Grothendieck Topologies and Deformation Theory II

Compositio Mathematica 106: 321±348, 1997. 321

c 1997 Kluwer Academic Publishers. Printed in the Netherlands.

Grothendieck topologies and deformation Theory II

D. GAITSGORY School of Mathematical Sciences, Tel-Aviv University, Ramat-Aviv, Israel e-mail address: [email protected]

Received 13 February 1996; accepted in revised form 7 March 1996

Abstract. Starting from a sheaf of associative algebras over a scheme we show that its deformation theory is described by cohomologies of a canonical object, called the cotangent complex, in the derived category of sheaves of bi-modules over this sheaf of algebras. The passage from deformations to cohomology is based on considering a site which is naturally constructed out of our sheaf of algebras. It turns out that on the one hand, cohomology of certain sheaves on this site control deformations, and on the other hand, they can be rewritten in terms of the category of sheaves of bi-modules. Mathematics Subject Classi®cations (1991): 14F20, 18F10. Key words: Grothendieck topologies, sheaves, deformations, derived categories.

0. Introduction 0.1. In the present paper we continue the study of deformation theory of algebras using the approach of [Ga]. We will extend the main results of [Ga] to the global case. Namely, we pose and solve the following problem: what cohomological machinery controls deformations of a sheaf of algebras over a scheme? This question has already been studied by many authors [Ill], [GeSch], [Schl].

0.2. Let ®rst A be an associative algebra over a ring. Consider the category of all

C (A) algebras over A, let us call it . One can observe that every question concerning

the deformation theory of A can be formulated in terms of this category.

(A) Our ®rst step will be to apply a linearization procedure to C ,inotherwords we will endow it with a Grothendieck topology and then we will consider sheaves

of abelian groups on it. It will turn out that deformations of A are controlled by cohomologies of certain sheaves on this site. Cohomologies arise naturally as classes attached to torsors and gerbes. All this was done in [Ga].

When A is no longer an algebra over a ring but rather a quasi-coherent sheaf

C (A) of algebras over a scheme X , the de®nition of must be modi®ed in order

to take into account possible localization with respect to X , since the appropriate

cohomology theory would incorporate algebra cohomology of A and scheme coho-

mology of X . In this case instead of working with the whole category of sheaves on our site, we single out a subcategory which we call the category of quasi-coherent sheaves. This category will have properties similar to those of the category of

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322 D. GAITSGORY A quasi-coherent sheaves of A-bimodules among all sheaves of -bimodules and it will be more manageable.

The second step will be to ®nd a connection between the category of sheaves on

(A) A C and the category of quasi-coherent sheaves of -bimodules on our scheme

X . This connection will be described by two mutually adjoint functors, which

would enable us to rewrite the cohomology groups that control deformations of A

(A)

in terms of cohomologies of some canonical object T of the derived category

T (A) of quasi-coherent sheaves of A-bimodules. The object will be called the

cotangent complex of A. Another approach to the construction of the cotangent complex in a slightly different situation was used by Illusie [Ill].

0.3. Let us now describe the contents of the paper. In Section 1 we present a brief exposition of some well known facts and results from the theory of sites. For a more detailed discussion the reader is referred to [Ar], [Gr]. In the remaining sections we will freely operate with the machinery of sheaves, cohomologies, direct and inverse images; therefore the reader is advised

to look through this section in order to become familiar with the notation.

C (A)

In Section 2 we de®ne the site X along with its variants for af®ne schemes. We introduce also the appropriate categories of sheaves and functors between them.

The central results are

A) (1) Theorem 2.3.3 with its corollaries, that insure that the category Sh ( is well de®ned (2) Theorem 2.5 that says that cohomologies of quasi-coherent sheaves computed inside the quasi-coherent category and inside the category of all sheaves give

thesameanswers. L

In Section 3 we introduce functors = and that establish connection between

(X ) A

the category Sh and the category qc -mod. Let us remark that it would be

C (A)

possible to work with the category of all sheaves on X without introducing quasi-coherent sheaves explicitly. We, nevertheless, decided to do that, since to our mind, introducing this category and basic functors that are connected to it re¯ects the nature of the things and clari®es the exposition.

Finally, Section 4 is devoted to deformation theory. Theorem 4.2 describes how

C (A)

to pass from deformations to cohomology of sheaves on X via torsors and gerbes, and in 4.3 we translate the assertions of this theorem to the language of

cohomology of quasi-coherent sheaves of A-bimodules.

0.4. The results of the present paper can be easily generalized to the case of algebras over an arbitrary operad (cf. [Ga]). We opted for treating the case of associative algebras only in order to simplify the exposition. One can also develop a similar theory for operad co-algebras.

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0.5. In recent years there have been a lot of interest in deformation theory. We have to mention the works [Ge-Sch 1,2], [Ma], [Ma-St], [St], [Fo]. Our approach is close to that of [Ill]. Let us also point out that one of the central ideas of the

present paper: to resolve an algebra A by free algebras (at least locally) goes back to Quillen and to Grothendieck [Qu].

1. Preliminaries on Grothendieck topologies 1.0. In this section we will review certain notions from the theory of sites. Proofs will be given mostly in cases when our exposition differs from the standard one.

1.1. Let C be a category possessing ®ber products. A Grothendieck topology (cf. [Gr]) on it (or a structure of a site) is a collection of morphisms that are called covering maps if it satis®es the following three conditions:

(1) Any isomorphism is a covering.

U ! V V ! W U ! (2) If : and : are coverings, then their composition :

W is a covering too.

U ! V V ! V

(3) If : isacoveringandif : 1 is an arbitrary morphism, then

U V ! V

the base change map 1: 1 1 is a covering. V

1.1.1. EXAMPLES

1. For any category C there exists the minimal Grothendieck topology: the only

C; ) coverings are isomorphisms. This site will be denoted by ( min . 2. Let Set be the category of sets. We introduce the structure of a site on it by declaring surjections to be the covering maps.

3. Let Seto be the category opposite to Set. We introduce a Grothendieck topol-

X ! Y

ogy by declaring : to be a covering if the corresponding map of sets

! X Y is an injection. 4. Constructions similar to the above ones can be carried out when the category

Set is replaced by an abelian category, in particular, by the category Ab, the category

of abelian groups.

C (X ) 5. Let X be a topological space. Let be the category whose objects are

®nite disjoint unions of open subspaces of X .

def

U; V ) = U V X:

Hom( maps from to compatible with the embedding to

2 (U; V )

Amap Hom is a covering if it is surjective.

X X C

6. Let a cite C have a ®nal object 0 and let be any other object of .We C

can de®ne a new cite X whose underlying category is the category of `objects of

X X

C over ', with morphisms being compatible with projections to . A morphism

C C

in X is declared to be a covering if it is a covering in .

comp3953.tex; 17/06/1997; 10:10; v.5; p.3

324 D. GAITSGORY

C F 1.1.2. Let C1 and 2 be two sites. A functor between the underlying categories is said to be a functor between sites if the following holds:

(1) F maps coverings to coverings.

C A; B D

(2) If A; B ; D are three objects in 1 with mapping to , then the canonical

(A B ) ! F (A) F (B ) C

map F is a covering in 2.

F (D )

We say that a functor F between two sites is strict if it preserves ®ber products, i.e. if the map in (2) is an isomorphism.

1.2. DEFINITION. A sheaf of sets (resp. of abelian groups) on a site C is a functor

o o

C Ab S between the sites and Set (resp. ), the latter considered with the topology speci®ed in the Example 3 above. Morphisms between sheaves are by de®nition natural transformations between

such functors. C DEFINITION. A presheaf of sets on C (resp. of abelian groups) is sheaf on when the latter is considered with the minimal topology. It is an easy exercise to verify that the above de®nition of a sheaf coincides with the traditional one. From now on, by a sheaf we will mean a sheaf of abelian groups. It will be left to the reader to make appropriate modi®cations for sheaves of sets. Note, that a sheaf of abelian groups is the same as a group-like object in

the category of sheaves of sets.

C ) The category of sheaves will be denoted by Sh( . This category possesses a

natural additive structure and is in fact an abelian category. If S isasheaf,andif

2 C; S (X ) (X; S )

X will be denoted by and will be called the set of sections

X (X; S ) ! (Y; S) Y ! X of S over .Themap for a map will be called the

restriction map.

