Deformations of quasicoherent sheaves of algebras · 2017. 2. 3. · V.A. Lunts / Journal of...

Journal of Algebra 259 (2003) 59–86

www.elsevier.com/locate/jalgebra

Deformations of quasicoherent sheaves of algebras

Valery A. Lunts1

Department of Mathematics, Indiana University, Bloomington, IN 47405, USA

Received 23 April 2001

Communicated by Michel Van Den Bergh

Abstract

Gerstenhaber and Schack [NATO Adv. Sci. Inst. Ser. C, Vol. 247, 1986] developed a deformationtheory of presheaves of algebras on small categories. We translate their cohomological description tosheaf cohomology. More precisely, we describe the deformation space of (admissible) quasicoherentsheaves of algebras on a quasiprojective schemeX in terms of sheaf cohomology onX andX ×X.These results are applied to the study of deformations of the sheafDX of differential operators onX.In particular, in caseX is a flag variety we show that any deformation ofDX, which is induced by adeformation ofOX, must be trivial. This result is used in [Lunts, Rosenberg, manuscript], where westudy the localization construction for quantum groups. 2002 Elsevier Science (USA). All rights reserved.

1. Introduction

Let X be a topological space,k be a field, andAX be a sheaf ofk-algebras onX. Wewould like to study infinitesimal deformations ofAX . Such deformations form ak-vectorspace which we denote by def(AX). In caseX = pt it is well known that the infinitesimaldeformations of (thek-algebra)A = AX are controlled by the Hochschild cohomologyof A. More precisely, def(A) = HH 2(A) = Ext2A⊗Ao(A,A). However, for a generalXandAX the situation is more subtle. More generally, given anAX-bimoduleMX we mayask for cohomological interpretation of exal(AX,MX)—the space of algebra extensionsof AX byMX (exal(AX,AX)= def(AX)).

E-mail address:[email protected] This research was partially supported by the CRDF grant RM1-2089 and by the NSA grant MDA904-01-1-

0020.

0021-8693/02/$ – see front matter 2002 Elsevier Science (USA). All rights reserved.PII: S0021-8693(02)00555-0

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60 V.A. Lunts / Journal of Algebra 259 (2003) 59–86

Gerstenhaber and Schack [GS] developed a deformation theory ofpresheavesofalgebras. Given a small categoryU and a presheaf of algebrasAU onU (i.e., a contravariantfunctor from U to the category ofk-algebras) they consider the space def(AU ) ofinfinitesimal deformations ofAU and give it a cohomological interpretation. Namely, givenanAU -bimoduleMU they define a natural exact sequence of complexes ofk-vector spaces

0→ T •a (MU )→ T •(MU ) → �T •(MU )→ 0.

The middle term is the total complex of thesimplicial bar resolutionof MU and

Hi(T •(MU )

) = ExtiAU⊗AoU(AU ,MU )

is the Hochschild cohomology ofAU with coefficients in MU . The cohomologyHi(�T •(MU )) is the cohomologyHi(U,MU ) of the nerve ofU (or the classifying spaceof U ) with coefficients inMU . Finally,

H 2(T •a (MU )

) = exal(AU ,MU );

in particular, H 2(T •a (AU )) = def(AU ). As a consequence they obtain a long exact

sequence ofk-spaces

· · · → Ext1AU⊗AoU(AU ,MU )→H 1(U,MU )→ exal(AU ,MU )

→ Ext2AU⊗AoU(AU ,MU )→H 2(U,MU )→ ·· · .

Returning to our problem of trying to interpret cohomologically the space exal(AX,

MX), we may proceed as follows. LetU be the category of (all or some) open subsetsof X. From the sheaf of algebrasAX and its bimoduleMX we obtain the correspondingpresheavesAU andMU . At this point there are two natural questions.

Q1. Is exal(AX,MX) equal to exal(AU ,MU )?Q2. Can we interpret the spaces Exti

AU⊗AoU(AU ,MU ) and Hi(U,MU ) as sheaf

cohomologies onX or X ×X?

The answers to these questions in general are probably negative.In this paper we obtain positive answers to the above questions in caseX is a quasi-

projective scheme overk and AX and MX are quasicoherent sheaves onX, whichsatisfy some additional conditions (the pair(AX,MX) must be admissible in the senseof Definition 4.7 below). In this case there is a natural quasicoherent sheaf of algebrasAe

Y

on the product schemeY = X × X (this is the analogue of the ringA ⊗ Ao for a singlealgebraA). Moreover, theAX-bimoduleMX gives rise to aAe

Y -moduleMY ; in particular,theAX-bimoduleAX defines anAe

Y -moduleAY . If U is the category of allaffineopensubsets ofX, then we prove that

exal(AX,MX)= exal(AU ,MU ),

V.A. Lunts / Journal of Algebra 259 (2003) 59–86 61

and

ExtiAU⊗AoU(AU ,MU )= ExtiAe

Y

(AY ,MY

), H i(U,MU ) =Hi(X,MX).

In particular, we obtain the long exact sequence

· · · → Ext1AeY

(AY ,MY

) →H 1(X,MX) → exal(AX,MX) → Ext2AeY

(AY ,MY

)→H 2(X,MX)→ ·· ·

which allows us to analyze the space exal(AX,MX). One of the implications is thatexal(AX,MX) behaves well with respect to base field extensions. It is easy to describethe morphisms

H 1(X,MX)→ exal(AX,MX)→ Ext2AeY

(AY ,MY

)explicitly. Note that ifX is affine thenHi(X,MX)= 0 for i > 0 and hence exal(AX,MX)

= Ext2AeY(AY ,MY ). Moreover, in this case

Ext•AeY

(AY ,MY

) = Ext•AX(X)⊗AoX(X)

(AX(X),MX(X)

)and thus

exal(AX,MX) = exal(AX(X),MX(X)

).

In the special case whenAX = OX and MX is a symmetricOX-bimodule, theisomorphism

ExtiOY(OX,MX)= ExtiAU⊗Ao

U(AU ,MU )

was proved by R. Swan in [S].We apply the above results to analyze def(AX) in caseX is a smooth quasiprojective

variety overC andAX =DX—the sheaf of differential operators onX. In this case

ExtiAeY

(AY , AY

) =Hi(Xan,C).

If in additionX is D-affine (for example,X is affine) thenHi(X,DX) = 0 for i > 0and hence

def(DX)=H 2(Xan,C).

In the last section we studyinduceddeformations ofDX , i.e., those which come fromdeformations of the structure sheafOX. In particular ifX is a flag variety we show thatevery induced deformation ofDX is trivial. This result is used in the work [LR], where westudy quantum differential operators on quantum flag varieties.


2. Preliminaries on extension of algebras and Hochschild cohomology

2.1. Extensions of algebras

Fix a fieldk. An algebra means an associative unitalk-algebra. Fix an algebraA; Ao isthe opposite algebra andAe := A ⊗k A

o. An A-module means a leftA-module; anA-bimodule means anAe-module.

Fix an algebraA and anA-bimoduleM. Consider an exact sequence ofk-modules

0 → M →Bε−→A → 0

with the following properties:

• B is an algebra andε is a homomorphism of algebras. (HenceM is a 2-sided idealin B.)

• TheB-bimodule structure onM factors through the homomorphismε and the resultingA-bimodule structure onM coincides with the given one. (In particular, the square ofthe idealM is zero.)

Definition 2.1. An exact sequence as above is called analgebra extensionof A by M. Anisomorphism between extensions

0 →M → B → A → 0 and 0→ M → B ′ → A → 0

is an isomorphism of algebrasα :B → B ′ which makes the following diagram commuta-tive:

0 M

id

B

α

A

id

0

0 M B ′ A 0.

An extension issplit if there exists an algebra homomorphisms :A →B such thatε ·s = id.ThenB =A⊕M with the multiplication(a,m)(a′,m′)= (aa′, am′+ma′). The collectionof isomorphism classes of algebra extensions ofA byM is naturally ak-vector space whichis denoted exal(A,M). The zero element is the class of the split extension.

Given a map ofA-bimodulesM → M ′ the usual pushout construction for extensionsdefines a map

exal(A,M)→ exal(A,M ′).

Given a homomorphism of algebrasA′ → A the pullback construction for extensionsdefines a map

exal(A,M)→ exal(A′,M).


Thus exal(·, ·) is a bifunctor covariant in the second variable and contravariant in the firstone.

In caseM =A the space exal(A,A) can be considered as deformations of the first orderof the algebraA. Let us describe this space in a different way. Putk1 := k[t]/(t2). Considerk1-algebrasB with a given isomorphismθ : grB → A ⊗k k1. (The algebraB has thefiltration {0} ⊂ tB ⊂ B and grB denotes the associated graded.) The isomorphism classesof such pairs(B, θ) form a pointed set which we denote by def(A). The distinguishedelement in def(A) is represented by the algebraB =A⊗k k1.

We claim that exal(A,A) = def(A) (hence def(A) is a k-vector space). Indeed, given(B, θ) as above we obtain an exact sequence

0→ tB =A → B → A → 0,

which gives a well defined map from def(A) → exal(A,A). Conversely, given an algebraextension

0 →M =A→ B →A → 0,

define the multiplicationt :B → B by t · 1B = 1A ∈ M. This makesB a k1-algebra anddefines the inverse map exal(A,A)→ def(A).

The above description of exal(A,A) allows us to define the set defn(A) of nth orderdeformations ofA as the collection of isomorphism classes ofkn := k[t]/(tn+1)-algebrasB with an isomorphism ofkn-algebras grB → A ⊗k kn. Thus def1(A) = def(A) =exal(A,A). The algebraB = A ⊗k kn represents thetrivial deformation. Note thatB istrivial if there exists ak-algebra homomorphisms :A →B, which is the left inverse to theresidue homomorphismB → A. Indeed, thens ⊗ 1 :A⊗k kn → B is an isomorphism ofkn-algebras.

Note that the quotient homomorphismB → B/tnB defines the map defn(A) →defn−1(A). Denote by defn0 (A) ⊂ defn(A) the preimage in defn(A) of the trivialdeformation in defn−1(A).

Lemma 2.2. There exists a natural identificationdefn0 (A)= def(A). In particular,defn0 (A)has a natural structure of ak-vector space.

Proof. LetB ∈ defn(A) be such thatB/tnB =A⊗k kn−1. Consider the obviousk-algebrahomomorphismA → A⊗k kn−1 and the induced pullback diagram

0 tnB

id

B ′ A 0

0 tnB B A⊗k kn−1 0.

ThenB ′ represents an element in def(A). We get a map defn0 (A)→ def(A).


The inverse map def(A) → defn0 (A) is defined as follows. GivenB ′ ∈ def(A) considerthe projectionA⊗k kn−1 →A and the corresponding pullback diagram

0 A

id

B A⊗k kn−1 0

0 A B ′ p

A 0.

ThenB is akn-algebra as follows:

t : (b′,0)→ (0, tp(b′)

), t :

(0, tn−1a

) → (tp−1(a),0

).

This proves the lemma.✷Corollary 2.3. Assume thatdef(A)= 0. Thendefn(A)= 0 for all n.

Proof. Induction onn using the previous lemma.✷2.2. Hochschild cohomology

The space exal(A,M) has a well-known cohomological description. Namely, there is anatural isomorphism

exal(A,M)= Ext2Ae (A,M).

Let us recall how this isomorphism is defined. Consider the bar resolution

· · · ∂2−→B1∂1−→B0

∂0−→ A → 0,

whereBi = A⊗i+2 and

∂i(a0 ⊗ · · · ⊗ ai+1)=∑j

(−1)ja0 ⊗ · · · ⊗ ajaj+1 ⊗ · · · ⊗ ai+1.

Note thatBi ’s are naturallyAe-modules and the differentials∂i are Ae-linear. HenceB• → A is a free resolution of theAe-moduleA. Thus for anyAe-moduleM

H • HomAe (B•,M)= Ext•Ae (A,M).

Note that HomAe (Bi,M) = Homk(A⊗i ,M).


Given an algebra extension

0 →M → B →A → 0,

choose ak-linear splittings :A →B and define a 2-cocycleZs ∈ Homk(A⊗2,M) by

Zs(a, b)= s(ab)− s(a)s(b).

Differentk-splittings define cohomologous cocycles, hence we obtain a map exal(A,M)→Ext2Ae (A,M) which is, in fact, an isomorphism.

The spaces Ext•Ae(A,M) are called the Hochschild cohomology groups ofA with

coefficients inM. In particular, Ext•Ae (A,A) = HH •(A) is the usual Hochschild coho-mology ofA. Note that the space Ext0

Ae(A,M)= HomAe (A,M) coincides with the centerZ(M) of M:

Z(M)= {m ∈M | am=ma ∀a ∈ A}.

The space Ext1Ae(A,M) classifies the outer derivations ofA into M. Namely, a map

d :A → M is a derivation if d(ab) = ad(b) + d(a)b. It is called an inner derivation(defined bym ∈ M) if d(a) = [a,m]. Denote by Der(A,M) (respectively Inder(A,M))the space of derivations (respectively inner derivations). Then

Ext1Ae(A,M)= Outder(A,M) := Der(A,M)/ Inder(A,M).

Remark 2.4. Consider the split extensionB = A ⊕ M ∈ exal(A), i.e., the multiplicationin B is (a,m)(a′,m′) = (aa′, am′ + ma′). Then an automorphism of this extension is analgebra automorphismα ∈ Aut(B) of the form

α(a,m)= (a,m+ d(a)

),

whered :A → M is a derivation. In other words the automorphism group of the trivialextension is the group Der(A,M).

2.3. Deformation of sheaves of algebras

Let X be a topological space andA be a sheaf ofk-algebras onX. Let Ao denote thesheaf of oppositek-algebras andAe = A⊗k Ao. Given anAe-moduleM we may repeatthe above definition for algebras and modules to define the space of algebra extensionsexal(A,M). In particular, an algebra extension ofA by M is represented by an exactsequence of sheaves ofk-vector spaces

0 →M → B ε−→ A→ 0

such thatB is a sheaf ofk-algebras andε is a homomorphism of sheaves of algebrassatisfying the properties of the Definition 2.1 above. A split extension is the one admitting


a homomorphism of sheaves of algebrass :A → B such thatε · s = id. In particular, a splitextension must be split as an extension of sheaves ofk-vector spaces.

In caseM = A we may again define the set defn(A) of nth order deformations ofA,so that def1(A) = def(A) = exal(A,A). Let again defn0 (A) ⊂ defn(A) be the subsetconsisting ofnth order deformations which are trivial up to ordern − 1. Then repeatingthe proof of Lemma 2.2 we get defn

0 (A) = def(A). In particular, defn0 (A) is naturallya k-vector space and def(A) = 0 implies defn(A) = 0 for all n.

3. Review of Gerstenhaber–Schack construction

In the paper [GS] the authors develop a deformation theory of presheaves of algebrason small categories. We will review their construction in a special case which is relevant tous. Namely letX be a topological space andU be the category of all (or some) opensubsets ofX. Let A = AU be a presheaf of algebras onU , i.e., A is a contravariantfunctor fromU to the category of algebras. We denote bykU the constant presheaf ofalgebras:kU (U) = k for all U ∈ U . Let A-mod be the Abelian category (of presheaves)of left A-modules. The presheaf of algebrasAe = A ⊗Ao is defined in the obvious way:Ae(U)=A(U)⊗k A0(U). In caseA= kU for M ∈ kU -mod we denote ExtikU (kU ,M)=Hi(U,M).

Fix anA-bimoduleM (i.e.,M ∈ Ae-mod). The group exal(A,M) is defined exactlyas above in the case of a single algebra and its bimodule. We are going to give a naturaldescription of the group exal(A,M) in terms of homological algebra in the category ofpresheaves onU . In particular, we will construct a canonical map

exal(A,M)→ Ext2Ae (A,M).

First recall some constructions from [GS].

3.1. Categorical simplicial resolution

Let C = CU be a presheaf of algebras onU . Given U ∈ U denote its inclusioniU :{U} ↪→ U . The obvious (exact) restriction functor

i∗ :C-mod→ C(U)-mod, K �→ K(U)

has a right exact left adjoint functoriU ! :C(U)-mod→ C-mod,

iU !K(V )={C(V )⊗C(U) K, if V ⊂ U ,0, otherwise.

Thus if K is a projectiveC(U)-module, theniU !K is a projective object inC-mod.In particular, the categoryC-mod has enough projectives (it also has enough injectives(see [GS])).


If the categoryU has a final objectU , thenC = iU !C(U) is projective inC-mod. Inparticular, then

ExtiC(C,K)= 0, for all K ∈ C-mod, i > 0.

ForN ∈ C-mod define

S(N ) :=⊕U∈U

iU !i∗UN

with the canonical map

εN :S(N ) → N .

ClearlyS is an endo-functorS :C-mod→ C-mod with a morphism of functorsε :S → Id.Define a diagram of functors

· · · s2 ∂1−→ s1∂0−→ s0

∂−1=ε−−−−→ Id → 0,

wheresi = Si+1 and∂i = εsi −S(∂i−1). This diagram is a complex, i.e.,∂i∂i−1 = 0, whichis exact. So forN ∈ C-mod we obtain a resolution

· · · → s1(N ) → s0(N ) →N → 0.

Explicitly we have

sk(N )=⊕

Uk⊂···⊂U0

iUk!i∗Uk. . . iU0!i∗U0

N .

If N is locally projective (i.e.,N (U) is a projectiveC(U)-module for allU ∈ U ), then thecomplexs•(N ) consists of projective objects inC-mod. So in this case forM ∈ C-mod wehave

HomC(s•(N ),M

) = R Hom•C(N ,M).

3.2. Simplicial bar resolution

Consider the bar resolution of the presheaf of algebrasA:

· · · → B1 → B0 → A,

whereBi = A⊗i+2 (this is a direct analogue of the usual bar resolution for algebrasdescribed above). The presheavesBi are locally free Ae-modules, but usually notprojective objects inAe-mod. So the simplicial resolutions•B• of B• is a double complexconsisting of projective objects inAe-mod. For anAe-moduleM denote byT ••(M) the


double complex HomAe (s•B•,M), and letT •(M) = Tot(T ••(M)) be its total complex.We have

ExtiAe (A,M)=Hi(T •(M)

).

Consider the double complexT ••(M). It looks like

∏U

Homk

(A(U)⊗A(U),M(U)

) ∏V⊂U

Homk

(A(U)⊗A(U),M(V )

)∏U

Homk

(A(U),M(U)

) ∏V⊂U

Homk

(A(U),M(V )

)∏U

Homk

(k,M(U)

) ∏V⊂U

Homk

(k,M(V )

),

where the left lower corner has bidegree(0,0). The vertical arrows are the Hochschilddifferentials while the horizontal ones come from the simplicial resolution.

Let T ••a (M) ⊂ T ••(M) be the sub-double complex which is the complement of the

bottom row. Put

T •a (M)= Tot

(T ••a (M)

), Hn

a (A,M) :=Hn(T •a (M)

).

Note that the complexT •(M)/T •a (M) is just Homk(s•(kU ),M). Hence we obtain the

long exact sequence

→Hna (A,M)→ ExtnAe (A,M)→ Hn(U,M) →Hn+1

a (A,M)→ ·· · .

In caseM is a symmetricA-bimodule, i.e.,am = ma for all a ∈ A, m ∈ M, the abovesequence splits into short exact sequences [GS, 21.3]

0 →Hna (A,M)→ ExtnAe (A,M)→ Hn(U,M)→ 0.

3.3. The isomorphismexal(A,M)�H 2a (A,M)

Let the extension

0→ M → B →A → 0

represent an element in exal(A,M). Choose localk-linear splittingssU :A(U) → B(U).Let us construct a 2-cocycle inT ••

a (M). Namely, put

Z0,2(a, b)= sU (ab)− sU (a)sU (b), U ∈ U, a, b ∈A(U),

Z1,1(a)= sV rAU,V (a)− rBU,V sU (a), V ⊂U, a ∈ A(U),


whererAU,V :A(U) →A(V ), rBU,V :B(U)→ B(V ) are the structure restriction maps of the

presheavesA andB. Then(Z0,2,Z1,1) is a 2-cocycle inT ••a (M) and the induced map

exal(A,M) →H 2a (A,M)

is an isomorphism [GS, 21.4]. The inverse isomorphism is constructed as follows. Let(Z0,2,Z1,1) be a 2-cocycle inT ••

a (M). For eachU ∈ U put B(U) = A(U) ⊕ M(U)

as ak-vector space; define the multiplication inB(U) by (a,m)(a′,m′) = (aa′, am′ +ma′ + Z0,2(a, a′)). We makeB the presheaf of algebras by defining the restriction mapsrBU,V :B(U)→ B(V ) to berBU,V (a,m)= (rAU,V (a), r

MU,V (m)+Z1,1(a)).

In particular, we obtain the 5-term exact sequence

· · · → Ext1Ae (A,M)→ Hn(U,M) → exal(A,M)→ Ext2Ae (A,M)→ H 2(U,M).

4. Admissible quasicoherent sheaves of algebras and bimodules

Definition 4.1. Let Z be a scheme andAZ be a sheaf of unitalk-algebras onZ. We saythatAZ is aquasicoherentsheaf of algebras if there is given a homomorphism of sheavesof unital k-algebrasOZ → AZ which makesAZ a quasicoherent leftOZ-module. NotethatAo

Z is then a quasicoherent rightOZ-module. Denote byµ(AZ) ⊂ AZ-mod the fullsubcategory of leftAZ-modules consisting of quasicoherentOZ-modules.

Fix a quasiprojective schemeX over k with a sheaf of unitalk-algebras onAX . LetAo

X be the sheaf of opposite algebras andAeX = AX ⊗k Ao

X . An AX-module means aleft AX-module; anAX-bimodule means anAe

X-module. PutY = X × X with the twoprojectionsp1,p2 :Y → X. We have the sheaves of algebrasp−1

1 AX andp−12 Ao

X on Y

and hence also their tensor productp−11 AX ⊗k p

−12 Ao

X .Assume thatAX is quasicoherent. Then we can take the quasicoherent inverse images

p∗1AX andp∗

2AoX (using left and rightOX-structures, respectively). Put

AeY := p∗

1AX ⊗OYp∗

2AoX.

Note that for affine openU,V ⊂ X, AeY (U × V ) = AX(U) ⊗k AX(V ). This is a quasi-

coherent sheaf onY with a natural morphism of quasicoherent sheaves

β :OY → AeY ,

which sends 1 to 1⊗ 1. We also have the obvious morphism of sheaves ofk-vector spaces

γ :p−11 AX ⊗k p

−12 Ao

X →AeY .

Definition 4.2. We say that the quasicoherent sheaf of algebrasAX satisfies condition(∗)if Ae

Y has a structure of a sheaf of algebras so thatβ andγ are morphisms of sheaves ofalgebras.


Note that ifAX satisfies condition(∗) then, in particular,AeY is a quasicoherent sheaf of

algebras onY . It seems that the algebra structure onAeY as required in the condition(∗), if

it exists, should be unique. In any case, there is a canonical such structure in all examplesthat we have in mind.

Examples. 1. The condition(∗) holds if the sheaf of algebrasAX is commutative. Moregenerally, if the image ofOX lies in the center ofAX .

2. Assume that char(k)= 0 andX is smooth. Then(∗) holds for the sheafAX =DX ofdifferential operators onX. In this case

p∗1DX ⊗OY

p∗2DX =DY .

Let ωX be the dualizing sheaf onX. ThenDoX = ωX ⊗OX

DX ⊗OXω−1X and hence

AeY = p∗

1DX ⊗OYp∗

2D0X = p∗

2ωX ⊗OYDY ⊗OY

p∗2ω

−1X .

Let MX be anAX-bimodule. Then, in particular,MX is anOX-bimodule.

Definition 4.3. We say thatMX satisfies the condition(+) if for an open affineU ⊂X andf ∈O(U) we have

MX(Uf )=O(Uf )⊗O(U) MX(U)⊗O(U) O(Uf ).

Remark 4.4. The sheaves of algebrasAX in Examples 1, 2 above satisfy the condition(+)

when considered asAX-bimodules.

Lemma 4.5. LetAX be a quasicoherent sheaf of algebras which satisfies the condition(∗),and letMX be anAX-bimodule which satisfies the condition(+). ThenMX defines a(unique up to an isomorphism) Ae

Y -moduleMY onY such that for an affine openU ⊂X

MY (U ×U)=MX(U).

We haveMY ∈µ(AeY ).

Proof. Choose an affine open covering{U} of X. Then the affine open subsetsU × U

form a covering ofY . Fix one such subsetV = U × U . The sheaf of algebrasAeY is

quasicoherent, hence by Serre’s theorem below we have the equivalence of categories

µ(Ae

V

) �AeY (V )-mod.

The sheafMX defines anAeY (V ) = AX(U) ⊗k AX(U)-moduleMX(U), hence defines

a quasicoherentAeV -moduleMV . If V ′ = U ′ × U ′ ⊂ V , then the condition(+) for

MX implies thatMV |V ′ = MV ′ . Hence the local sheaves glue together into a globalquasicoherentAe

Y -moduleMY . The last assertion is obvious.✷


Theorem 4.6. LetZ = SpecC be an affine scheme,AZ—a quasicoherent sheaf of algebrason Z. Put A = Γ (X,AX). Then the functor of global sectionsΓ is an equivalence ofcategories

Γ :µ(AZ) →A-mod.

Its inverse is∆ defined by

∆(M)=AZ ⊗A M.

BothΓ and∆ are exact functors.

Proof. The point is that for anA-moduleM the quasicoherent sheaf∆(M) is indeedan AZ-module. The rest follows easily from the classical Serre’s theorem about theequivalence

qcoh(Z)� C-mod. ✷Definition 4.7. We call a quasicoherent sheaf of algebrasAX admissibleif it satisfiesconditions(∗) and(+) (as a bimodule over itself). We call anAX-bimoduleMX admissiblein it satisfies condition(+). We say that(AX,MX) is an admissible pair if bothAX andMX are admissible.

Remark 4.8. The sheaf of algebrasAX as in Examples 1, 2 above is admissible.

Let us summarize our discussion in the following corollary.

Corollary 4.9. Let (AX,MX) be an admissible pair. Then

(i) AX defines is a quasicoherent sheaf of algebrasAeY on Y such that for affine open

U,V ⊂X, AeY (U × V )=AX(U)⊗k AX(V )o;

(ii) MX defines a sheafMY ∈ µ(AeY ) such that for affine openU ⊂ X, MY (U × U) =

MX(U).

Proof. This follows immediately from Definition 4.2 and Lemma 4.5.✷We will be able to give a cohomological interpretation of the group exal(AX,MX) for

an admissible pair(AX,MX).


5. Cohomological description of the group exal(AX,MX) for an admissible pair(AX,MX)

Let X be a quasiprojective scheme overk and (AX,MX) be an admissible pair. Wewill consider the group exal(AX,MX) of algebra extensions ofAX by MX . Note that ifan exact sequence

0 →MX → BX →AX → 0

is such an extension, then we do not require the sheafBX to be quasicoherent, or even anOX-module.

Denote byU = Aff (X) be the category of all affine open subsets ofX. Given a sheafFX

onX we denote byj∗XFX the presheaf onU , which is obtained by restriction ofFX to affine

open subsets. We will usually denotej∗XFX = FU if it causes no confusion. In particular,

we obtain presheaves of algebrasAU = j∗XAX , Ae

U :=AU ⊗AoU (Ae

U �= j∗XAe

X).

Lemma 5.1. Then there is a natural mapexal(AX,MX) → exal(AU ,MU ) which is anisomorphism. In particular,def(AX) = def(AU ).

Proof. Given an exact sequence of sheaves onX

0 →MX → BX →AX → 0,

which represents an element in exal(AX,MX) we obtain the corresponding sequence

0→ MU → BU → AU → 0

of presheaves onU . This last sequence is exact becauseMX is quasicoherent. Hence itrepresents an element in exal(AU ,MU ). So we obtain a map

exal(AX,MX)→ exal(AU ,MU ).

Vice versa, let

0 → MU → B1 → AU → 0

represent an element in exal(AU ,MU ). Denote by+ the (exact) functor which associatesto a presheaf onU the corresponding sheaf onX. Then(AU )+ =AX , (MU )+ =MX andhence we obtain an exact sequence

0 →MX → B+1 →AX → 0

which defines an element in exal(AX,MX). This defines the inverse map

exal(AU ,MU )→ exal(AX,MX). ✷


Let Db(AeY ) and Db(Ae

U ) denote the bounded derived categories ofAeY -mod and

AeU -mod, respectively. LetDb

µ(AeY )(Ae

Y ) ⊂ Db(AeY ) be the full subcategory consisting

of complexes with cohomologies inµ(AeY ). Denote byj∗

Y :AeY -mod→ Ae

U -mod the leftexact functor defined byj∗

Y (F)(U) :=F(U ×U), U ∈ U . Consider its derived functor

Rj∗Y :Db

(Ae

Y

) →Db(AeU).

Theorem 5.2. The functor

Rj∗Y :Db

µ(AeY )

(Ae

Y

) → Db(AeU)

is fully faithful. Equivalently, forM,N ∈ µ(AeY ) the map

j∗Y : ExtnAe

Y(M,N )→ ExtnAe

U

(j∗YM, j∗

YN)

is an isomorphism for alln.

Proposition 5.3. The map

j∗X :Hn(X,MX)→ Hn(U,MU )

is an isomorphism for alln.

Let us first formulate some immediate corollaries of the theorem and the proposition.

Corollary 5.4. There exists a natural exact sequence

Ext1AeY

(AY ,MY

) → H 1(X,MX)→ exal(AX,MX) → Ext2AeY

(AY ,MY

)→ H 2(X,MX).

In particular, if X is affine thenexal(AX,MX)= Ext2AeY(AY ,MY ). If MX is a symmetric

AX-bimodule, then we get a short exact sequence

0 → exal(AX,MX) → Ext2AeY

(AY ,MY

) → H 2(X,MX) → 0.

Proof. Indeed, this follows from Lemma 5.1, Theorem 5.2, Proposition 5.3 and results ofSection 3. ✷

Recall the following theorem of J. Bernstein.

Theorem 5.5 [Bo]. LetZ be a quasicompact separated scheme,CZ—a quasicoherent sheafof algebras onZ. Then the natural functor

θ :Db(µ(CZ)

) → Dbµ(CZ)(CZ)

is an equivalence of categories.


Corollary 5.6. Assume thatX is affine. Then

exal(AX,MX) � exal(AX(X),MX(X)

).

Proof. PutAX(X) =A, MX(X)= M. We have

exal(A,M)= Ext2A⊗Ao(A,M).

By Serre’s theorem

Ext2A⊗Ao(A,M)= Ext2µ(AeY )

(AY ,MY

).

By Bernstein’s theorem

Ext2µ(AeY )

(AY ,MY

) = Ext2AeY

(AY ,MY

).

Finally, by Corollary 5.4 above

Ext2AeY

(AY ,MY

) = exal(AX,MX). ✷Question. Under the assumptions of the last corollary letB be a sheaf of algebras onXrepresenting an element in exal(AX,MX). Is B = AX ⊕ MX as a sheaf ofk-vectorspaces?

6. Proofs of Theorem 5.2 and Proposition 5.3

Proof of Proposition 5.3. Let kU be the constant presheaf onU and s•(kU ) → kU beits categorical simplicial resolution (Section 3). It is a projective resolution ofkU , whichconsists of direct sums of presheavesiU !k. Hence

Hi(U,MU )= Exti (kU ,MU)=Hi Hom•(s•(kU ),MU).

Consider the exact functor(·)+ from the category of presheaves onU to the category onsheaves onX. Thenk+

U = kX—the constant sheaf onX. The functor(·)+ preserves directsums and(iU !k)+ = kU—the extension by zero of the constant sheaf onU . SinceMX isquasicoherent, for an affine openU ⊂X we haveHi(U,MX) = 0 for all i > 0. Thus

Hi(X,MX)= Exti (kX,MX)=Hi Hom•(s•(kU )+,MX

).

It remains to notice that

Hom(kU ,MX)= Γ (U,MX)= Hom(iU !k,MU ).

This completes the proof of the proposition.✷


Proof of Theorem 5.2. Let us formulate a general statement which will imply thetheorem. LetZ be a quasicompact separated scheme overk. Let Aff(Z) be the categoryof affine open subsets ofZ andW ⊂ Aff (Z) be a full subcategory which is closed underintersections and constitutes a covering ofZ. LetAZ be a quasicoherent sheaf of algebrasonZ. Denote byAW the corresponding presheaf of algebras onW . Let

j∗Z :AZ-mod→AW -mod

be the natural (left exact) restriction functor.✷Proposition 6.1. In the above notation, the derived functor

Rj∗Z :Db

µ(AZ)(AZ)→Db(AW)

is fully faithful.

Proof. By Bernstein’s theorem the natural functor

θ :Db(µ(AZ)

) →Dbµ(AZ)

(AZ)

is fully faithful. So it suffices to prove that the compositionRj∗Z · θ is fully faithful. The

functorj∗Z :µ(AZ) → AW -mod is exact. LetM,N ∈ µ(AZ). It suffices to prove that the

map

j∗Z : Ext•µ(AZ)

(M,N )→ Ext•(j∗ZM, j∗

ZN)

is an isomorphism.

Step 1.Assume thatZ is affine andZ ∈ W . Then by Serre’s theoremµ(AZ) � AZ(Z)-mod. ReplacingM by a left free resolution we may assume thatM =AZ . But then

Exti (AZ,N ) = Exti(AZ(Z),N (Z)

) ={N (Z), if i = 0,0, otherwise.

On the other hand,j∗ZAZ =AW is a projective object inAW -mod (Section 3) and

Hom(AW , j∗

ZN) = Hom

(AW (Z), j∗

ZN (Z)) =N (Z).

So we are done.

Step 2. Reduction to the case whenZ is affine.Let iU :U ↪→ Z be an embedding of someU ∈W . Denote byAU the restrictionAZ|U .

We have two (exact) adjoint functorsi∗U :µ(AZ) → µ(AU), iU∗ :µ(AU) → µ(AZ). ThefunctoriU∗ preserves injectives.

Choose a finite coveringZ = ⋃Uj , Uj ∈W . Then the natural map

N →⊕j

iUj∗i∗UjN


is a monomorphism. So we may assume thatN = iU∗NU for someU ∈ W andNU ∈µ(AU). Then we have

Ext•(M, iU∗NU)= Ext•(i∗UM,NU

).

We need a similar construction on the other end. LetiU :WU ↪→ W be the embedding ofthe full subcategoryWU = {V ∈ W | V ⊆U}. LetAWU

be the restriction ofAW to WU .We have the obvious functori∗U :AW -mod→ AWU

-mod and its right adjointiU∗ definedby

iU∗(K)(V ) :=K(V ∩U).

Both i∗U andiU∗ are exact andiU∗ preserves injectives. ForK ∈ AWU, L ∈ AW we have

Ext•(i∗UL,K

) = Ext•(L, iU∗K

).

Note that the following diagrams commute

µ(AZ)

j∗X

i∗Uµ(AU)

j∗U

AW -modi∗U AWU

-mod

,

µ(AZ)

j∗X

µ(AU)iU∗

j∗U

AW -mod AWU-mod

iU∗

(herej∗U is the obvious restriction functor). Hence the following diagram commutes as

well:

Ext•(M,N )j∗Z

Ext•(j∗ZM, j∗

ZN)

Ext•(M, iU∗NU) Ext•(j∗ZM, iU∗j∗

UNU

)

Ext•(i∗UM,NU

) j∗U

Ext•(j∗Ui

∗UM, j∗

UNU

).

But j∗U is an isomorphism by Step 1 above. Hencej∗

Z is also an isomorphism.✷

7. A spectral sequence

Let X be a quasiprojective variety and(AX,MX) be an admissible pair. ForN1,N2 ∈µ(Ae

Y ) we will construct a spectral sequence which abuts to Ext•Ae

Y(N1,N2). In particular,

we will get an insight into the group Ext2Ae

Y(AY ,MY ).


Lemma 7.1. Any object inµ(AeY ) is a quotient of a locally freeAe

Y -module.

Proof. LetK ∈ µ(AeY ). ConsiderK as a quasicoherentOY -module. As such it is a quotient

of a locally freeOY -moduleQ (we can takeQ = ⊕OY (−j)). Then theAY -module

AY ⊗OYQ is locally free and surjects ontoK. ✷

Let P• → N1 be a resolution ofN1 consisting of locally freeAeY -modules. From the

proof of the last lemma it follows that there exists an affine coveringV of Y such thatfor eachV ∈ V and eachP−t the restrictionP−t |V is a freeAe

V -module. We may (andwill) assume that eachV ∈ V is of the formU ×U for U from an affine open coveringUof X. Choose one such affine coveringV . Let C•(P•) → P• be the correspondingCechresolution ofP•. This is a double complex consisting ofAe

Y -modules, which are extensionsby zero from affine open subsetsV of freeAe

V -modules. Thus

H •Ae

YHom

(Tot

(C•(P•)

),N2

) = Ext•AeY(N1,N2).

The natural filtration of the double complexC•(P•) gives rise to the spectral sequencewith theE2-term

Ep,q

2 = H p(V,ExtqAe

Y(N1,N2)

),

which abuts to Extp+q

AeY(N1,N2).

In particular, in caseN1 = AeY , N2 = MY this spectral sequence defines a filtration of

the group Ext2AeY(Ae

Y ,MY ). Namely, there are maps

α1 : Ext2AeY

(Ae

Y ,MeY

) → H 0(V,Ext2AeY

(Ae

Y ,MeY

)),

α2 : ker(α1)→ H 1(V,Ext1AeY

(Ae

Y ,MeY

)),

α3 : ker(α2)→ H 2(V,Ext0AeY

(Ae

Y ,MeY

)).

Recall that forV = U × U ∈ V by Bernstein’s and Serre’s theorems we haverespectively

Γ(V,ExtqAe

Y

(Ae

Y ,MeY

)) = Γ(V,Extq

µ(AeY )

(Ae

Y ,MeY

))= ExtqAX(U)⊗Ao

X(U)

(AX(U),MX(U)

).

7.1. Cohomological analysis of the groupexal(AX,MX)

Consider the exact sequence

H 1(X,MX)ε−→ exal(AX,MX)

ρ−→ Ext2AeY

(Ae

Y ,MY

).

Let us describe the morphismsε andρ explicitly.


SinceMX is quasicoherent the cohomology groupH 1(X,MX) is isomorphic to theCech cohomologyH 1(U,MX). Given a 1-cocycle{mij ∈ MX(Ui ∩ Uj) | Ui,Uj ∈ U}define an algebra extension

0→ MX → B →AX → 0

as follows: on eachU ∈ U the sheafB|U is a direct sum of sheavesMX|U andAX|U withthe multiplication

(m,a)(m′, a′) = (ma′ + am′, aa′).

That is, locallyB is a split extension. Define the glueing algebra automorphisms

φij :BUi∩Uj

∼−→ BUi∩Uj , φij (m,a)= (m+ [a,mij ], a

).

This defines the mapε :H 1(X,MX) → exal(AX,MX).Now assume that an algebra extensionB represents an element in exal(AX,MX).

Considerρ(B) ