m o r p h i s m

to define

f, w h i c h we w i l l

confusion: P f:

f, fY(K" )

> K"

the trace map for

call

pf

to avoid

338

Let

K"

be a residual complex on Y.

Since it is an

inJective complex, the natural map

~f,. of'(K') of the functor

f'

> f{'O(K')

into its derived functor

isomorphism in D+(X) [I.5.1].

E~f,:

f~

is an

Hence also

> ~f~Q(K') - fY(~')

f,(K') - ~of'(I af.f~Q(K" )

Trff, is the trace of [III 6.5].

> Q(K'),

By

f.f'(K') is a Cousin complex with respect

to the filtration Z', and

K"

is an inJective Cousin

complex, so by [III 3.2], this map in the derived category is represented by a unique map of complexes

f.f'(K')

> K"

.

Composing with the inverse of the isomorphism E~f, above,

339

we obtain the desired map

f.fY(K') If

f: X

morphisms,

>Y,

>K" and

.

g: Y

>Z

are two finite

then there is a commutative diagram

(gf).(gf)Y

P~f

>

1

(viii)

Pg

Pf

g.f.fYgY

> g,gY

which follows from [III 6.6] and the functoriality of the above construction.

Remark.

We have given the above definition in some

detail to establish its relation w i t h the trace map defined for the derived category in Chapter III. course define

pf

One could of

much more quickly by the usual "evaluation

at one" map, without passing to the derived category, but w e need the functorial properties below. Theorem ~.2.

As above, we work in the category of

locally noetherian preschemes,

and morphisms of finite type

34o such that the dimension of the fibres is bounded.

There

exists a unique theory of trace, consisting of the data a) below, subject to the conditions a)

For each morphism Trf:

of functors from of

f: X

f.fA

TRA 1 and

TRA 2.

> Y, a morphism

> 1

Res(Y) to the category of graded sheaves

~y-modules (where 1 denotes the forgetful functor:

consider a residual complex

~

simply as a graded

~y-module). TRA 1).

If

f

and

g

are two consecutive morphisms

of finite type, then there is a commutative diagram Tr gf

I

Trg

Cf,g

> g.gA

g.f.fAgA

TRA 2).

If

1

f

is a finite morphism, then

is the one we already know, i.e., notation of Theorem 3.1.

Trf

Trf = pfSf, using the

341 Lemma

[EGA IV w ?] Let

4.3.

of finite type, its fibre,

let

x s X

and let

Z

the r e d u c e d i n d u c e d neighborhood

V

f: X

> Y be a m o r p h i s m

be a point w h i c h is c l o s e d in

be the c l o s u r e of

structure.

of

y = f(x)

x

in X, w i t h

T h e n there is an open Z n f-l(v)

such that

is

finite over V~ Proof. the r e d u c e d

Replacing induced

on

X

and

by

Z

strucutre,

X and Y are i n t e g r a l and f is dominant.

X

schemes, Since

with

B

quotient extension

A

> B

X

and

and

in algebra:

Let

K

be the

B | A K is a finite

T h e n there exists an element Af

B

f ~ O

is a finite A f - m o d u l e .

To prove this statement, in

is local

b e an i n c l u s i o n of i n t e g r a l domains,

field of K~

set of elements

with

point of X,

Y are affine,

statement

field of A, and a s s u m e that

B |

f(Z)-

the q u e s t i o n

a finitely generated A-algebra.

in A such that

by

x is the g e n e r i c

furthermore

Y, w e m a y a s s u m e

Y

w e r e d u c e to the case w h e r e

thus we r e d u c e to the f o l l o w i n g Let

and

let

such that

bl,-.-,b n

be a finite

3~2

l)

the

2)

the elements

b. 1

generate

as an A-algebra,

B

b.| 1

generate

B~K

and

as a K-vector

space. Then

for each

i,J,

bibj|

6 B@K,

so we can w r i t e

n

bibj|

bk | hJk

Aij k E K.

k=l Let

f

be a common

1 Y be a morphism, T h e n there

and let K"

is a unique m a p

> K~ such that

i

is an open subset of Y,

is a closed subscheme

of

finite over U, we have a

commutative

so the

is in fact a ring.

w

whenever

Aijk,

Af-module.

complex on Y.

Trf: of graded

by the

a set of g e n e r a t o r s

is a finitely g e n e r a t e d Lemma

for all the

Aij k are all in Af,

Then the

B~

Af-algebra,

denominator

diagram

U

> f-z(u)

343

#g

g.gY(K')
J(y)

Indeed, x is closed in its fibre if and only

tr.d. k(x)/k(y) = O, i.e., if and only if

Proposition 3.4 above. collection of maps

Trf

Hence ,x

Trf

as above.

certainly gives rise to a On the other hand, if

y' 6 Y is a point not equal to y, with there is no non-zero map of

d(x) = d(y), by

f.J(x) into

d(y') ~ d(x), then

J(y' ),

because

d(y') > d(y) by Proposition 3.4, and so

y' ~ [y]-.

f.J(x) has support in [y]-).

is determined by the

maps

Trf, x

Thus

Trf

(Note that

above, and these can be given arbitrarily.

We will now construct maps

Tr

x E X which is closed in its fibre.

f,x

as above for each point

345

Given an

x 6 X closed in its fibre,

choose by Lemma 4.3 an open neighborhood

i

Z

> f-l(v )

V of y = f(x) such that Z = [x]-Qf-I(v) is finite over V.

Let

I

be the ideal

V

of Z with the reduced induced structure, and let In .

Z n'

For

n

=

1,2,

''0 , be the subscheme of

X

zn'

n < n' we have a closed

immersion

inn'

: Z

n

> Z

defined by

n' , and

Z

one can verify (:) using VAR l,

z

t

~. f

\ VAR 2, and (viii) above, that the

gn

V

following diagram is commutative:

#gn'Cin ' f'~i n gn.gYn(K")

K"

> f.in iYnfA(K')

Y iY,fA(K ") n

f*in'.Jnn'*Jnn'

%gn''Cin' ,f'%in.

Y (K')

gn '.gn '

"ipjnn'

,,.>f.in

~ / ~ iy fA(K.)

f.f~(K')

Pin,

(v)

346 One can also w r i t e down another diagram with three indices n, n' and n" w h i c h shows that the sheaves the maps

PJnn'I

form a direct system as n varies,

diagram shows that the

9i n

f~f~(~)

and the

the

are all injective

pi n

f~in i~fA(K" )

and

and this

map this direct system into

pgn#gnCin,f#in maps.

map it to K'. F u r t h e r m o r e Thus we can pass to the limit

and obtain a map of a certain subsheaf of

f~f~(~)

to K~.

Looking at the effect of this construction on the component f~J(x) of f~fAK', we see that the n system, via the inclusion

th

term of the direct

pl n, is just f ~ H o m ~ v ( d,/ IX n

J(x)),

2%

and hence the limit is on Z.

f~J(x)

itself,

since J(x) has support

Hence the map we obtain is defined on all of

and we decree this to be

f~J(x),

Trf, x.

To complete the proof of the lemma, we must check various things.

First,

that

Trf, x

is well-defined,

depend on the choice of the open set V. Tr

f,x

i.e., does not

That is clear,

depends only on the stalks at x and y, and all of our

constructions

are compatible w i t h localization.

Second, we must check the property of the lemma. W and

since

U

be as in the statement.

So let

It will be sufficient

check the diagram on the component of x for each

x 6 W.

Z be the closure of x, with the reduced induced structure

to Let (which

347 w i l l b e finite over U).

Let

J

b e the i d e a l of Z in W, and

!

for e a c h by

jn.

n ~ 1,2j..., Let

Z

n

Z

n

b e as above.

i m m e r s i o n s as shown. the r e a d e r

let

b e the s u b s c h e m e of Then

there

are

W e leave to

/

(') to w r i t e down a 7,

W

defined

closed

W

> f-1(u)

/

|

h u g e c o m m u t a t i v e d i a g r a m of Z ' s n

Z n

and

Z''s n

w h i c h in the limit

g i v e s the d i a g r a m of the lemma on the c o m p o n e n t of x. T h i r d w e m u s t c h e c k the u n i q u e n e s s ,

b u t this is clear

from the c o n s t r u c t i o n . P r o o f of theorem. Trf in

For a m o r p h i s m

to b e the m a p g i v e n b y the lemma. K~

W - X,

> Y, w e d e f i n e

It is c l e a r l y f u n c t o r i a l

In c a s e f is a f i n i t e m o r p h i s m , w e c a n take U ~ Y a n d so that

T r f = pfSf, w h i c h p r o v e s c o n d i t i o n

T o p r o v e T R A i, let g: Y

f: X

f: X

> Z b e two m o r p h i s m s .

> Y and

TRA 2. i

w

>

x

It is

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

V

l

....

>Y

T R A 1 for a s i n g l e r e s i d u a l c o m p l e x K" 7.

on X a n d for a s i n g l e x 6 X w h i c h is c l o s e d in its fibre over Z. a s s u m e that W =

Ix}-

T h e q u e s t i o n is local,

is finite over Z, a n d that

so w e m a y

V = {y]-,

348

where y = f(x), is finite over Z.

Finally it is sufficient

to check the commutativity for a given element a E r(J(x)). We choose a subscheme structure enough so that

a

W

n

is in the image of

on

W

with

iYJ(x)o

n

large

Now one can

check (~) the required commutativity using the property of the lemma, and (viii), VAR I, and VAR 2 above. The uniqueness of since

TRA 1 and

Trf

is clear, as it was in the lemma~

TRA 2 imply the condition of the lemma. q.e.d.

349 ~5.

Behavior with respect to certain base changes. In this section we show that the functor

morphism

Trf

fA

and the

are compatible with certain base changes which

take residual complexes into residual complexes. Definition. preschemes

A morphism

f: X

>Y

of locally noetherian

is residually stable if a)

f

is flat

b)

f

is integral

c)

the fibres of

[EGA II f

6.1.1],

and

are Gorenstein preschemes

[v w Examples. 2.

A n open immersion is residually

A composition of residually

residually

then

i.

stable X

and

(use Y

stable.

stable morphisms

is

[V 9.6]).

.

If

are the spectra of fields

f

is residually stable if and only if

K

k ~ K, is algebraic

over k. 4.

If

f: X

> Y is residually

x E X is closed in its fibre, fibres are zero-dimensional

since

stable, then every point f

is integral.

Gorenstein preschemes.

Thus the

35o

Proposition morphism, then

and if u:

g: X' = X ~ Proof.

If

5.1.

f: X ---~Y is a residually stable is a morphism of finite type,

Y' ---->Y Y'

Clearly

is also residually

~ >Y' g

is flat and integral.

the fibres of g are Gorenstein,

let

and consider the map of fibres

v: X'y.

Gorenstein

since

f

is residually

stable. To show that

y' 6 Y, let >Xy.

stable.

y = u(y'), Now

Xy

is

The fibres of

v

are tensor products of fields, one of which is finitely generated Hence

(namely

X'y'

is Gorenstein

Lemma 5.2. I

k(y')/k(y)),

Let

be an A-module.

residue

field

k

A Then

of

A

hence Gorenstein

[V 9.5].

[V 9.6]. be a noetherian I

local ring, and let

is an inJective hull of the

if and only if

a)

I

has support at the closed point of Spec A,

b)

HOmA(k,I ) ~ k, and

C)

There is a sequence of ideals

which form a base for the length ( A / ~ n ) =

Proof.

If

are immediate,

I

~-adic

~i --m ~2 _D ...

topology,

A

such that for all n,

length ( H O m A ( A / ~ , I)).

is an injective hull of k, then

since

of

HOmA(.,I ) is a dualizing

a),b),c)

functor for

351

modules which

of finite

is primary

for

Conversely,

let

and let

J

k

> J

extends Indeed,

for s u f f i c i e n t l y integer.

I

be an A - m o d u l e

m4(xy) = O, so ~(xy) = x.~(y), N o w choose

J

given

~:

I

> J.

the n a t u r a l I claim

y 6 I, y + O~ we have

is an

Let

x 6 4~

Hence

n-1

a sequence

n

%

map is

~Mny = 0

be the least

for w h i c h

~(xy) = @(xy)

~(y) ~ O, and

such

xy ~ O.

But

But

~ O.

is injective.

of ideals

~i

-D

~2

_D

...

as

in

Then we note that ~(H~

~ Hom(A/~n,J)

and both have the same length, Therefore

they are equal.

the submodules #

a),b),c),

By b), w e can find an

is injective,

to a map

xy 6 k. so

for any ideal

satisfying

hull of k.

large n, by a).

Then there

c) holds

.

k c I, and since

injective.

c).

~

In fact,

be an inJective

injection @:

length.

is also

surJective.

is an inJective

hull of

namely the length of

But

of elements

and

J

annihilated

by

Thus k.

,

I

I

A/~

n

.

are the union of ~n'

is isomorphic

so we see that to

J, and so

352

Proposition morphism, f*(K')

5-3.

and let

K"

is a residual

Proof.

Let

Clearly

sheaves w i t h coherent

f:

X

> Y be a r e s i d u a l l y

be a r e s i d u a l complex f*(K')

complex

on

Y.

stable

Then

on X. is a complex of q u a s i - c o h e r e n t

cohomology.

We have only to c h e c k

that

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

f*(Kp)

JCx), xEX

P

Thus we reduce to the following x E X,

let

k = k(y)

be the local r i n g of of

k' = k(x)

x.

an inJective

b

c)

Ho

hull of

I |

of y. B

Let

k

of the lemma above.

over

Since

A, we have

at the closed point of Spec A

For each n,

4a

B =

x

is an injective

k, I )~= k

length Where

~ y

hull of

hull

over B.

I has support

)

For each

statement:

be an injective

A =

Then

We apply the criteria

a)

I

y = f(x), and let

over the local ring

local

(A/4~n) = length

is the m a x i m a l

(HOmA(A/~n , I)),

ideal of

A.

I

is

353

N o w since

x

is closed in its fibre,

at the closed point of Spec B.

Since

I |

B

has

support

is flat over A, w e

B

have

I@AB) ~ B/~B.

HornB(B/mB,

Therefore

HOmB(k' , I@AB ) ~ HOmB(k' , B / ~ B ) ~ k ' since tPst

B/~B for

is

an Artinian

(B/~nB)

Thus the conditions an injective

Artin

ring A.

u: Y'

> Y

k'

5.4.

over

Let

Y

T h e n one can where

By

(HOmB(B/~

note

flat over A,

Y'

be the spectrum

find a r e s i d u a l l y

is the spectrum closed r e s i d u e

~MA'

=

~4', and w i l l do.

A

I|

is

of a local stable m o r p h i s m

of a local A r t i n

ring

field.

> A'

such that

k' = the a l g e b r a i c Indeed,

since it is Artinian,

over that of Y.

Gorenstein.

and

[EGA O777 10.3.1 ] one can find a local A r t i n

Y' = Spec A'

It is integral

, I|

B.

rang A' and a local h o m o m o r p h i s m

algebraic

Finally

of the lemma are fulfilled,

with algebraically Proof.

= length

hull of

Proposition

Then

ring.

each n,

length

A'

Gorenstein

,

The only

Y'

A'

closure

is flat over

and its residue

is of k. Y.

field is

fibre is a field, w h i c h

is

354

Now we come to the behavior of residually stable base change. f: X

> Y

fA

and

u: Y'

U' X !

>X

>Y

be a residually stable morphism. Let

If '

X' = X ~

y.

Y', and let

under

Let

be a morphism of finite

type, and let

Trf

u '

be

f'

U

>Y

as shown. If

f

is a finite morphism,

f'Yu*

z

derived from [III 6.3]

there is a natural isomorphism

> u*f y

(v)

(cf. section 2 above).

On the other hand if

f

is smooth 9 there is a natural

isomorphism z

.

f' u

> u ,.fz

-

(VI)

derived from [III 2.1]. Theorem 5.5.

For every morphism f: X

type, and every residually stable morphism the notations above) there is an isomorphism d

such that

u9

:

f'Ju*

--

> u'*f A

>Y u: Y'

of finite > Y (using

355

I)

If

morphism, 2)

v: Y"

then If

d

>Y'

uv, f

f,g

= d

is another r e s i d u a l l y

u,

fd v

c

"

are two consecutive morphisms

type, then the isomorphisms and

,f'

d

stable

of finite

are compatible with

Cf,g

f',g'

3)

If

is a finite morphism,

f

compatible with 4)

If

f

V

then

via the isomorphisms

is a smooth morphism,

then

d

define

d

u,f u,f

for

f

f

following the c o n s t r u c t i o n of

theorem.

Let

~f,~f,

.

I,II,III and IV of w

u

and

and check its properties, fA f

given in ~3. be as in the previous

Then there is a commutative

diagram

>

f~f .A u ~

u

Trf, du,f

f~u .fd

is

u,f

finite or smooth using 3) and 4).

for a r b i t r a r y

Theorem 5.6.

d

.

Show first that V and VI are

compatible with the isomorphisms Define

is

u,f

Sf,#f,

compatible with VI via the isomorphisms Suggestion of Proof (:).

d

=

u*f.f ~

Then

356 Proof (~). pf

and

V

Show first that the analogous diagram of

is valid for

construction of

Trf

in

~,

f

finite.

Then follow the

w

to show that it is true in general.

357

C H A P T E R VII,

w

THE D U A L I T Y T H E O R E M

C u r v e s over an A r t i n ring. In this section w e w i l l make e x p l i c i t the r e s i d u a l

c o m p l e x on a curve over the s p e c t r u m of an A r t i n ring~ we w i l l i d e n t i f y the trace map of [VI w r e s i d u e of a differential.

w i t h the

closed residue

field 9 we w i l l prove that the sum of the r e s i d u e s i,e. 9 the trace m a p is a m o r p h i s m of complexes. case w i l l be used in the following

noetherian

theorem

preschemes~

the r e s i d u e s

A

with residue

Y

is k.

of l o c a l l y

smooth over Y9 w i t h r e l a t i v e x E X is r a t i o n a l

over

be the s p e c t r u m

field k 9 and we let

be a smooth curve over Y (i.e. 9 a c o n n e c t e d

k(x)

special

in turn implies that the sum of

this section w e w i l l let

of a local A r t i n ring

closed point

This

section to prove the

for a proper m o r p h i s m which

is zero 9

is zero on any p r o p e r curve.

Throughout

prescheme

"classical"

Then 9 in the case of the p r o j e c t i v e

line over an A r t i n ring w i t h a l g e b r a i c a l l y

general residue

and

irreducible

dimension Y

one).

if its r e s i d u e

In that case one can find a local p a r a m e t e r

w i t h the following properties:

X

A field t E ~x

358

an

(o)

O ' / t --- A .

(l)

t c %-

(2)

t is a non-zero-divisor in

(3)

~t

(~)

The total quotient ring

n

~.~

l,t

-I

Proposition 1.1.

on

a).

Y.

fz(~)

,t

Let

be an injective hull of complex

k

-2

K

of

f: X ---->Y over

A, so that

~

Let

I

is a residual

is the complex 0

w = ~/y

>

I I ixCHx1(I| %3)

xEX closed

is the sheaf of relative 1-differentials,

is the generic point of X,

H1 x

be as above.

Then

in(i | "~)

meaning the

~x is generated as

,...

-i

where

1,t,...,t n-1

is a free A-module with basis

by

9~ U x - m o d u l e

x

given

i~,ix

module spread

out

is the notation of [II ~7], as a constant

sheaf,

and

is a local cohomology group [IV ~l]. b).

fz(~)

c).

If

is an inJective resolution of

X E X

f~(~) | ~[1].

is a point rational over Y, then

359

HI(I ~A WX) ~ I ~A WX | K/~X where

K

is the total quotient ring of ~

to the stalk

~x (which is equal

at the generic point, hence independent of

x). Proof.

a). follows directly from the definition of

fz

[VI 82], using [IV 83] and [IV.I.F]. b). follows from [V 8.3],[V.7.3], and [IV.3.1].

c).

tEO'

Suppose that

x

is rational over Y, and let Then by [V.4.1] we can calculate

be a local parameter.

the local cohomology as

Hxl( I | But

t

l i ~ Ext~ ( ~ / t n, I~AWx) . n/ -x

WX) =

is a non-zero-divisor in

~x' so we can calculate the

Ext with the resolution t n

0

x

>V

>

~/t n

>O ,

x

and find that EXt~x ( ~ t

n, I|

X)

~

I |

w # t n ( I|

) 9

The map in the direct system is multiplication by t, and

360

li~

~t

n

=~'

)

=N

I |

K/~"x

,

n

so we have Hl(i|

N o w let

Z

be a closed

of X, c o n c e n t r a t e d x

o

6 X, r a t i o n a l

write ring,

subscheme X

at a closed point

over Y.

Then we can

Z = Spec B, w i t h B a local A r t i n finite over A.

inJective take

Wx | K/~x

~

calculate

hull of

k

Let over

to be a residual explicitly

[VI 82] b e t w e e n

I

be an A

as above,

complex

the residue

iYfZ(~)

and

on

Y.

and W e propose

isomorphism

gY(~).

Since

to

(IV) of Z

is an affine

scheme, we will use modules, instead of sheaves, for convenience. Using

the results F(gY(~))

above,

we have

= HomA(B,I )

F(i yfz([)) = Hom6(B, I |

Wx

| K/~ O

. . Indeed,

where X O

since

Z

is c o n c e n t r a t e d

at the

361

point

x

there are no homomorphisms

o9

of

~Z into iD(I|

i

or into

ix(HI(I|

Let

t

for

x ~ xO .

be a local parameter at

the image of Finally,

)

t

Xo "

Let

under the structural morphism

note that

WXo

is a free

so we can represent elements of

bo E ~B ~

> B.

~ - m o d u l e with basis

I |

Wx O | K / ~

be

dt,

as

Z aitidt i

(where we have identified

and for each closed point

fA(~) with

x E X, rational over Y, its

restriction F(Trf,x):

to the stalk at

x

I |

is given as follows:

local parameter, and write

U =

Z

>r

Wx @ K/~x

u E I |

let

Wx @ K/~x

t E

~x

be a

as

aitidt

iX u

of X, finite over

as above,

suppose

363

that

a. = O 1

for

Our element

u

i < -r.

is then i d e n t i f i e d

% E Hom O. -X

(B,

I |

defined b y

@(1) = u.

isomorphism

to a certain

and by d e f i n i t i o n

But by the lemma above,

u

Since

as

where

of

in

t-ldt

a_l , as required.

of taking

has a l r e a d y been

it, we can state as a the coefficient

of the I local parameter

is in strong contrast w i t h

defined

by the residue

@(i)I is the c o e f f i c i e n t

that the process

II w

the element

~(i)

=

the trace m o r p h i s m

is independent

[16, Ch.

w i l l do.

K/O"X)

and we are here callculating

corollary

r

,

.%(1) = u, so our trace is

in

with

~t

of the trace m o r p h i s m we then have

r(Trf,x)(U)

defined,

Wx |

Z = Spec

This corresponds

E HornA (B,I)

Remark.

Then

the a p p r o a c h

t

chosen.

t-ldt This

of Serre's book

the riesidue of a d i f f e r e n t i a l

a_l , and a tedious

of

proof is r e q u i r e d

is

to show that

36# it is independent To establish

of the local parameter

the r e l a t i o n b e t w e e n

[loc.

his approach

cit.

Prop.

5].

and ours, we

make the following Definition.

Let

wEM|

be a m e r o m o r p h i c Then

at

be an A-module,

x.

of

differential

w

and let

w~

for each closed point

the residue t

M

on

X

with

coefficients

x E X, rational

at x, as follows.

in

M.

over Y, we define

Choose

a local parameter

Let

w x eM~

be the image of

A ~x ~ K / ~ x

w

w~ = w x |

under the n a t u r a l m a p

-

> Wx~Kl~ X

9

Write Wx =

Z

a.ltldt

i0 xgX closed w e obtain the X

following

exact

sequence

of global

sections

on

: O

o

> r(Xo,~)

> -n

>

] I

%|

>o .

xEX o closed For each closed point

x 6 X , w e can choose a local parameter O

of the form t-Cx, w i t h

c x E A.

that

(uniquely)

w

can be w r i t t e n

W = f(t)dt

+

~

xgX O closed

Z

ix '

~[Trf

jCy)

where we take those

d

x'

>

II

J(y')

,

which are immediate specializations

of x, and those y' which are immediate specializations

of y.

Clearly it is enough to consider each

so we

y'

separately,

must show the commutativity of the diagram

x--->~' fCx' ):y'

Ir

Trf

J y)

d

Given a particular element subscheme structure on where

i: Z

> X

Z

=

Ix]-

>

J(y' )

a q F(J(x)), we can put a so that

is the inclusion.

Let

a E F(i~iAfAK ") g: Z

>Y

be

371

the c o m p o s i t i o n Tri = Pi using

TRA

and

of complexes.

the generic

f

that

f

f-l(u)

i.e.,

of

Trf x

and so

Tr

is i r r e d u c i b l e

and

Suppose

maps

is closed

[VI 3.4]

the fibre of

of

so by y'

y'

each

x'

X

by

x

is

[EGA III 4.4.11]

there

such that the r e s t r i c t i o n

in w h i c h case

Thus we m a y assume Trf = pf

which

is

and we are done.

that

f,(J(x))

x

is not closed

to zero.

in its fibre.

If no immediate

then also

Trf.d

f,(J(x))~

and there is nothing

to prove.

So suppose

immediate

specializations

of

Then we have

is

is discrete.

in its fibre,

fibres.

is a

g

we m a y r e p l a c e

is a finite morphism.

of complexes,

Case 2. Then

U

itself is finite,

a morphism

to show that

X

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

is a proper morphism,

to

(see [VI ~4])

In other words,

is an open n e i g h b o r h o o d of

is a finite m o r p h i s m

of complexes

to the case

in its fibre~ f

i

point.

N o w by r e a s o n

Also

since

1 it w i l l be sufficient

thus reduce

closed

Then

is a m o r p h i s m

morphism Z

fi.

x'

x

f(x) = f(x')

and we must check that the diagram

are closed = y

specialization

is zero on that some

in their

for all such

x'

372

d

f,(J(x))

d

o

is commutative.

J(y)

>

A s above, w e fix an e l e m e n t

then can r e p l a c e structure.

>II X >X'

X

by

Ix]-

a E F(J(x)),

and

with a suitable subscheme

T h u s w e r e d u c e to the case

X

irreducible and

x its g e n e r i c point. N e x t w e m a k e the b a s e e x t e n s i o n

Spec

> Y, a n d thus Y

r e d u c e to the case

y

is a c l o s e d p o i n t of

is c o m p a t i b l e w i t h l o c a l i z a t i o n . and Y,

f f

N o w since

m a p s the g e n e r i c p o i n t of

X

factors t h r o u g h a c l o s e d s u b s c h e m e

Y'

> Y

of c o m p l e x e s , the case

since e v e r y t h i n g

X

is n o e t h e r i a n ,

to the c l o s e d p o i n t of

b y a s u i t a b l e power of the m a x i m a l i d e a l of i:

Y,

Y'

of

~y

Y

.

defined

Since

is a finite m o r p h i s m ,

its trace is a m o r p h i s m

so w e can r e p l a c e

by

Y = Spec A with

A

Y

Y', and so r e d u c e to

a local A r t i n ring, a n d

i r r e d u c i b l e of d i m e n s i o n one over

Y.

X

373

We refer

to

[EGA IV w

for the following

two

results: a)

A proper

projective, b)

scheme of d i m e n s i o n

and

A projective

over a local ring

A

(The a d v e n t u r o u s proof of his own, For

a)

scheme

can replace

[EGA III

and admits

For

b)

divisor

one puts

these results,

and

X

by a

[EGA III 4.7.i]. curve over

of s u f f i c i e n t l y

high degree

first in a large p r o j e c t i v e

successively

we see that

a finite m o r p h i s m

for a finite morphism,

with

A

an artin ring.

closed.

is faithfully

flat,

in the extended

onto

X

~.

we reduce

W e now invoke

to the case w h e r e

algebraically

are done.

the r e f e r e n c e

"

into smaller

spaces.)

known

to reduce

2.6.2]

~ n

into ~ n A

to the case of a n o n - s i n g u l a r

N

Using

dimension

reader

~ A ' and then projects

projective

of r e l a t i v e

a finite m o r p h i s m

any positive

is v e r y ample.

X

admits

using

one reduces

a field, w h e r e

space

i over an A r t i n ring is

Since

the theorem

is

1 X = ~A'

[VI 5.4] and

[VI 5.6]

field of

the base

so that it is enough

situation.)

over

to the case

the residue

(Note that

is p r o j e c t i v e

A

is

extension

to prove

But this is C o r o l l a r y

Y'

i" Y

the theorem

1.6,

so we q.e.d.

Y,

374

w

The duality theorem

for proper morphisms.

In this section w e prove the long-awaited

duality theorem

for a proper m o r p h i s m of locally n o e t h e r i a n preschemes We will Then Dy

K"

suppose the existence of a r e s i d u a l complex

K" and

and DX

fA K" on

Y

give rise to pointwise dualizing and

X, respectively,

f: X on

>Y. Y.

functors

and we express the

duality theorem as

> Dy(RfJ')

Dx(F') for

F"

Dqc(X).

B e f o r e p r o v i n g the theorem, we must

map

Trf

of

[VI 4.2] agrees in the case of a finite or

p r o j e c t i v e m o r p h i s m w i t h the trace maps and

Trpf

of

So let

K~

Trff

f: X

> Y Y

6.5]

is n o e t h e r i a n and has finite Krull dimension. Y.

Then we construct an

isomorphism

as follows:

[III

be a finite m o r p h i s m of preschemes,

be a residual c o m p l e x on

~f:

of

[III 4,3],

and assume that Let

show that the trace

Qf~fAK~

--

> Rf~f~QK"

375

Qf.fA K.

~f~

> Rf.QfAK" --

(I)

'f

>

= f.Q#K "

>

(2) > Rf.QEf~QK"

> Rf.f~QK"

,

where as usual, Q denotes the natural map from complexes to elements of the derived category; [I.5.1];

$f

~f.

is the isomorphism of

is the isomorphism of [VI 3.1c]; the map (I) is

the definition of

~

[VI, w

and (2) is the isomorphism of

[VI 1.1c] (here is where we need the hypothesis that

Y

is

noetherian of finite Krull dimension). P_roposition 3.1. with

Y

Let

f: X

> Y

be a finite morphism,

noetherian of finite Krull dimension,

a residual complex on

Q f. fA K"

l

Y.

and let

K"

be

Then there is a commutative diagram

uf

l

>

L.

Rf frQK" Q

QK"

where

Trf

[III 6.5].

is the trace of [VI 4.2] and Trff is the trace of

376 Proof.

Follows immediately from TRA 2 [VI 4.2] and the

definition of

Now let

pf [VI w

Y

be a noetherian prescheme of finite Krull

dimension, and let

n f: X = ~ y

> Y

be the structural map

of an n-dimensional projective space over Y. residual complex on 8f:

Y.

uf

~

>Rf.

Let

finite Krull dimension, let projection, and let

K"

be a

QK"

above, using

[VI 3.1d], the definition of Proposition 3.2.

K"

Then we define an isomorphism

Qf.f~K"

similar to the map

Let

~ Y

~f , the map

[VI

~f

of

82], and [VI 1.1c].

be a noetherian prescheme of X = P~,

let

f: X

> Y

be a residual complex on Y.

be the Then

there is a commutative diagram

Qf.fAK"

Bf

> af.f'QK"

/

~rpf

Q

QK" where of

Trf

[III

is the trace of

~.3].

[VI 4.2], and

Trpf

is the trace

377

Proof.

Choose a section

s:

Y

> X of f, and consider

the following diagram:

Q c

QK"

Q Tr s~f~/ Qf, s,sAfAK . s ~ Qf, fA K. [VI 3.1b] [VI 4.2]

* S, fQ

QK"

Q Tr f~ [VI 4.2]

Tr fsQ

> af.as.s ~ f~Q~" [III 8.1] =

QK"

Trp fQ

> af.f~Q~" [III 6.5] =

> QK" [III 4.3]

where the notations have the sources indicated, and the second vertical arrow is obtained by sandwiching middle of the four isomorphisms which define

u

s

8f 9

in the Now we

make the following observations: i).

The composition of the upper row of arrows is the

identity on 2).

QK'.

This follows from TRA 1 [VI 4.2].

The composition of the lower row of arrows is also

the identity on QK'. 3). from

This is the statement of [III iO.l].

The left-hand square is commutative.

This follows

VAR 5 [VI 3.1] and the definition of the isomorphisms

US,8 f and (IV) of [VI w

378 4).

The middle square is commutative9

This is

Proposition 3.1 above. 5). hence

#s,f

Trf s

6) 9

and

are isomorphisms by construction,

Trpf

is an isomorphism.

Cs, f

is an isomorphism,

are isomorphisms.

hence

and

Q Trf

(Note incidentally we have used Theorem 2.1

above that Trf is a morphism of complexes, consider

Q Tr s

in order to

Q Trf in the first place. )

T).

We conclude finally that the right-hand square is q.e.d.

a commutative diagram of isomorphisms.

Now we come to the duality theorem itself.

Let

f: X

> Y

be a proper morphism of noetherian preschemes of finite Krull dimension.

Let

K"

be a residual complex on Y.

(Note that

the existence of a residual complex imposes a slight restriction on the preschemes ~.onsidered [VI 1.1] and [V w Dy (rasp. Dy) the functor and by

R Ho~(-,

Dx (rasp. DX) the functor

(rasp9 R Hom'(9

.)

QK') (rasp9 R Hom'(-, QK9

R Ho~(',

QfAK')

QfAK')) 9 Then composing the morphism of [II9

with ~f. and Q Trf, we obtain the duality morphism ef:

We denote by

Rf. s

)

> s

)

379

for

F" 6 D-(X).

and using

Applying the functor

~F(Y,-)

to both sides,

[II 5.2] and [II 5.3] we obtain a global duality

morphism

> Dy(RfJ')

f: DxCF')

Taking the cohomology of this, we get morphisms i el:

E

i, . . Xtx[F ,Qf&K )

> xt (RfJ" ,QK')

Theorem 3.3 (Duality Theorem).

Let

f: X

>Y

be a

proper morphism of noetherian preschemes of finite Krull dimension,

and let

duality morphisms for all

K"

be a residual complex on Y.

ef, ef, and efi

Then the

defined above are isomorphisms

F" 6 D- (X). qc

(Note that the hypothesis

of finite Krull dimension is

needed only for the definition

of ~f (cf.

restrict to bounde" complexes

F" E Dqc(X), we can state the

theorem assuming only that and that the fibres of

f

X

and

Y

[II 5.5]).

If we

are locally noetherian,

are of bounded dimension.

The proof

is the same.) Proof.

We proceed in several steps, eventually using

Chow's lemma to reduce to the case of projective know already.

space which we

38o a)

Clearly it is sufficient

isomorphism. Y

The question

to show that

dimension.

Y

is noetherian,

Using the lemma on way-out

A n y quasi-coherent

its coherent

subsheaves.

sheaves.

we reduce to

sheaf F on X.

sheaf on F is the direct limit of

sheaves.

N o w since

in the first variable

it is a quotient

of

Thus using the lemma on

Now since

are quasi-coherent, d)

If

~

a residual

then for any

f: X

RHom" transforms

to direct products,

case of a single coherent

and

functors,

functors again, w e reduce to the case of a direct sum

of coherent

c)

we m a y

and of finite Krull

In particular,

a direct sum of coherent way-out

In particular,

affine,

the case of a single quasi-coherent b)

is an

is local on Y, so we may assume

is the spectrum of a local ring.

assume that

@f

Y

sheaf

F

is affine,

on

and

g: Y

we reduce to the

X.

and all the sheaves considered

it is enough to show >Y

>Z

ef

is an isomorphism.

are two proper morphisms,

complex on Z, and if we take

F" E D~(X)

DX(F') ....

we have a commutative

egf

....

Dy(af.F')

direct sums

K" = g A ~ , diagram

> Dz(ag.RfJ

-)

381 We deduce

the following

elementary

but essential

consequences:

(i) SO is

@

If gf

(ii)

(iii) arguments,

and

e

@

and

@

g

so is If

Of

@f

then

and @

where

proven

for all

are isomorphisms

for all

F"

induction

i: Z

> X

of X, different

We now apply Chow's the existence

on X, we may assume of the form

is a closed

non-empty coherent

>

X, w h i c h

g: Z

immersion

Lemma

[EGA II 5.6.1]

of a scheme

F

> Y, of

Z

X, consider

X'

g

>X

\,

Y-morphism

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

on

the

X' p r o j e c t i v e

the natural

I

Y

on a

open subset U of X (we may assume X 4 ~:). sheaf

of

from X.

over Y, together w i t h a p r o j e c t i v e g: X'

for every complex

6 Dc(X).

for every m o r p h i s m

onto a subscheme

to deduce

are isomorphisms

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

g

g = fi, and

f)

gf

@gf

By n o e t h e r i a n

theorem

for all arguments,

.

the form R__f.F" w i t h

e)

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

g

. If

arguments,

@f

map

Given a

382

u:

> Rg.g*F , and embed it in a triangle

F

G

U

F

Then since SO

g

8f(G ~ )

~

>

Rg.g

F

is an isomorphism on U, G" has support on X-U~ and

is an isomorphism by our induction hypothesis

Thus it will be sufficient to show that isomorphism.

the case

g) embed

f

for all arguments,

Y

the projection.

(i) above, we can treat separately.

n

~y

X

Now

i

i

and

J

P

Using Y

p

is a finite morphism,

so

e.

1

is an

isomorphism by Proposition 3.1 above, and [III 6.7]. @

and

eg

is affine, we can

in a suitable projective

> Y

is an

and so we reduce to

space over Y, say f = pi, with p: ~

e).

is projective.

Since X

@f(~g.g*F)

Using (iii) above, we reduce to showing

are isomorphisms

efg

.

Also

is an isomorphism by Proposition 3.2 above, and [III 5.1]. P q.e.d.

383

W e can now pull ourselves up by our bootstraps,

and

|

obtain a theory of f" and

Trf

for complexes with coherent

cohomology and schemes admitting dualizing complexes. Corollary 3.4.

W e consider the category of

noetherian preschemes which admit a dualizing complex, and we consider morphisms of finite type.

(a).

For every such morphism

Then

f: X

> Y, there is a

!

theory of

f"

consisting of a functor

f" :

>

plus the data 2)-5) and properties VAR 1 - VAR 6 of [III 8.7] (only leave out the w o r d "embeddable" wherever it occurs). (b).

For every such proper morphism

f: X

> Y, there

is a theory of trace consisting of a functorial morphism !

Trf:

with the properties

Rf.f"

> 1

TRA 1 - TRA 4 of [III 10.5]

(only leave

out the phrase "proJectively embeddable" wherever it occurs).

384

(c).

For every such proper morphism

f: X

> Y, the

duality morphism I

@f:

=Rf* R ~ ( F ' , =

f'G')

> R Ho~(Rf.F',

obtained by composing the morphism of

[II 5.5] with

in

Trf

F" E Dqc(X)

the second place, is an isomorphism for G" E D:(Y).

G') ,

and

(Compare [III ii.I].)

Proof.

(a)

observe that

Let

QK"

K"

be a residual complex on Y (and

is then a dualizing complex on Y, and QfAK"

a dualizing complex on X).

Let

Dy

and

~X

be as in the

theorem above, and define |

f'(G') = Dx(~f*Dy(G'))

for all

G" E Dc+(Y). Note that

of it is defined.

Dy(G')

E Dc(Y), SO that

Here is where we need that

QK"

dualizing complex, not just pointwise dualizing.

Lf* =

is a Observe also

|

that

f"

is independent of the choice of

The construction of the isomorphisms [VI 3.1]. and

For the isomorphisms

[III 2.4], respectively.

and verifications of

df

and

K" (use [V 3.1]). Cf,g

is easy, using

ef, we use [III 6.9b]

The details of these constructions,

VAR 1 - VAR 6 are left to the reader (:).

Observe, by the way, that this part of the Corollary does not depend on the duality theorem, and could have come just the construction of

fJ

[VI 3.1].

after

385 (b)

To define

G" E Dc+(Y) , and use [II 5.10],

Trf, let

[VI ~.2], Theorem 2.1 above, and [V 2.1]:

Rf.f'G"

=

Rf. =R --Ho--~(Lf*Dy(G'= -), QfAK')

l

m ~ ~

[II 5.101

_R_ H o ~ r

R__f.QfAK~ QTrf

~

G

~

> R Ho

,,

The verification

G'), QK')

of TRA 1 is clear, using TRA 1 of [VI 4.2].

For TRA 2 we use [III 6.9c]. (c)

Details left to reader (:).

To prove the duality formula, we reduce as in part b)

of the proof of the theorem above, to the case of complexes with coherent cohomology.

So let

F" E Dc(X)

and

F"

G" E D+(Y). c -

Consider !

Rf. R= HOmxCF" , f'G')

.

(1)

!

we

~X

ite

F" - s

is a dualizing

and

f'G" = Dx(Lf*Dy(G') ).

functor, it transforms

R Hom

NOW since

into

of the duals of the arguments, with the order reversed 9 (I) becomes

~ Hom Thus

386

(2)

=Rf. D x R HO____~(Lf*Dy(G'), Dx(F" )) o

Applying the duality theorem above, this becomes

_by

R

which in turn, since

_DX(F')) L f*

(3)

is a left adJoint of

Rf. [II 5.10],

is isomorphic to

Dy R Homy(Dy(G'),

=Rf.Dx(F'))

.

Now applying duality to the second argument,

(4) becomes

Dy =R Homy(Dy(G ~ ), Dy(=Rf.F') )

and since

Dy

is dualizing,

this is

R Homy(Rf.F'~ G')

.

The reader may check that our chain of isomorphisms is indeed ~f, which proves (C)o Proposition 3.5 (Compatibility of Local and Global duality). Let

f: X

and let

> Y K"

be a proper morphism of noetherian schemes,

be a residual complex on Y.

closed point, with local ring dualizing complex (Qf~K~

on

A =

~

x

.

Let

x 6 X

Let

R~

A, and assume that

be a

be the R"

is

387

,A._

normalized

[V 66].

Let

I = F (R').

Let

F" E D--(X). C-

X

Then

-

the diagram X

R F (F') =

X

ay(e f)

is commutative,

where

,.. Ho__~(~f. Ho~(~',Qf~K" ) ,QK ")

8

is the local duality isomorphism X

[V 6.2], u is the natural map of derived functors obtained from the inclusion F

c f., x--

8

af..o~(F'.Qf'K')

is obtained

from the stalk map

> .o,i(F~.R" )

and the trace map

Trf,x:

and

Dy8 f

f.J(x) = I

> K" ,

is the transpose by

isomorphism of Theorem 3.3.

Dy

of the global duality

(Note we write four times Hom"

instead of R Hom', because the second argument

in each case

is injective.) Proof. Remark.

Immediate

from the definitiQns of the maps in question.

This compatibility

proof of "Lichtenbaum's theorem" remark in middle of p. 103].

is the one needed to complete the [LC theorem 6.9, see parenthetical

388

w

S_mooth mor~hisms. In this section we give the special case of the duality

theorem for a proper smooth morphism of locally noetherian preschemes.

In this case we can eliminate the hypothesis

that our preschemes admit residual complexes. The results below are valid practically without change for Cohen-Macaulay morphisms one defines

~/y

(see

[v.9.7]).

In that case

to be the unique cohomology group of

I

I

f'(~y).

The functor

can glue the sheaves defines

~

f"

is defined only locally, but one to obtain a global one.

WX/Y

as in the smooth case.

= f* | WX/y[n]

One then We leave

the details to the reader. Throughout this section,

f: X

> Y

will be a proper,

smooth morphism of locally noetherian preschemes,

and we will

suppose for simplicity that the fibres are all of the same dimension,

say n.

W e denote by

relative n-differentials s and by DC'd

,,

>

the sheaf fu

DCx)

given by f~(G') = f*(G') | WX/y[n ] (compare

[III,

the functor

n ~/y

of

389 Theorem 4.1.

For every proper, smooth morphism f: X ---~Y

of relative dimension n of locally noetherian preschemes, there is a morphism

9 f: Rnf.(~x/Y) with the properties Proof.

> ~y

b)-g) of [III ll.e].

We will first consider the case where

noetherian and admits a dualizing complex.

Y

is

Then by Corollary 3.4,

we have

f:(~y) = f~(~) = ~x/y[n] , and we have a trace map |

Trf:

~f.f" (Yy)

>

~y

.

Taking the cohomology in degree n, we obtain a map 7f:

as required.

Rnf. (WX/y)

> ~y

The proofs of the properties b)-g) are similar

to loc. cit. except for c), which we will leave to the reader. For the general case, the question is local by c), so we may assume

Y = Spec A is the spectrum of a noetherian ring

We consider flat base extensions

ui:

Yi ------>Y' where

Yi

A. is

390

the spectrum of a complete local noetherian ring over A. Bi

By the Cohen structure theorem [I~,

Bi, flat

(31.1)],

each

is a quotient of a regular local ring, hence admits a

dualizing complex, and so the theorem holds for for each

i

Y... 1

Thus

we have a morphism

7f.: I

Rnfi. (WXi/Yi )

Furthermore, by c), if

vii:

i

> Yj

Yi

compatible with the morphisms

7fi = vij*Tfj

> ~y

ui,uj,

is a morphism

we have

9

By the lemma below, applied to the A-module Homy(Rnf. (~X/y) 9 ~y) there is a unique

Tf: Rnf.(WX/y) such that this

7f

7f i = ui*Tf

'

> ~y

for all i.

has the properties

(details left to reader~).

By virtue of its construction,

b)-g), which completes the proof

391

Lemma ~.2. an A-module.

Let

A

be a noetherian ring, and let

We consider the category (Bi)iE I

w h i c h are complete noetherian local rings, morphisms of A-algebras.

~:

M

of

M

be

A-algebras

flat over A, and

Then the natural map

>

M

. >,~i. M . ~i

> MI

> ~/M

~ / ~ . , ~ B i -.-~pm ( ~ / ~ i

> o

9

392

Now

~2

is biJective,

because

= k(~), and so the

natural map

is biJective.

On the other hand,

~3

is inJective,

so

is biJ ective. Now let

M

be an arbitrary A-module of finite type.

Then we can find a filtration O - M O _CM 1 _c ... c M r - M

whose quotients ~i E Ass M.

Mi/Mi_ 1

are of the form

A/~ i with

By using the 5-1emma and induction on r, we

reduce to the previous case. Remark.

One sees from the proof that it would be

sufficient to consider only is either maximal, Corollary ~. 3. f: X

>Y

Bi

of the form (A~) ^

or an element of

a)

where

Ass M.

For every proper,

smooth morphism

of noetherian preschemes of finite Krull dimension.

There is a trace morphism Trf:

Rf.~G"

~ G"

393

for

G" E D b (Y), satisfying qc

where

TRA I-TRA ~ of [III iO.5] (but

TRA 4 is valid for arbitrary base extension). b)

The resulting duality morphism

,_f: R_f.~ Ho~(F', f~G') is an isomorphism for Proof.

Define

F" E Dqc(X) Trf

Details left to reader'

and

> ~= Bo~(Rf.F',G') G ~ E Dqc(Y).

by the projection formula and

7f.

394 Index of Definitions

acyclic complex bounded complex Cartan-Eilenberg resolution catenary Cech complex codimension codimension function co-finite type module Cohen-Macaulay c o m p ~ x Cohen-Macaulay morphism Cohen-Macaulay-i ring Cohen-Macaulay sheaf cohomological functor complex Cousin complex Cousin complex of F ~-functor depth derived category derived functor duality morphism dualizing complex dualizing functor embeddable morphism F-acyclic family of supports filtration associated to a dualizing complex finite cohomological dimension finite inJective dimension finite Tor-dimension Gorenstein complex Gorenstein morphism Gorenstein prescheme Gorenstein ring homotopy hyperext Koszul complex Krull dimension local complete intersection local hyperext

1.5 1.2 1.7 V.7 III.3 IV.I V.7 V.5 IV.3 V.9 V.9 IV.2 I.l 1.2 IV.3 IV.2 I.l IV.2 I.~ 1.5 IIIo5 V.2 V.5 III.8 1.5 IV.I V.7 1.5 1.7 II.4 IV.3 III.l, V.9 V.9 V.9 1.2 1.6 III.7 V.2 III.l II.3

395

local hype~ Tor localization of a category localizing subcategory locally noetherian category local parameter mapping cone multiplica~ive system noetherian object normalizedl dualizing complex pointwise bounded below pointwise dualizing complex pointwise finite inJective dimension proJectively embeddable rational point reflexive complex regular pr~scheme relative d~fferentials resldual c9mplex residuallystable morphism residue o4 a differential residue symbol sheaf of families of supports smooth morphism translation functor triangle ! triangulated category way-out functor Z/Z'-skel~ton

ll.~ 1.3 1.5 II.7 VII. 1 1.2 1.3 I1.7 V.6 V.8 V.8

v.8 III. I0 VII. 1 V.2 V.2 III.l VI. 1 V1.5 VII.I zzz.9 IV.I III.l I.l I.l I.l 1.7 IV. 2

396 Index of Notations

Of course there is the usual collection of variable notations:

A,B

functors or sheaves, of them,

f,g

X,Y

Hom(F,G)

for preschemes,

for morphisms of preschemes,

of functors on sheaves, a group, as

F,G

for abelian categories or rings,

Hom(F,G),

or Tor(F,e),

derived category,

as

etc.

x,y

for

for points

~,~

for morphisms

In general no underline denotes

one underline denotes a sheaf, as two underlines denotes an object in a R Hom'(F~

or

F'eG ", and a dot

denotes a complex. In the list of stable notations below, we have distinguished five categories: subscripts,

~atin alphabet,

other alphabets,

superscripts,

and arbitrary symbols.

Latin alphabet Ab acc Ass

B Coh Coz D

D(A) D(x) dcc dim Dual

the category of abelian groups ascending chain condition associated primes boundaries of a complex category of coherent sheaves category of Cousin complexes dualizing functor derived category = D(Mod(X)) descending chain condition dimension category of dualizing complexes

V.2

II.l VoO I.~ II~ V.2 VI.1

397

E Ext Ext fid

FR1-FR5

IV.2,IV.3 1.6

associated Cousin complex

ii.3 1.7.6,H.7.2o

finite inJective dimension axioms of multiplicative systems cohomology

H Hom Hom i x Icz category of inJective Cousin complexes identity map id im image J(x),J(x,x') standard inJectives

k(x) residue field of a point x K.(f)sK'(f;F ) Koszul complexes ker kernel axioms of direct systems LI-L3 L left derived functor (by analogy) ~f~r locally free of finite rank Lno category of locally noetherian preschemes .od(X) category of ~x-modules pfid Ptwdual

Q Qco

Qis R Res Spec Supp T Tot Tr

1.3

1.5 II.5 III.8 IIol

objects of a category pointwise finite inJective dimension category of pointwise dualizing complexes functor into derived category category of quasi-coherent sheaves quasi-isomorphisms right derived functor residue symbol category of residual complexes residue of a differential spectrum of a ring support of a sheaf or module translation functor trace map

II.7 1.2 III.7

m=

Ob

1.3 I.I,IV.I 1.6 II.3 II.7

v.8 VI.I 1.3 II.l I.~ 1.5

iii.9 VI.I VII.I

II.7 I.l II.~ III. IO,VI. ~,VII. 1

398

TRI-TR~ TRA tr. do Trf Trp VAR Z

axioms of a triangulated category axioms of trace transcendence degree trace for a finite morphism trace for a p~oJective morphism axioms for f" or fA cycles of a complex

I.l III iO.5,VI 4.2 III.6 III~ III 8.7,VI 3.1

Greek and other al~habets 7

trace for smooth morphism

F

global sections of a sheaf change of differentials fundamental local isomorphism duality morphism

n @

>n

">n w

f A

III 3.4pIII ll.2j v i i ~.1

zzz 1.5 I I I 7.3 III.5,III.6,III.ll, V . 6 , VII~

map of a functor to its derived functor truncated complex

z.5

truncated complex

1.7

residue isomorphism sheaf of differentials sheaf of differentials structure sheaf of a prescheme

III o8 IIIol IIIol

1.7

maximal ideal of a ring prime ideal of a ring affine space projective space the rational numbers the integers

Subscripts KA,(A) , DA,(A)

z.4

D (X)

IZ.1

A !

as in

c

as

lid

as in

c D(X)fid

fTd

as in

D(X)fTd

in

Z.7

II.4

399

Gor(Z" )

as in as in

qc X

as in

f as as in in as in

IV.3

D(X)Gor(Z" ) localization at a prime ideal

Dqc(x) &: local Fx

x

II.l ring at a point x

stalk of a sheaf at x local cohomology

IV.I IV.I

Z

as in

I,II

as in

RI 9 ~II

1.6

>n

as in

a > n ~>n ':

1.7

as in

f. :

II.2

as in

1.2,I.~

as in

K b, Db: b o u n d e d .th X i: I term of a complex .th Hi : I c o h o m o l o g y group RiF : .th derived functor

Y

as in

fY

VI.2

Z

as in

i

as in

F',X':

as in

C":

as in

K(A)':

as in

f*:

as in

K + 9D + ,= R+ :

as in

K~,D-,~-:

direct image

Superscripts b

as in as in

ee

+

VI.2 denotes a complex double complex

1.7

opposed category

inverse image bounded below b o u n d e d above

II.~ 1.2,I.~,I.5 1.2,I.4,I.5

400 |

as in f"

'

Intr.,III.8,VII.3

xzx.6

as

in

f~

as

in

A

as

in

fa

v

as

in

Lv :

--

as

in

Ix]--:

N

as

in

~:

sheaf associated to a module

^

as in

~:

completion of local ring

IXI.2 VI.3 dual sheaf

II.5.16,III.l

closure

Arbitrary Symbols {

restriction

>>

enough larger than

x

product

|

tensor product direct sum product disjoint union or sum [

]

f

t

shift operator closed interval reference to Bibliography or other chapter

1.2

V.8 Intro.

union

U

the set of

e e o o

>

morphism

>

effect of a map on elements >

distinguished

side of triangle

morphism being constructed as in

F.G:

composition of functors

1.1

4Ol

BIBLIOGRAPHY

GT

M. Artin, "Grothendieck Topologies", mimeographed seminar notes, Harvard (1962).

i.

H. Bass, "InJective dimension in noetherian rings", T.A.M.S. 102 (1962), 18-29.

.

"On the ubiquity of Gorenstein rings", Math. Zeitschrift 82 (1963), 8-28. H. Cartan and S. Eilenberg, "Homological Algebra", Princeton University Press (1956).

M

.

.

.

.

G

C. Chevalley, "Introduction to the theory of algebraic functions of one variable", Amer. Math. S,c. Surveys (1951). Eckmann and Schopf, "~ber inJektive Moduln", Archly der Math. ~ (1953). P. Gabriel, "Des Cat6gories Ab~liennes", Math. Fr. 90 (1962), 323-448. Giraud, thesis

Bull. S,c.

~to appear)

R. Godement, "Topologie alg~brique et th~orie des faisceaux", He/mann, Paris (1958). .

T

D. Gorenstein, "An arithmetic theory of adJoint plane curves", T.A.M.S. 72 (1952), 414-436. A. Grothendieck, "Sur quelque points d'alg~bre homologique", Tohoku Math. J. IX (1957),

119-221.

.

, "Theor~mes de dualit~ pour les faisceaux alg4briques coh~rents", seminaire Bourbaki, no. 149, Secr. Math. I.H.P. Paris (1957).

.

, "The cohomology theory of abstract algebraic varietie~ in Int. Cong. of Math. at Edinburgh, 1958, Cambridge Univ. Press (1960), lO3-118.

4o2

EGA

A. Grothendieck, "El~ments de G6om~trie Alg~brique", Publ. Math. I.H.E.S. Paris ~,8,11,17,20,24,

28 (196o ff). LC

9 "Local Cohomology", mimeographed seminar notes by R. Hartshorne, Harvard (1961).

SGA

9 "S~minaire de G6om6trie Alg~brique", notes polycopi6s, I.H.E.S. Paris (196o-61; 1962).

IO.

, "Residus et dualitY", pr~-notes pour un "S4minaireHartshorne" manuscript (1963).

11.

R. Hartshorne,

"A property of A-sequences",

Math. Fr.

Bull. Soc.

94 (1966).

12.

S. Mac Lane, "Categorical Algebra '~ Colloquium lectures given at Boulder, Colo. Aug. 27-30 (1963) at the 68th summer meeting of the AMS.

13.

E. Matlis, "InJective modules over noetherian rings", Pac. J. Math. 8 (1958), 511-528.

14.

M. Nagata, "Local Rings", Interscience Tracts no. 13, J. Wiley and Sons, N.Y. (1962).

15.

J.-P. Serre, "Un th~or~me de dualitY", Comment. Math. Helvet. 29 (1955), 9-26.

FAC

, "Faisceaux alg~briques coh~rents", Annals of Math. 61 (1955), 197-278.

16.

, "Groupes algebriques et corps de classes", Paris, Hermann (1959).

17.

18.

J.-L. Verdier, "Th~or~me de dualit~ de Poincare", Comptes Rendus 256 (1963), 2084-2086. , thesis

(to appear)

403 19.

O. Zariski, "Complete linear systems on normal varieties and a generalization of a lemma of Enriques-Severi~ Annals of Math. 55 (1952),

552-592. 20. t

9 "Algebraic sheaf theory", Scientific Report on the second summer institute, Part III, Bull. Amer. Math. Soc. 62 (1956), 117-141.

,,

ZS

O. Zariski and P. Samuel, "Commutative Algebra", 2 vols., van NostrRmd, Princeton (1958,1960).

21.

M.F. Atiyah and R. Bott, "A Lefschetz fixed point formula for elliptic differential operators" 9 Bull. Amer. Math. Soc. 72 (1966), 2A5-250.

22.

A. Grothendieck, "De IRham cohomology of algebraic varieties", to appear, Publ. Math. I.H.E.S.

APPENDIX

:

COHOMOLOGIE A SUPPORT PROPRE

ET CONSTRUCTION DU FONCTEUR

fl.

par P. DELIGNE (I)

Verdier a montr@ que d~ns le cam topologique, le formalisme de la dualit6 de Poincar@ me ramenait ~ des probl~mem locaux en haut (voir ~I~ ). Pour transpomer ma construction au cadre sch@matique, il faut disposer d'une th@orie de la cohomologie "~ support propre" pour lem faimceaux cob@rents. Sauf mention explicite du contraire, tous lee pr@mch@mae consid@r6s mont noeth@rienm et los pr@faisceaux quamicoh6rents.

n ~ I. Le sorite des pro-objets.

Proposition I. Soit

C

une

U -cat@6orie (2) oG existent los

(I) Ceci est une version compl@t@e d'une lettre de P. Deligne R. Hartshorne (lettre du 3 Mars 1966). Les notes de has de page ont @t@ ajout@es par le copiste. (2) U

d@migne un univerm fix@ dans toute la suite.

405

limites inductives finies. Soit

h

un foncteur

(ens).

C ~ --~

Les conditions suivantes son~ @quivalentes : (i)

h

est limite inductive, solon un petit (3) ensemble or-

donn6 filtrant, de foncteurs repr@sentables. (ii)

h

est limite inductive, selon u m p e t i t e

cat6~orie fil-

trante, de foncteurs repr@sentables. i

(iii) h

transforme

lim

fini~s en

lim

finies, et il existe

un petit ensemble d'ob~ets telique tout @14ment d'un

h(X)

se

factorise par l'un d'eux.

Los implications viales et

(iii) ~

l a c a t ~ g o r i e des cat6gorie de

C

(i) ~

(ii)

(F, ~ )

(ii) ---~ (iii)

standard (h

pour

stable par

est limite des

~ s h(F)

lim

et

hF(L)

solon

dans l a s o u s -

(ii) zz~ (i). Si ~

s o n t deux categories filtrantes 9 un foncteur lim

F

hF

finie engendr@e par le petit

ensemble donn6). Reste ~ prouver que

dit cofinal si

sont tri-

F : ~"

>~

et % est

est le foncteur final. Pour tout

T G :~--, ~

, on a alors

lim G =

lim GF

9 I1 s'agi@ de prouver:

i

lemme. Pour route petite cat@gerie filtrante

~

, il existe un

fonc%eur cofinal d'un petit ensemble ordonn~ filtrant dans La premiere projection : ~ • N teur cofinal (N

(3)

muni de l'ordre naturel)

"petit" = " 6 U"

.

~ ~

~

, est un fonc-

; ceci permet de se

.

406

ramener

supposer

Obj

, pr6ordo

n6 par

Eo= (X,Y)

r

,

n'a pas de plus grand @l@ment. On ordonne par inclusion l'ensemble

E

des sous-cat@gories finies de

~

ayant tm seul

objet final (fini signifie d'ensemble de flAohes fini). On d@finit um foncteur de

E

d~ns ~

en associant ~ chacune de oes

cat@gories son objet final. Sous les hypotheses faites,

E

est

filtrant et ce foncteur cofinal. Les foncteurs v@rifiant les conditions 6quivalentes de la proposition I sont les Ind -objets de une sous-oat6gorie pleine par limite inductive.

In~ C

C s ~

de

Ind C

Hom

C (C ~

9 lls forment (Ens))

, stable

est un foncteur (cat)--~(oat).

Si les limites induotives filtrantes existent dans

C

tout

Hom (h, hx)

est corepr@-

, de

dans

h 6 Ind C

, le foncteur

X

sentable, d'oh un foncteur, not6

~ lim

En partioulier, on a toujours un foncteur et tout foncteur

C

Ind C

. Dans

> Ind ~

~ Ind~ Ind C

=

lim

faut et suffit qu'il ~ransforme filtrantes en

lim

Ind C C

, si

,

X.

Pour qu'un fonoteur sur

lim

Ind C ,

; en partioulier,

"lim"

~ une sous-cat6gorie pleine de

"lim"

~

9

, les limites inductives fil-

est un syst~me inductif filtrant dans

(Xi)i~ I li~ X.

C

Ind Ind C

C

se prolonge en un foncteur

trantes sont exactes, et seront not@es identifiant

Ind C

, pour

Ind C

soit repr@sentable,

lim finies

en

lim

il

finies,

filtrantes et que sa restriction ~

C

satisfasse ~ la condition de petitesse de la prop. I (iii) (en

407

effet, la proposition I montre lalors que sa restriction ~ est un

C

Ind -objet, auquel il est partout @gal vu la condition

sur les limites). Ce qui pr@c~de, sauf la prop. I (iii) et l'assertion pr@c@dente reste vrai en utilisant partout des limites de suites (ou d@nombrables, c'est la mSme chose). On d@finit par dualiti@ la cat@gorie objets (sous-cat@gorie pleine de

Proposition 2.

Soit

X

pro C

Hom (C, (Ens)) ~

des pro9

un pr@sch@ma %uasi-compact quasi-s@par@

(non n@cessairement noeth@rien)i. La cat@6orie des faisceaux %uasi-coh@rents sur

X

est @%uivalente ~ la cat@~orie des

Ind -

ob~ets de la cat@gorie des faisceaux %uasi-coh6rents de pr@sentation finie sur

X

.

La fl@che est

"lim"~.

i >

lim ~.

1

~

. Hi

~

est de

1

p.f. (presentation finie), pour tout syst~me inductif filtrant (~i)is I

'

Homx ( ~ '

li-~m~i) = lim H o m x ( ~ , ~ i )

collement fini qui commute ~

lim L ).

(par un re-

, on se ram~ne au cas affine).

Le foncteur pr4c@dent est donc pleinement fiddle. Son image est stable par limites inductives filtrantes et scmmes finies. Ii suffit de prouver que pour tout tout

s E ~x

~ il existe

~

x~X

, tout

~

sur

X

de p.f. et une fl@che de ~

d~ns l'image de laquelle sei trouve

s

, et dans

:

sera limite d'images de fai~ceaux de p.f. et chacune d'elles !

quotient d'un faisceau de p.f. ~ar une limite de sous-faisceaux F

408

de type fini. Soit sur un voisimage quasi-compact fl@che

f : ~---,~,

de prolonger ~

~

et

de p.f.

f

&

X

,

s

U

e_~n ~

% et

X

a.ans ~IU ~

Uw V

si

V

est affine, ce UO V

, '~ un faisceau sur 9 I1 est possible

, d6finis sur

Soit

~

duit fibr6 de

~,~

s'envoie darts

~

X

l'inclusion de et ~

sur

( ~ U

k

V

:

nier comme quotient de 0 n

~

X

~I

. Si on repr6sente q~1

de p.f;, l'un d'eux prolongs d@j~

~n

dana

U

ae

X

e_~t f

prolonger

une ~

et

6tant de p.f.).

% ~ .

sous-faisceaux de type fini, comme

type fini de

une

,an faisceau de p.f. sur un ouvert %uasi-compact

d'un sch@ma affine

flSche de f

~

x

dans l'image. Ii suffit

qui revient k effectuer un prolongement de

Soit

de

entier, st, proc@dant pas ~ pas,

il suffit de savoir prolonger ~

Lsmme.

U

et ~ I prolonge

le pro~

et

comme limits de ses

est quasi-compact et . Si on ~epr@sente ce der-

par une limite de sous-faisceaux de

, on volt de mSme qu'on peut remplacer

~I

par un faisceau de p.f.

Cor I. (~ coh)

Pour qu'un foncteur contravariant additif ~

Ab

F

:

soit repr@sentable, il faut et suffit Qu'il

soit exact k gauche et transforme

lim

filtrante @n

lim

fil-

trante.

Cot 2.

Tout faisceau de p.f. d~fini sur un ouvert quasi-compact

4O9

U

de

X

se prolonge en un faisceau de p.f. sur Soit

3% une cat~gorie ab61ienne.

X

On aura A travailler

de faqon essentielle dans

pro D~j~

, sous-cat6gorie

de

"lim" K.

oG la oohomologie de

pro D ( ~ )

form6e des

1

est uniform6ment born4e avec teu~

pro

D(A)

~

Proposition 3-

(4)

tif

de foncteurs

HP(K)

A

sur

f : K ~

~ HP(L)

9

HP

Hom(~,K)

se prolonge en un fonc-

forment un s3st~me oonserva-

pro Dbj~ L

une fl@che de

pro D b ~

soient des isomorphismes

Ii s'agit de v6rifier que pour tout

Hom(K,.) ~

Hp

K

.

Les fonoteurs

Soit les

pro

i

pleine

M

~ans

est un isomorphisme.

telle que

(dans

pro ~

).

Db(d~) I o "lim"K~ 9

P6sons

On dispose d'une suite speotrale convergente

ExtP(HqK~ ,M) ~ mite

lim

E x t P ( H q K ~ ,M) ~

vergente oar tour (ioi lim : lim

ExtP+q(K~ ,M)

q

d'oG par passage ~ la

lim E x t P + q ( K ~ ,M)

, encore con-

est uniform~ment born@. Utilisant que tout fonc-

Ext) passe aux pro-o~jets et le fonoteur

Ind Ab

~ Ab

ExtP(HqK,M) ~

, on peut r ~ c r i r e

:

lim ExtP+q(K,M) =

Hom

(K ~-p-q~ ,M)

Cette suite spectrale reste d'ailleurs ME D(J%)

li-

vraie pour

. La proposition r6|3ul~e aussitSt de l'existence

de

ces suites.

(4)

i.e. si

versibles,

u u

est une flSche telle que les est inversible.

HP(u)

soient in-

410

n e 2. Lemmes fondamentaux.

Rappelons que les pr6sch6mas consid6r6s sont noeth6riens.

Proposition ~. (Th6orAme ~e dualit6 pour une immersion ouverte). Soient

U

un ouvert de

X

,

j

la flSche d'inclu-

sion, "~ un i~6al d6finissant

le ferm6 compl6mentaire,

faisceau coherent sur

~

un prolongement

U

et

%uasi-ooh6rent

coh6rent de

~

sur

~

X

un

9 Soit

9 On a

(5) La f l g c h e injective

lim H o m x ( ' ] n ~ , ~ )

: si l'image par

dans le compl6mentaire sance de

J

, soit

de

~k

f U

de

k

~ n~

dans ~

, et l'image de X

assez grand, route section

~ n+k~

s

de ~ k

U) qui se prolongs en une section de

Remplagant

~

~k~

a son support

est nulle.

affine et soit

image (sur

par

set

, elle est annul6e par une puis-

Supposons maintenant Pour

---, Homu(~,~ )

sur

X

sur

a une

X

, on peut supp0ser disposer d'un

d/agramme 0

, R '

O . ~ R1

(5)

~0 n

~ :3 ---~0

II

, 0 n ~-~ ~

Cette formule est @videmment bien connue, cf. p. ex. la

th~se de P. GABRIEL.

@

411

o0 los fl~ches pointillges sont d6finies sur Pour

~

U

9

assez grand, en vertu de Krull et du fair que

est conoentr6 hors be permet de prolonger

U f

, on a en

R/RO R I

R D ] ~ . ~ n c R I . , ce qui

~ : ~--~.

Dans le cas g6n6ral

U.

un recouvrement affine fini

1

de

X

Les

et lim

Uij k

des recouv-rements affins finis des

Uij = U.DU..I

commutent aux produits finis, d'o~ :

o----.~o%(~Y,j~ ) --i, TT 9 Eo% T

~'~

o ~ lira .- Ho.x(J n ) ,j ' ~1 ~ y

ijk

Eomui (~n ~' ~) ~

ij~ ~ lim__~homui jk(~ n,~,(~

ce qui ach~ve la d6monstration. La proposition 4 donne une formule explicite pour le fonoteur che &

j~

(coh

sur

U)

, pro (coh sur

9 On notera oe foncteur

Jl

X)

"adjoint" & gau-

("prolongement par z@ro")

(6) ; il r6sulte de Krull qu'il est exact ; cola r@sulte aussi de ce que,

X

@rant noeth6rien, un injeotif be la oat6gorie des

faisceaux quasi-ooh6rents sur jectif quasi-coh6rent sur

U

X

, restreint h

U

, reste in-

9

Proposition 5. (Ind~pendance be la oompaotification). U ~--~ X

Soit

f : X

~Y

tun morphisme propre, indui-

sant un isomorphisme entre l'ouvert

(6)

U

de

Y

Cette terminologie et notation confliotent ~videmment aveo

ceux g6n~ralement admis (livre de Gedement, SGA 1962 ...). Le leoteur pr6f~rera peut-Stre lire

~!

au lieu de

Jl

4.12

et

r-1(u)

et soit

9 Soit

~ = f~3

(i)

un faisceau d'id6aux d@finissant 9 Si

s_~ k> 0

~

est cob@rent sur

X

, le systSme projectif

Y - U ,

, on a

Rkf@u n ~

est essen-

tiellement nul (i.e. d@finit le proobjet nul) (ii) s__~i k = 0

, pour

n

assez grand,

f~un+1~

= ~.f~$n~

On:ha (~O

Oo

~-~ R k f ~ n ~ dono pour

n

~__~n -module de type fini (EGA III) n=O

est un

n=O

assez grand,

~j | R k f ~ n ~ --. R k f ~

jectif ; (ii) en r~sulte. Si il existe

N

tel que

,

k~O

j~Rkf~um~

Rkf~

n+1~

est s u r -

est nul sur

U

soit nul (quel que soit

! n).

Le diagramme JP@Rkf~n~

prouve que pour

N

---, R k f , ~ n + P ~

assez grand et

Rkf~U n +N~

n ~ 3- D@finition de

Rkf.un

Hf!

Un morphisme

~

k>O ~

0

(si

n

,

est nul, soit

(i)

.

.

f

(resp un couple de morphismes compo-

sables, resp. un triple ...) sera dit compactifiable respectivement

no)

trouver des diagrammes commutatifs

si on peut

x~-~ X ~

/

f

~

x~X