C ! C

1.3. Let F : 1 2 be a functor between sites. We have then the natural functor

F (C ) ! (C )

(called direct image) :Sh 2 Sh 1 . This functor is always left exact.

F F (C ) !

The functor has a left adjoint (called the inverse image): :Sh 1

C ) F Sh( 2 . The functor is always right exact. The following standard facts are often used in the theory of sheaves:

PROPOSITION

F Z ! F (X )

(1) The functor is right exact if for any covering there exists

Y ! X F (Y ) ! Z

a covering : , endowed with a map : such that the

(Y ) ! Z ! F (X ) F ( )

composition F coincides with .

F (2) The functor F is left exact if the functor is strict in the sense of 1.1.2.

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GROTHENDIECK TOPOLOGIES AND DEFORMATION THEORY II 325

F Y ! X

(3) Assume that F is strict. Then the functor is faithful if for any : and

Z ! F (Y ) Z ! F (Y ) ! F (X ) : such that the composition is a covering,

is a covering too.

1.3.1. EXAMPLES

C; ) ! C 1. Let Forget: ( min be the canonical functor of sites. The above constructions yield the embedding functor from sheaves to presheaves and its left adjoint, which is called the functor of associating a sheaf to a presheaf. It is a good

exercise to describe the associated sheaf explicitly. C

2. Let pt be the category of one object and one morphism. If is a site, for

X 2 C pt pt ! C pt

any we have a functor X : , that sends the unique object of to

X Z pt Z = pt (Z)

. We have the canonical constant sheaf on . Let by de®nition X . X

This sheaf will be called the constant sheaf corresponding to X . By de®nition we

(Z ;S)=(X; S ) S 2 (C )

have: Hom X functorially with respect to Sh .

Analogous construction produces also a sheaf ConstX in the category of sheaves

of sets. A sheaf of sets is called representable if it is of the form ConstX for some

X .

C ! C X 2 C

3. Let F : 1 2 be a functor between sites and let 1.Then

F (Z ) ' Z :

F (X )

4. Recall the situation of 1.1.1 Example 6. We have the natural embedding

i C ! C Y ! Y X

functor : X and its right adjoint Cart: . Note, that the functors

i S ! S jC X

and Cart are canonically isomorphic. We denote this functor by

C Y 2 C X

and call it the functor of restriction of a sheaf to X . By de®nition, for

(Y; SjC ) ' (Y; S)

we have X .

X ! X S ! S jC

If now 0 is a covering, the functor X is exact and faithful.

C )

1.4. Cohomology of sheaves. Along with the abelian category Sh ( one consid-

( (C ));D ( (C ));D ( (C ))

ers also the corresponding derived categories D Sh Sh Sh

( (C )) (C ) and D Sh . It can be shown [Ar], [Gr] that the category Sh has enough

injective objects. In particular, any left exact functor admits a right derived functor.

i i

2 C; R (X; S ) H (X; S ) If X will be denoted by .

1.5. Torsors and Gerbes S

1.5.0. Let now our category possess a ®nal object X0 and let be a sheaf of abelian

i i i

(S ) H (X ;S) ' R (X ;S) groups. H will denote 0 0 .

1.5.1. Before de®ning torsors and gerbes in the sheaf-theoretic context we need to

recall several de®nitions.

Let be an abelian group and let act on a set . We say that isatorsor

over if this action is simply transitive. Torsors over a given group form a rigid

comp3953.tex; 17/06/1997; 10:10; v.5; p.5

326 D. GAITSGORY

! = monoidal category (cf. [DM]) under 1 2 1 2 with the anti-diagonal

action of . M

Let now O be a monoidal category and let be an arbitrary category. We say M

that O acts on if we are given

M ! M

(1) A functor Action: O .

O M ! M (2) A natural transformation between the two functors O

Action

O O M O M

Action ?

? Action

O M M

such that the obvious `pentagon' identity is satis®ed.

O X 2 M O ! M

We say that M is a gerbe bound by ,ifforany the functor

! (A X )

given by A Action is an equivalence of categories.

M O M

If O is a groupoid and if is a gerbe bound by ,then is also a groupoid

(M ) (O )

and 0 is a torsor over 0 . S 1.5.2. A sheaf of sets is called a torsor over if

(1) S viewed as a group-like object in the category of sheaves of sets acts on the

X 2 C; (X; S ) (X; ) object ,i.e.ifforevery acts on in a way compatible

with restrictions.

2 C; (X; ) (X; S ) (2) For every X is a torsor over , whenever the former is

nonempty.

X (X; )

(3) For some covering X of 0,theset is nonempty.

(S ) S

Let TorsC denote the category of torsors over . From 1.5.1 we deduce that it is a groupoid and that it possesses a structure of a rigid monoidal category.

( (S )) H (S )

LEMMA. The group 0 Tors C is canonically isomorphic to .

(Z ;S) X

Proof. In fact, we claim more: Consider the category ExtC 0 , whose

! S ! E ! Z ! objects are short exact sequences 0 X0 0 and whose morphisms

are maps between such sequences that induce identity maps on the ends.

(S )

We claim, that this category is canonically equivalent to Tors C .

! S ! E ! Z !

Indeed, for any such extension 0 X0 0 we associate a torsor

X X

by setting for every over 0

(X; ) = Z ' (X; Z ) ! (X; E ): splittings: X0

This functor is easily seen to be an equivalence of (monoidal) categories and

comp3953.tex; 17/06/1997; 10:10; v.5; p.6 GROTHENDIECK TOPOLOGIES AND DEFORMATION THEORY II 327

1 1

( (Z ;S)) ' (Z ;S) ' R (Z ;S)

X X 0 Ext X0 Ext 0 Hom 0

(X ;S)) ' H (S ): 2 R ' 0

1.5.3. We are heading towards the de®nition of gerbes, but ®rst we need to recollect

the notion of a stack.

X X 2 C

Let C be a site with a ®nal object 0. Suppose that for each we

(X ) Y ! X

are given a category G , for each map : we are given a functor

G G(X ) ! G(Y ) Y ! X Z ! Y

: and for each composition of maps : and : ,

G G ) G

we are given a natural transformation , such that all the data are

compatible with respect to two-fold compositions.

G G(X ) ! G(Y ) Y ! X

Remark. Functors : for : will be called the

2 G(X ) ! sjY 2 G(Y ) restriction functors and will be often denoted as s . Such a collection of categories and of functors is called a presheaf of categories. It is said to be a sheaf of categories (or a stack) if moreover the following two

axioms are satis®ed:

X 2 C C

(1) Let , and let us consider the category X as in 1.1.1 Example 6. Let

;s G(X )

also s1 2be two objects of . We can consider the presheaf of sets on

(Y; Y ! X ) 2 C ! (G (s );G (s )):

: X Hom 1 2

2 C

We require that this presheaf is a sheaf for each X .

Y ! X Y

(2) Let : be a covering. Consider the category of descent data on

s 2 G(Y )

with respect to X , whose objects are pairs andanisomorphism

G (s) ! G (s) p ;p Y Y Y p

p1 2 ,where 1 2are the two projections from to , X

such that the above isomorphism satis®es the obvious cocycle condition on X

the three-fold ®ber product of Y with itself over . Morphisms in this category

! s

are de®ned to be maps s1 2 compatible with isomorphisms between their

Y G(X )

pull-backs on Y . We have the obvious functor from to this category X of descent data. We require that this functor is an equivalence of categories.

EXAMPLES

G(X ) = C

1. Consider the presheaf of categories : X .

LEMMA. This presheaf of categories is a stack if and only if

X 2 C Y 2 C C Z 2 C !

X X

(1) For every , X , the presheaf on given by

(Z; Y )

HomC is a sheaf.

S C Y 2 C X

(2) For every sheaf of sets on X the fact that for some covering

X; S jC S

Y is representable implies that is representable as well.

comp3953.tex; 17/06/1997; 10:10; v.5; p.7 328 D. GAITSGORY

The proof follows directly from the de®nitions.

All the sites in this paper will satisfy the conditions of the above Lemma.

S G(X )= (S jC ) X

2. If is a sheaf of groups, we can de®ne Tors C (torsors over

S jC C X X in the category ). This presheaf of categories is always a stack, which we

denote by TorsS .

X S

1.5.4. Gerbes. Let once again C be a site with a ®nal object 0 and let be a sheaf C of abelian groups. Let G be a stack on endowed with the following additional

structure:

G(X ) (S jC ) X

(1) Each is acted on by the monoidal category TorsC .

Y ! X

(2) For each : we are given a natural transformation between two

(S jC ) G(X ) ! G(Y ) X

functors TorsC

G ! ( G );

X Y S

Action Action Tors

which is compatible with the natural transformations of 1.5.1(2) and with

composition of restrictions.

X 2 C; G(X ) (S jC ) X

Suppose that for each is a gerbe over TorsC and that there

X G(X ) G exists a covering X of 0 such that is nonempty. We say then that is a

gerbe bound by S .

Functors between gerbes bound by a sheaf of abelian groups S and natural

transformations between such functors are de®ned in a natural fashion. G

Remark.LetS be a sheaf of abelian groups and let be a stack such that

;s 2 G(X) Y X s

if s1 2, there exists a covering of such that the pull-backs of 1

G(Y ) G S

and s2 on become isomorphic. Then is a gerbe bound by if for every

2 G(X ); (s) (X; S ) X s X; s Aut is isomorphic to functorially in and in .

EXAMPLES

S (C ) S

1. Let be in Sh . A basic example of a gerbe bound by is the stack Tors S

of 1.5.3 (Example 2) above. G

It is an easy observation that a gerbe is equivalent to Tors S if and only if

(X )

G 0 is nonempty.

! S G

2. Let S1 2 be a map of sheaves of abelian groups. If is a gerbe bound

G S

by S1, we can construct an induced gerbe bound by 2.

! S ! K ! K ! Z !

3. (cf. [D-Ill], [BB]) Let 0 0 1 X0 0 be an exact

K K ! K

sequence of sheaves on C .Let denote the 2-complex 0 1.Tothis

(K ) S 2-complex we can associate a gerbe G bound by in a canonical way by

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GROTHENDIECK TOPOLOGIES AND DEFORMATION THEORY II 329

G(K )(X )= ! K jC ! E ! Z ! X

setting: the category of extensions 0 1 X 0 C

of sheaves over X endowed with a map of complexes

- - - - -

K jC E Z X 0 0 0 X 0

id id

? ?

- - - - -

S K jC K jC Z

X X

0 0 X 1 0

(K ) It is easy to verify that G de®ned in this way is indeed a gerbe. The following assertion follows from the de®nitions:

LEMMA

(G(K )(X )) ' ];K jC ): (Z [

X X

( (C )) 0 HomD 1

Sh X

K ! K

If now : is a quasi-isomorphism of 2-complexes, we get a canoni-

) ) (K G(K cal functor between the corresponding gerbes G and . This means that

the operation of assigning a gerbe to a 2-complex is well de®ned on the derived

( (C )) category D Sh .

1.5.5. The following proposition is not dif®cult to prove:

! G(K )

PROPOSITION. The assignment K establishes a one-to-one cor-

D ( (C )) respondence between the set isomorphism classes of objects K in Sh

(K ) ' S with nontrivial cohomologies only in degrees 0 and 1, such that H ,

H (K ) ' Z G S X0 and the set of equivalence classes of gerbes bound by .

In particular, since the set of isomorphism classes of 2-complexes of the above

2 2

(Z ;S) ' H (S) G S type in the derived category is Ext X0 ,toanygerbe bound by

(S ) G(X ) we can associate a well de®ned class in H that vanishes if and only if 0

of this gerbe is nonempty.

C (A)

2. The site X

2.0. As it has been explained in the introduction, our bridge between deformations

C (A)

and cohomology is based on considering sheaves on the site X which we are about to de®ne. Throughout this paper, by a scheme we will mean a separated scheme. It is not dif®cult, however, to generalize all our results to the case of an

arbitrary scheme.

f X ! Y f f

If : is a morphism of schemes, and us will denote the usual

inverse and direct image functors on the categories of quasi-coherent sheaves of

(X ) O (Y ) O and modules.

comp3953.tex; 17/06/1997; 10:10; v.5; p.9 330 D. GAITSGORY

Remark. For an af®ne scheme we will usually make no distinction between a

quasi-coherent sheaf and its global sections.

X X

2.1. Let be a scheme and let ZarX denote the Zariski site of , whose objects

are disjoint ®nite unions of open subsets of X and whose morphisms are maps of

X A

schemes over . A morphism in ZarX is a covering if it is surjective. Let be a

quasi-coherent sheaf of associative algebras on X .

C (A)

2.1.1. DEFINITION of X .

(U; B ;) U 2 B

X U

Objects: triples U , with Zar , is a quasi-coherent sheaf of

U B ! AjU

associative algebras on and : U is a map of sheaves of associative

jU A U algebras. Here A is the restriction of on . When no confusion can be made,

we will omit .

((V; D ); (U; B )) (j; ) j 2 U

Morphisms: Hom V is a set of pairs ,where

(V; U ) D ! B jV j U Hom and : V (restriction by means of ).

Zar X

C (A)

The category X is easily seen to have ®ber products.

(j; ) (V; D ) ! (B; B ) j U

Topology: : V is said to be a covering map if is a

covering in Zar X ,andif is an epimorphism.

(X; A) Sometimes when no confusion can be made we will write A instead of

for X being an af®ne scheme.

X X = (R ) C (A)

2.1.2. Variant.When is an af®ne scheme Spec , the site X will

new new

(A) C (A)

be often denoted by C to emphasize the difference between and X

old X

(A)

C : X

old

(A)

DEFINITION of C

B A Objects: R -algebras with a map to .

Morphisms: Algebra homomorphisms commuting with structure maps to A.

Topology: Covering maps are de®ned to be just epimorphisms of algebras.

(A) C (A) X 2.1.3. ShX will denote the category of sheaves of abelian groups over .

new

X = (R ) (A) For X af®ne, Spec , this category will also be denoted by Sh ,

old X

A) whereas Sh ( will denote the category of sheaves of abelian groups over

old X

(A)

C . X

new

(U; B ) 2 C (A)

EXAMPLE 1. Let U . According to 1.3.1(2) we can consider the

new X

Z 2 (A)

U;B )

sheaf ( Sh .

X U

EXAMPLE 2. A similar construction can be carried out in the case of an af®ne

old

= (R ) C (A) R V (V )

scheme X Spec for . For a projective -module , let Free X

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old

(V ) 2 C (A) denote the free associative algebra built on V .IfnowFree ,the

old X

Z (X )

V ) sheaf ( is a projective object of Sh . This is because every covering of

Free A

V )

Free( admits a section.

C (A) G((U; B )) = U

2.1.4. LEMMA. The presheaf of categories over X given by :

C (B ) U

U is a stack.

C (B ) ' C (A)

U X

Proof. Observe ®rst of all, that U and by 1.5.3(1) we

(U;B )

must check that the following two conditions are veri®ed: U

(U; B ) 2 C (A) C (A)

X X

(1) For every U , the presheaf of sets on given by

(V; D ) 2 C (A) ! ((V; D ); (U; B ));

V X V U

(A)

HomC X

is in fact a sheaf.

S C (A)

(2) If is a sheaf of sets on X becomes representable after restriction to

(U; B ) 2 C (A) (X; A) X some U covering , then it is representable.

The ®rst point is obvious. In order to treat the second one, let us decompose the

(U; B ) ! (A; X )

map U as a composition

(U; B ) ! (U; AjU ) ! (X; A);

U = X

and it becomes suf®cient to treat separately the case when X is af®ne with

B ' AjU 2

and the case when U . In both cases, the assertion is straightforward.

Y ! X

2.2. Let f : be a morphism of schemes. Let us be given quasi-coherent

X A Y

sheaves of algebras A on and on . Assume also be given a map of sheaves

0 0

A ! f (f; ) (Y; A ) A)

of algebras : ( . We say then that is a map from the pair

X; A)

to the pair ( .

f; ) f

2.2.1. We have a functor denoted ( or just

0 0

C (A) ! C (A ) (U; B ) (U Y; f (B ) (A jU Y )):

Y U

X : goes to

X X

f (AjU ) us

This functor is strict if f is ¯at. Y In the case when both X and are af®ne schemes, we have also the functor

old old 0

(A) ! C (A )

C . Having said this, we possess the following collection of

X Y Y

functors between categories of sheaves on X and on .

f (A ) ! (A)

Y X

(1) :Sh Sh .

f f (A) ! (A )

X Y

(2) The left adjoint of : :Sh Sh . By 1.3, this functor is exact

Y ! X A ! f (A)

if f is ¯at and is moreover faithful if is onto and is us

epimorphic. 0

old 0 old

X Y f (A ) ! (A) A !

(3) (for and af®ne) :Sh Sh . This functor is exact if

Y X

(A)

f is an isomorphism, by 1.3(1). us

comp3953.tex; 17/06/1997; 10:10; v.5; p.11

332 D. GAITSGORY Y

(4) (also for X and af®ne) The left adjoint of the previous functor, denoted by

f Y ! X

f . This functor is also exact if is ¯at and is moreover faithful if is

! f (A)

onto and A is epimorphic, by 1.3.

(A) (A)

2.3. Our next goal will be to de®ne a certain subcategory Sh in ShX

which we will call the category of quasi-coherent sheaves. The category Sh ( X

will have properties analogous to those of the category of quasi-coherent sheaves of

(X ) O (X ) O -modules inside the category of all sheaves of -modules over a scheme

X . It will turn out that for an af®ne scheme, the category of quasi-coherent sheaves

old

is equivalent to Sh ( . X

old

C (A) ! 2.3.1. Let X be an af®ne scheme. We have the natural inclusion functor

new X

(A) B ! (X; B ) C : . In this case there is the direct image functor denoted

X new old

! ) (A) ! (A)

(new old :Sh Sh given by

X X

B; ( ! )(S )) = ((X; B );S);

( new old

! ) ( ! )

and the left adjoint of (new old , denoted by old new .

! ) ( ! )

The functor (old new is exact and the functor new old is left exact.

( ! ) ( ! )

R new old will denote the right derived functor of new old .

(Y; A ) ! (X; A) X Y 2.3.2. Let now f be a map with and af®ne.

LEMMA

new 0 old

f ( ! ) ( ! ) f (A ) ! (A)

(1) The functors new old and new old :Sh Sh

X Y

are canonically isomorphic.

! ) f f ( ! ) (2) The same for the functors (old new and old new from

old new 0

A) (A )

Sh ( to Sh .

X Y Proof. To prove the ®rst statement it suf®ces to observe that each of the two

functors identi®es with the direct image functor corresponding to compositions:

0 0

old new new 0 old old new

(A) ! C (A) ! C (A ) C (A) ! C (A ) ! C (A )

C and that

X X Y

X Y Y are naturally isomorphic.

The second statement follows from the ®rst one by adjunction. 2

! ) 2.3.3. We will now describe the functor (old new more explicitly. The next result can be considered as an analog of Serre's Lemma.

THEOREM. Let X be an af®ne scheme. Then the adjunction morphism of functors

! ) ( ! ) (

Id old ! new old old new is an isomorphism.

Sh ( X

Proof of the Theorem.

comp3953.tex; 17/06/1997; 10:10; v.5; p.12 GROTHENDIECK TOPOLOGIES AND DEFORMATION THEORY II 333

new

C (A) Step 1.LetX be an af®ne scheme and consider a full subcategory X aff

new

C (A) (U; B ) U

of formed by pairs U with af®ne. This subcategory carries X a natural (induced) Grothendieck topology. Let emb be the embedding functor

new new

(A) ! C (A)

emb: C . By de®nition, this is a functor between sites in the X X aff sense of 1.1.2. The following lemma is easy to prove.

new new

(C (A)) ! (C (A) )

LEMMA. The functor emb :Sh Sh is an equivalence X X aff

of categories. In particular

' ' : emb emb Id and emb emb Id

old

2 (A)

Step 2. Let us start now with a sheaf S Sh and consider the following X

0 new

C (A) presheaf S on .

X aff

j (U; B ) ! (X; A)

For : U we set

((U; B );S ) ' ((U; B );j (S)):

U U

We claim that this presheaf is in fact a sheaf.

j (U; B ) ! (X; A) j (U ;B ) ! (X; A)

U i i U

Let : be a covering, and let : i denote

i + (U; B ) (X; A) (X; A)

the map of the 1-fold ®ber product of U with itself over to .

Without restricting generality it suf®ces to check that the complex

0 0 0

! ((X; A);S )) ! ((U; B );S ) ! ((U ;B );S ); U 0 U 1 1

is exact at ®rst two places. Since the functor of taking sections is left exact, it is enough to prove the following lemma:

LEMMA. The canonical complex (called the Cech complex of S with respect to

(U; B )

U )

S ! j j (S ) ! j j (S ) ! j j (S ) ! ;

0 !

0 0 1 1 2 2

is exact.

Proof of the Lemma.

(X; A) ! (U; B )

Step 1. Assume ®rst, that the map U admits a section. In this case our complex is exact since we can write down an explicit homotopy operator.

Step 2. In the general case, by 2.2.1(4), it is enough to prove that our complex is

exact after applying the functor j . However, when we do that, the complex obtain is

Ï j (S ) (U; B ) (U ;B ) U

the Cech complex of over U with respect to 1 1. Therefore, we

(U ;B ) ! (U; B ) U ®nd ourselves in the situation of Step 1, since the projection 1 U1

comp3953.tex; 17/06/1997; 10:10; v.5; p.13 334 D. GAITSGORY

admits a section. 2 0

Step 3. We claim now, that the sheaf S constructed above is in fact canonically

( ! )(S )

isomorphic to emb new old . For this we must establish an isomorphism

S ; (M )) ! (S; ( ! )(M ));

new ( old

C (A) )

Hom ( emb Hom new old

Sh Sh ( X

aff X

new 0

M 2 S A)

for any Sh ( . But this is clear from from the construction of .

! )

In order to ®nish the proof of the theorem it remains to observe that (new old

S ) ' S 2 emb ( , by de®nition.

2.3.4. Let us now present several corollaries of the above theorem.

old

( ! ) (A) COROLLARY 1. Assume X to be af®ne. The functor old new realizes Sh

new X

as a full Abelian subcategory of Sh ( stable under extensions. X

This is a formal consequence of the Theorem. Y COROLLARY 2. Let X and be af®ne and consider the canonical morphism of

functors

( ! ) ! ( ! ) f ;

f new old new old

f f

that is given by the adjunction map of the functors and .

new

A) Then it is an isomorphism when applied to objects of Sh ( of the form

old X

( ! )(S ) S 2 A)

old new for Sh ( . X

Proof of Corollary 2.

( ! )( ! )(S ) ' f (S ) Indeed, on the left hand side we get f new old old new

whereas on the right-hand side we get

! ) f ( ! )(S ) '

(new old old new

! ) ( ! ) f (S ) ' f (S ); (old new old new

and it is easy to verify, that under these identi®cations the above natural transfor-

(S ) 2

mation yields the identity morphism on f .

= (R ) COROLLARY 3. Let X Spec be an af®ne scheme.

i old

( ! )( ! )(S )= S 2 (A) i > (1) R new old old new 0forany Sh and for any 1.

old newX

! ) D ( (A)) ! D ( (A))

(2) The functor (old new : Sh Sh if fully faithful.

X X Proof of Corollary 3.

comp3953.tex; 17/06/1997; 10:10; v.5; p.14

GROTHENDIECK TOPOLOGIES AND DEFORMATION THEORY II 335

S ! K ! K ! To prove the ®rst point, we must show that if 0 ! 1 2 0isan

new old

A) S 2 (A)

exact sequence of sheaves in Sh ( with Sh , the sequence

S ! (( ! )(K )) ! (( ! )(K )) ! ; 0 ! new old 1 new old 2 0

old

A) is exact in Sh ( .

X old

2 C (A)

For this it suf®ces to check that if B is a free algebra built on

! (B; S) ! (B; K ) !

a projective R -module, the complex of sections 0 1

) ! B; K ( 20 is exact.

old new

A) (A)

Now, since Sh ( Sh is stable under extensions, X

1 X

(Z ;S)=

X;B )

Ext old ( 0 implies

Sh ( X

1 1

((X; B );S) ' (Z ;S)= ;

H new new

X;B )

Ext ( 0

A) (A)

Sh ( Sh

X X

and the assertion follows.

The second point readily follows from the ®rst one. 2

2.3.5. Let us summarize some of the above results into a proposition:

new

S 2 (A)

PROPOSITION. Let X be af®ne, Sh . The following conditions are X equivalent:

old

2 (A)

(1) S Sh .

(Y; A ) ! (X; A) Y X

(2) For some pair f : with af®ne and faithfully ¯at over

0 old

A ! f f (S ) (A ) A)

and with ( surjective, belongs to Sh . us

new Y

(j (U; U ) ! (X; A)) 2 C (A) U

(3) For every : B with af®ne, the canonical

( ! )(S ) ! ( ! ) j (S ) map j new old new old is an isomorphism.

The proof follows immediately from Corollaries 1, 2 of the Theorem.

(A)

2.4. We arrive now to the de®nition of a quasi-coherent sheaf in ShX .

new

S 2 (A) DEFINITION. Let X be ®rst an af®ne scheme. A sheaf Sh is said to be

old X

quasi-coherent if it belongs to Sh ( .

X S 2 (A)

Let now be an arbitrary scheme. A sheaf Sh X is said to be

new

(U; B ) 2 C (A) U

quasi-coherent if for every pair U with af®ne, the restric-

S j(U; B ) S (U; B ) U tion U (cf. 1.3.1(4)) of to is quasi-coherent in the sense of the previous de®nition.

LEMMA

(A)

(1) Quasi-coherent sheaves form a full abelian subcategory in Sh X , stable

under extensions. We will denote it by Sh ( . X

comp3953.tex; 17/06/1997; 10:10; v.5; p.15

336 D. GAITSGORY

f (Y; A ) ! (X; A) f

(2) Let : be a map. Then the direct image functor maps

qc qc

A ) (A)

Sh ( to Sh .

Y X

(Y; A ) ! (X; A) f

(3) Let once again f : be a map. Then the functor also maps

qc qc

A) (A )

Sh ( to Sh .

X Y Proof. The ®rst point follows readily from the de®nition and from the fact that

the analogous statement holds for X af®ne, by 2.3.4(1). Y To prove the third point we may assume that both X and are af®ne and then Lemma 2.3.2 implies the statement.

In order to prove the second point, as in the usual theory of schemes, it is Y suf®cient to check the statement in the case when both X and are af®ne.

new 0

! )(S ) 2 (A ) Let (old new Sh . By 2.3.5(3), we must show that for every

newY

(j (U; U ) ! (X; A)) 2 C (A) U

: B with af®ne, the canonical map

j ( ! ) f ( ! )(S ) !

new old old new

( ! ) j f ( ! )(S );

new old old new

is an isomorphism.

0 0

f U Y Y U

Indeed, let j and denote the maps from to and to and we have

j ( ! ) f ( ! )(S ) '

new old old new

j f ( ! ) ( ! )(S ) '

new old old new

0 0

j f (S ) ' f j (S ) '

0 0

f ! ) ( ! ) j ( (S ) '

new old old new

0 0

! ) f ! ) ' ( j (

new old old new

( ! ) j f ( ! );

new old old new

where the above isomorphisms follow by applying Lemma 2.3.2 and Theorem

2.3.3. 2

A) !

By analogy with the af®ne situation, we will denote the inclusion functor Sh (

(A) ( ! ) X

ShX also by old new . Note, however, that when is not af®ne there is

(A) ( ! ) (A) !

no functor analogous to new old :ShX Sh possessing good prop- X

erties.

f f

As before, restrictions of the functors , to the category of quasi-coherent

f f

sheaves will be denoted by , . By de®nition, we have

f ( ! ) ' ( ! ) f ; old new old new

and

( ! ) ' ( ! ) f : f old new old new

comp3953.tex; 17/06/1997; 10:10; v.5; p.16

GROTHENDIECK TOPOLOGIES AND DEFORMATION THEORY II 337

(A) (X; B ) 2 C (A) Z 2

X;B )

2.4.1. EXAMPLE. Let ,then ( Sh .

qc qc

(A) (B )

2.4.2. The category Sh has enough injective objects because every Sh U

X U

with U af®ne does (cf. [Ar], [Gr]).

( ! ) D ( (A)) ! D ( (A))

2.5. We have the natural functor old new : Sh ShX that

D ( (A)) D (A) X

sends Sh to the subcategory qc Sh that consists of objects of

D ( (A))

ShX with quasi-coherent cohomologies. !

THEOREM. The above functor induces an equivalence of categories: (old

b b

) D ( (A)) ! D (A)

new : Sh Sh X .

qc X

Corollary 3 of Theorem 2.3.3 implies the assertion for X af®ne, as well as the

following fact:

f (Y; A ) ! (X; A) R f

PROPOSITION. Let : be a map. Then the natural map

( ! ) ! ( ! ) R f

old new old new is an isomorphism of functors.

Proof of the Proposition. Y

In order to prove this statement, we may assume that both X and are af®ne.

2 (A )

A standard devissage shows that it is suf®cient to show that if I Sh is an

R f ( ! )(I )= i>

injective object, old new 0for 0.

R f ( ! )(I )

We know, that old new is the sheaf associated to the presheaf

i 1 1

(U; B ) ! H ((f (U );f (B ) A jf (U ));I):

U U

(A )

ShY

f (A)jf (U ) us

However, we know by 2.3.4(3) that

i 1 1

((f H (U );f A jf (U ));I) '; (B )

(A )

ShY

(U ) f (A)jf

i 1 1

((f (U );f (U ));I)= ; H (B ) A jf

qc U

0 0

A )

Sh (

Y 1

(A)jf f (U )

us 2 since I is injective.

Proof of the Theorem.

Since any complex is glued from its cohomologies, it suf®ces to prove that for

S ;S 2 A) Sh ( the map

1 2X

! ) qc (S ;S ) !

(old new :Ext 12

Sh (

! )(S ); ( ! )(S ));

Ext ((old new old new

A) ( 1 2

ShX is an isomorphism.

comp3953.tex; 17/06/1997; 10:10; v.5; p.17

338 D. GAITSGORY

j U ! X U

Choose : to be a covering in ZarX with af®ne. Choose also an

(S ) ! I I (AjU ))

embedding j where is an injective object of Sh .

2 Y

S ! j (I ) K

is an injection and let denote the cokernel. The above proposition

implies

i i

(S ;f (I)) ' (S ;f (I)) = ;

Ext Ext 0

1 ( 1

(A)

Sh ShX X

and we have a commutative diagram for every i>1

! )

(old new

i 1 1

(S ;K) (( ! )(S ); ( ! )(K ))

Ext qc 1 Ext old new 1 old new

(A)

Sh ShX

? ?

! )

(old new

i i

;S ) (( ! )(S ); ( ! )(S )) S

Ext ( Ext old new old new

12( 1 2

(A)

Sh ShX

i = ; The needed assertion follows by induction on i,sincefor 01 it is already

known (Lemma 2.4(1)). 2

3. A-Bimodules and sheaves on CX(A)

3.0. In this section we will study the connection between the category of quasi-

A C (A)

coherent sheaves of -bimodules and that of quasi-coherent sheaves on X .

The material here is parallel to the one of Section 3 in [Ga]. The category of

A A

quasi-coherent sheaves of -bimodules will be denoted by qc -mod.

3.1. Let us recall several de®nitions from [Ga]. If B is a quasi-coherent sheaf of

X I B

algebras on a scheme , we denote by B the sheaf of -bimodules given by

I = (B B ! B )

B ker (the map here is the multiplication).

M B (B; M )

(X )

If is a quasi-coherent sheaf of -bimodules, we will denote by O

(I ;M) B

the group HomB . A

3.2. Let now X be a scheme and let be a quasi-coherent sheaf of associative

(A) X = A !

algebras on . We will construct a localization functor : qc -mod Sh :

M 2 A =(M ) C (A) X

Let qc -mod. Consider the presheaf on given by

((U; B ); =(M )) = (B ;MjU):

U U

O (U )

The following Lemma follows directly from the de®nition.

LEMMA. This presheaf is in fact a sheaf.

comp3953.tex; 17/06/1997; 10:10; v.5; p.18 GROTHENDIECK TOPOLOGIES AND DEFORMATION THEORY II 339

3.2.1. If X is an af®ne scheme, similar constructions can be carried out in the

old old

(A) =

category C . In this case we denote the localization functor by . X

LEMMA. The functors

old qc

( ! ) = = A ! (A);

new old and : qc -mod Sh X

are canonically isomorphic.

The proof follows immediately from the de®nition.

(M )

3.3. Let us describe = in a slightly different way. Consider the sheaf of algebras

A M X (X; A M ) 2 C (A)

over , X . This is a group-like object in this category

((U; B ); (X; A M )) = (B ;MjU); U

Hom U

=(M ) ((U; B ); =(M ))) = ((U; B ); (X; A M )) U

and isasheafgivenby U Hom .

(M )

In other words, = is a group like object in the category of sheaves of sets with

=(M )=

M ConstA , as a sheaf of sets. (cf. 1.3.1(2)).

3.3.1. PROPOSITION

(Y; A ) ! (X; A) Y X

(1) Let f : be a map with and af®ne. Then the functors

old old old

= f f = A (X ) ! (Y ); 0

and : qc -mod Sh

us A

are canonically isomorphic.

Y f (Y; A ) ! (X; A)

(2) For any X and : the functors

= f f = A (X ) ! (Y ); qc

and :-mod Sh A us are canonically isomorphic.

new old

X = A ! (A) (A)

(3) If is af®ne the functor : qc -mod Sh takes values in Sh .

X X

(4) Let once again X be af®ne. Then the functors

ol d

( ! ) = = A !

old new and : qc -mod

new

(X ); (X ) A (X ) ! A

Sh : qc -mod Sh A

are canonically isomorphic.

X = A ! (X ) (A) A

(5) For arbitrary the functor : qc -mod Sh takes values in Sh .

= A ! (A)

(6) The functor : qc -mod Sh is exact and faithful. X Proof of the Proposition. The ®rst two points are immediately deduced from

the following general Lemma:

C ! C A 2 C LEMMA 1. Let F : 1 2 be a functor between two sites. Let 1 be an

(abelian) group-like object so that the sheaf of sets ConstA possesses a structure

comp3953.tex; 17/06/1997; 10:10; v.5; p.19

340 D. GAITSGORY

(A)

of a sheaf of abelian groups. Suppose that F is an (abelian) group-like object

F (A) ! (F (A))

in C2 as well and that :End End is a homomorphism of groups.

F ( ) '

(A)

Then Const ConstF . ab

Proof of the Lemma 1.

Remark. Here the subscript ab is to emphasize that the inverse image functor is

taken in the category of sheaves of abelian groups.

2 (C )

Observe ®rst that for any S Sh 1 ,

n 2 Z; ( ;S)) = f 2 (A; S )jn = n ( )g

Hom ) Const for any

Shab 1

n n (A) 2

where on the right-hand side denotes the endomorphism Id A End .

S 2 ) (C

Therefore, for each Sh ab 2 we have

( ;F (S)) 'f 2(A; F (S ))jn

Hom ) Const

Shab 1

n 2 Z 'f 2(F (A);S)jn n ( )g

= for any

= n ( )g n 2 Z ' ( ;S);

(C F (A)

for any Hom ) Const

Shab 2

and that is what we wanted to prove. 2

The third point follows from (1), (2), 3.2.1 and 2.3.5(3). Now, (2), (3) imply (5), whereas (3), 3.2.1 and 2.3.3 imply (4).

In order to prove (6) we may assume X to be af®ne, and the assertion follows

from the following lemma, whose proof is a straightforward veri®cation.

X = (R ) V R (V ) 2

LEMMA 2. Let Spec and let be an -module. Let also FreeR

old

(A) V

C be the free associative algebra built on . Then the functors

M ! (V; M ) M ! ( (V ); =(M )) A ! Ab;

R qc HomR and Free : -mod

are canonically isomorphic. 2

f (Y; A ' f (X; A) A))

3.3.2. Let be a map from a pair ( to the pair .We

f A ! A qc

have then the direct image functor us : -mod -mod and the natural

transformation

= f ! f = :

X us Y

PROPOSITION (1) The above above map is an isomorphism of functors.

(2) Moreover, it induces an isomorphism of the derived functors

= R (f ) ! R (f ) = ! R (= f );

X us Y X us

comp3953.tex; 17/06/1997; 10:10; v.5; p.20 GROTHENDIECK TOPOLOGIES AND DEFORMATION THEORY II 341

def

D (A) ' D (A ) D ( (A)) qc

from qc -mod to Sh . X

Proof. Let us ®rst make the following observation:

! R B

LEMMA. Let R be a homomorphism of commutative rings. Let also be

M B R

an R -algebra, and be a -bimodule. Then

(B; M ) ' (B R ;M):

R R The ®rst assertion follows now, since

Lemma

((U; B ); = f (M )) ' (B ;f (M )jU ) '

U X us U us

O (U )

(f (B ;Mjf (U)) ' ((f (U );f (B ));f = (M)):

U U Y

O (f (U ))

us us Y

In order to prove the second statement, it suf®ces to assume that both X and are

f f

af®ne. In this case the functors us and are exact (cf. 2.2.1(3)), and since the

functor = is always exact, the assertion follows.

A ' f A)

3.3.3. Remark. When the condition ( is not satis®ed, the functor of

us 0

direct image on quasi-coherent sheaves of A -modules can still be de®ned but it will be neither left exact nor right exact. Therefore, the simply minded isomorphism 3.3.2(1) will be false in that situation. However, if one modi®es the de®nition of direct image in order to get a correctly de®ned functor in the derived category, the above isomorphism in derived categories will still hold.

3.4. PROPOSITION±DEFINITION

(A) = A ! L

(1) The functor : qc -mod Sh admits a left adjoint denoted by :

(A) ! A

Sh qc -mod.

f (Y; A ' f (X; A) A))

(2) Let be a map from a pair ( to the pair . The functors

(X ) ! A L f f (Y ); L Y

and X:Sh -mod

qc us A

are canonically isomorphic.

2 (A) A

Proof.ForanyS Sh we must construct a quasi-coherent -module

(S )

L , satisfying

L(S );M) ' (S; =(M ));

Hom ( Hom

(A)

A Sh qc

-mod X

functorially in M .

L(S )

Assume ®rst, that for a given S ,such exists. Proposition 3.3.2(1) implies

f (Y; f (A)) ! (X; A) L (f (S )) (Y; f (A))

then, that for every : ,Y over exists

us us

as well and satis®es

L (f (S )) ' f L(S):

Y us

comp3953.tex; 17/06/1997; 10:10; v.5; p.21 342 D. GAITSGORY

Observe that this, on one hand, implies (2) modulo (1) and on the other hand X

reduces the construction of the functor L to the case when is af®ne.

In these circumstances, every object in Sh ( is a quotient of a one of the

V )

type Free( .

S = Z L(S ) = F (V )

V )

However, for Free( , Lemma 2 of 3.3.1 implies that :

(V ) A V satis®es our requirements, where F denotes the free -module built on .The

proof follows in view of the following assertion:

C F C ! C SUB-LEMMA. Let C1 and 2 be two Abelian categories and let : 1 2

be an additive left exact functor between them. Suppose that F admits a partially

de®ned left adjoint functor which is however de®ned on a large collection of objects C

in C2 (i.e. any object in 2 is a quotient of a one from this collection). Then this 2 left adjoint is de®ned on the whole of C2.

3.4.1. Since the functor = is exact, it can be prolonged to a functor between the

def qc

(A)) D (A) ' D (A ) ! D ( qc

corresponding derived categories: qc -mod Sh which

X L

will be also denoted by =. Our next aim is to show that the functor (which is obvi-

L ( D (A)) ! D (A)

ously right exact) can be also derived into a functor L : Sh ,

= D (A)) X A) ! D (

which will be the left adjoint functor to : ( Sh .When is

af®ne, the argument of 3.4 proves also the existence of such L , since the sheaves

Z V O (X )

V )

Free( with being a projective -module form a set of projective genera-

tors of Sh ( . However, in order to treat the general case an additional argument X is needed, since objects of the derived category cannot be reconstructed just from

the local information. A 3.4.3. THEOREM. Let X be an arbitrary scheme and be a quasi-coherent sheaf

of algebras on A.Then

L L L D ( (A)) ! D A)

(1) The functor admits a left derived functor : Sh ( .

(2) L satis®es

L L(S );M ) ' (S ; =(M ));

Hom( Hom

(A)) S 2 D ( M 2 D A)

functorially in Sh and in ( .

Y; A ) ! (X; A) f Y ! X

(3) Let ( be a map such that : is ¯at and such that

A ! f A)

( is an isomorphism. Then

L L f ' f L L ;

Y X

0 qc

(A ) ( (A)) ! D D

as functors Sh X .

Y X

Remark 1. In (3), one can drop the assumptions that Y is ¯at over and

A ! f (A) f A !

that is an isomorphism. In this case the functors : qc -mod

us us

qc qc

f (A) ! (Y ) A

-mod and :Sh Sh 0 will have to be replaced by appropriate

X A derived functors.

comp3953.tex; 17/06/1997; 10:10; v.5; p.22

GROTHENDIECK TOPOLOGIES AND DEFORMATION THEORY II 343

Remark 2. The category Sh ( is lacking objects that would be acyclic for X

the functor L. The situation here is similar to that in [Bo], [Be], when one wants

to de®ne the direct image functor for D -modules. As in [Bo], there are at least two

ways to overcome this dif®culty: a more straightforward one is to go beyond the

category Sh ( and work with arbitrary sheaves. In this case there are enough X

acyclic objects for the functor L, but the drawback of this approach is that we

D (A) D (A )

will have to rely on the equivalence of the categories qc and -mod with quasi-coherent cohomologies as well as on Theorem 2.5. Another way is the one described below:

Proof of the Theorem.

L D ( (A)) !

Step 1. First we will present a construction of a functor: L : Sh

(A)

D .

Y; A ) Y S

Consider ®rst a pair ( with is af®ne and let be a quasi-coherent sheaf

0 0

C (A ) (S )

on Y . We will construct a canonical sheaf Can mapping surjectively onto

S (S ) L (S )=(B; S) Z

with Can being acyclic for the functor . Namely, Can B,

old 0

(A ) B

the sum being taken over isomorphism classes of objects in C with a free

(Y ) algebra on a projective O -module. This construction has the following two

properties:

! S (S ) !

(1) For any map of sheaves S1 2 there is a canonical map Can 1

S )

Can ( 2 .

0 0

f (Z; f Y Z A )) ! (Y; A )

(2) If : ( is a morphism of pairs with and af®ne, there

0 0

( (S )) ! (f (S ))

exists a canonical map f Can Can .

(A )

Thus any complex S of sheaves bounded from above in Sh admits a admits

S ) ! S a canonical quasi-isomorphism Can( by a complex consisting of sheaves

acyclic with respect to the functor =.

Let now S be a complex bounded from above on giving rise to an object of

D ( (A)) j U ! X U

Sh , and choose : to be a covering in ZarX with af®ne. Put

A ' j j U ! X i + U (A) i

and let also i : be the 1-st cartesian product of with

us X

itself over X . All these schemes are af®ne since is assumed to be separated.

U (S jU ) S jU

i i

For each i , ®x the canonical resolution Can of as above.

L( (j (S )))

Then for each i we can form a complex Can of quasi-coherent

AjU U i

sheaves of i -bimodules on .

i + U U p 6 k 6 i +

i There are 1 maps from i to , call them , and for each 1 1

1 k

we have a map of complexes

L( (j (S ))) !L( (j (S )));

p Can Can

k i i us 1

which is a quasi-isomorphism by 3.4(2), since the functor

qc qc

p (AjU ) ! (AjU );

i :Sh i Sh

k 1

AjU AjU

i i 1 is exact.

comp3953.tex; 17/06/1997; 10:10; v.5; p.23

344 D. GAITSGORY

The alternating sum of the maps p de®nes a complex of complexes

! j L( (j (S ))) ! j L( (j (S )))

0 us Can Can

1 1

( (j (S ))) ! ; j

! Can us

2 2

(S ) L A

or, in other words, a double complex in qc -mod, whose associated

(S )) L complex we denote by Ass( . Using Lemma 2.3.3 and Proposition 3.4(2) we see that the canonical map of

complexes

( (j (S ))) ! (L (S ))jU; L Can Ass

is a quasi-isomorphism. This implies that the functor

! (L (S ));

S Ass

( (A)) ! D (A)

is a well-de®ned functor D Sh , which we denote by

L(S )

L .

L(S ) Step 2. Let us prove now, that the functor L we have constructed satis®es

the adjunction property

L L(S );M ) ' (S ; =(M )): Hom( Hom ( )

For this we must construct the adjunction morphisms

!= (L (S )); S Ass

and

L (=(M ))) ! M : Ass(

This is done in the following way :

by Lemma 2:3 3

[j (j S )] [j j (S )] S i

Ass i Can Ass

i i

? :

by Lemma 3:3 2

[j =L( (j S ))] = [j L( (j S ))] i

Ass i Can Ass Can

i i

for the ®rst adjunction map, and

[j L(j (=(M )))] [j L (j (=(M )))] i

Ass i Ass Can

us us

i i

( )

by 3:4 2

? :

by Lemma 2:3 3

(L=(M ))] (M )] [j j [j j M i

Ass i Ass

i i us us

us us

comp3953.tex; 17/06/1997; 10:10; v.5; p.24 GROTHENDIECK TOPOLOGIES AND DEFORMATION THEORY II 345

for the second one. It is now easy to verify, that the adjunction maps constructed

)

above give rise to ( .

)

Step 3. The adjunction property ( established above implies that the functor

L(S ) L L we have constructed is a derived functor of as well as (3) of the The-

orem in view of Proposition 3.3.2. 2

X; A)

3.5. DEFINITION. Let ( be as before: a scheme with a quasi-coherent

T (A) D (A)

sheaf of algebras on it. We de®ne to be the object of qc given by

L(Z(X; A))

L .

H (T (A)) i> From the fact that L is right exact we infer that vanishes for 0

H (T (A)) = I

and that A .

(A) A M

T will be called the cotangent complex of .If is a quasi-coherent sheaf

i i

H (M ) (T (A);M)

of A-bimodules, we denote by the groups Ext .

O (X )

3.5.1. EXAMPLE. Suppose that A is ¯at over . It follows from the results

T (A) ' I X

of Quillen [Qu], that A . Indeed, this is true for af®ne, and then we apply 3.4.3(3).

4. Deformation theory 4.0. This section is almost a word by word repetition of [Ga], after we adopt certain

modi®cations connected with the fact that we are working over a scheme. +

i 1

X O (X ) O [t]=t O(X)

4.1.0. For a scheme , i will denote the sheaf .

X M

4.1.1. Let A be a quasi-coherent sheaf of associative algebras on and let A

be a quasi-coherent sheaf of A-bimodules endowed with a map of -bimodules

A ! M

: .

A; M )

Consider the category Extalg ( de®ned as follows:

(A; M ) (t (A; M ) ! (A; M )) =

Objects: O1-algebras ext such that ker :ext ext

t (A; M ) ! (A; M ))

im( :ext ext with ®xed isomorphisms

A; M )=t (A; M ) ' A (t (A; M ) ! (A; M )) ' M; ext( ext and im :ext ext

and such that under the above identi®cations the action of t

(A; M )=t (A; M ) ! (t (A; M ) ! (A; M )) ' M; ' A ext ext im :ext ext

coincides with .

Morphisms: Maps in this category are de®ned to be O1-algebra homomor-

' (t (A; M ) ! (A; M ))

phisms that induce identity maps on M im :ext ext and on

' (A; M )=t (A; M ) A ext ext .

comp3953.tex; 17/06/1997; 10:10; v.5; p.25

346 D. GAITSGORY

A; M )

LEMMA. The category Extalg ( above is canonically equivalent to the cate-

=(M ) C (A) (=(M ))

(A)

gory Tors C of -torsors on .

A; M )

Proof.Indeed,toanyext( as above, we can assign a sheaf of sets whose

(U; B ) 2 C (A) B ! (A; M )jU

X U

sections over U are algebra-homomorphisms ext

that respect the projection to A . This set of sections is clearly a torsor over

(B ;MjU) ' ((U; B ); =(M ))

U U

(U ) O .

The above assignment is an equivalence of categories by 2.1.4. 2

4.1.2. Let us de®ne the category Deform ( to have as objects quasi-coherent

O (X ) A A =A

i i i

sheaves of associative i -algebras , endowed with an isomorphism

O (X )

t ' A (A ; O (X )) =

such that Tor1 i 0. In other words, we need that

A ! (t A ! A ) ' (t A ! A ):

i i i

im : i ker :

O (X )

Morphisms in this category are just i -algebras homomorphisms respecting t

the identi®cations with A modulo . This category is obviously a groupoid. It is A

called the category of i-th level deformations of .

i +

i 1

A) ! (A)

We have natural functors Deform ( Deform given by reduction

i +

i 1 1

(A) (A) t A

modulo .If i is an object in Deform , we denote by Deform the A

category-®ber of the above functor. This category, which is obviously ai groupoid

A i +

too, is called the category of prolongations of i onto the 1-st level. +

i 1

A 2 (A) + Observe that for any object i Deform

1 A

) ' (A; A): (A +

i 1

(X )

Aut 1 O

Deform (

A i

4.2. We are now ready to state the main result of the present paper:

THEOREM

(A) (=(A))

(A)

(1) The category Deform is equivalent to the category Tors C of

=(A) C (A)

-torsors on X .

A 2 (A) G =(A) A

(2) To any i Deform one can associate a gerbe bound by

C (A) G ((X; A)) A

on X in such a way that is canonically equivalent to

i +

i 1

Deform ( .

A i

Remark. Observe that the ®rst point in the statement of the Theorem is a special = cases of the second one for i 0.

Proof. The ®rst point of the Theorem is a special case of Lemma 4.1.1 above

' A =

with M , id. G

We de®ne the gerbe A as follows:

G ((U; B )) O (U ) B

A U i U +

is the groupoid of -algebras i with an isomorphism

i 1 +

i 1

B =t B ' B (A )jU;

U U U i

+ i+

i 1 1

AjU

comp3953.tex; 17/06/1997; 10:10; v.5; p.26

GROTHENDIECK TOPOLOGIES AND DEFORMATION THEORY II 347 +

i 1

(t B ! B )= (t B ! B )

U U U U

+ i+ i+ i+ such that ker : i 1 1 im : 1 1 and such that the

map +

i 1

B ' B =t B ! (t B ! B );

U U U U U

+ i+ i+ i+

i 1 1 im : 1 1

AjU AjU ! (t B !

U +

factors through and gives rise to an isomorphism im : i 1

B )

U +

i 1 . +

i 1

G ((X; A)) ' (A)

By de®nition, A Deform .

G ((U; B )) ! G ((V; D )) (V; D ) ! (U; B )

U A V V U

Functors A for maps are

i i

V D B jV U

given by restricting sheaves to and taking ®ber products with V over .

(=(AjU ))

(B )

We must exhibit now the action of the category TorsC on

U U

G ((U; B )) U

A .

(=(M jU )) ' (B ;AjU)

(B )

Recall, that by 4.1.1, TorsC Extalg .

U U

(B ;AjU) (B ;AjU) U

Let ext U be an object of Extalg and we put

(B (B ;AjU)

+ U

Action i 1 ext

' (B (B ;AjU) ! B )= (t A ! B (B ;AjU)) :

+ U U i+ U ker i 1 ext im : 1 ext

B G ((U; B )) U

If now is a ®xed object of A , the inverse functor

U +

i 1

G ((U; B )) ! (=(AjU ));

A U

(B )

Tors C

U U

B 2 G ((U; B )) U

is provided by setting for any other object A the correspond-

U +

i 1

(V; D ) O (V )

i+ ing torsor to have as the set of sections over V the set of 1 -algebra

0 1

(B )j(V; D ) ! (B )j(V; D ) V

homomorphisms V that commute with the

i+ +

i 1 1 D

canonical projections to V . G

In order to ®nish the proof, we must show that A is a stack, but this follows i

from 2.1.4. 2

4.3. We will now translate the assertions of the above theorem into cohomological terms.

A) 1-st Level Deformations. The set of isomorphism classes of the groupoid Deform ( is canonically isomorphic to

1 1 1

(Z ; =(A)) ' R (T (A);A) ' H (A);

X;A)

Ext ( Hom

A A qc -mod

(cf. 1.5.2, 3.5, 4.2(1)).

A i

Prolongation of Deformations 1.If i is an -th level deformation, there exists a

(A)

canonical class in H which is zero if and only if there exists a prolongation

A A

+ i i 1 of . (cf. 1.5.5, 3.5, 4.2(2)).

Prolongation of Deformations 2. Suppose that for a given i-th level deformation +

i 1

(A) A

i the category Deform has an object. Then of this category is a torsor

A 0 i

comp3953.tex; 17/06/1997; 10:10; v.5; p.27 348 D. GAITSGORY

(A)

over the abelian group H . (cf. 1.5.1, 3.5, 4.2(2)).

O (X )

4.4. EXAMPLE. Suppose now that the sheaf A is ¯at as a sheaf of -modules.

A (I ;A)

From Example 3.5.1, it follows that deformations of are controlled by Ext A

(Ext s being taken in the category of quasi-coherent sheaves of A-bimodules) for

= ; i 1 2.

Acknowledgments The author is deeply indebted to J. Bernstein under whose supervision the work was written. Also I would like to thank L. Breen, A. Joseph, and S. Shnider for interesting and stimulating discussions.

References [Ar] Artin, M.: Grothendieck Topologies, Notes on a seminar by M. Artin, Harvard Math. Dept. Lecture Notes, 1962. [BB] Beilinson, A. and Bernstein, J.: A proof of Jantzen Conjectures, in the collection: I. M. Gelfand Seminar (part 1), series: Adv. in Soviet Math. 16 (1993). 2

[D-Ill] Deligne, P. and Illusie, L.: RelevementsÁ modulo p et decompositionÂ du complexe de De Rham, Inventiones Mathematicae 89 (1987). [DM] Deligne, P. and Milne, J.: Tannakien categories in the collection: Hodge Cycles, Motives, Shimura varieties, series: Lecture Notes in Mathematics 900 (1982). [Fo] Fox, T.: Introduction to algebraic deformation theory, Journal of Pure and Applied Algebra 84 (1993). [Ga] Gaitsgory, D.: Operads, Grothendieck topologies and Deformation theory, preprint (1995), in: alg-geom eprints, 9502010. [Ge-Sch1] Gerstenhaber, M.: The cohomology of presheaves of algebras, Trans. Amer. Math. Soc. 310(1) (1988). [Ge-Sch2] Gerstenhaber, M. and Schack, S. D.: Algebraic cohomology and deformation theory, in the collection: Deformation Theory of Algebras and Structures and Applications (II-Ciocco, 1986) (1988), Kluwer academic publishers, [Gr] Grothendieck, A.: SGA 4, parts I, II, III, Lecture Notes in Mathematics 269, 270, 305 (1972±1973). [Ill] Illusie, L.: Complexe Cotangent et Deformations,Â parts I, II, Lecture Notes in Mathematics

239, 283 (1972±73).

(m) [Ma] Markl, M.: A cohomology theory for A -algebras and applications, Journal of Pure and Applied Algebra 83(2) (1992). [Ma-St] Markl, M. and Stasheff, J.: Deformation theory via deviations, Journal of Algebra 170(1) (1994). [Schl] Schlessinger, M.: PhD Thesis, Harvard (1965). [St] Stasheff, J.: The intrinsic bracket on the deformation complex of an associative algebra, Journal of Pure and Applied Algebra 89(1, 2) (1993). [Qu] Quillen, D.: On the (co-)homology of commutative rings, in the collection: Applications of Categorical Algebra, Proceedings of Symposia in Pure Mathematics 17 (1970).

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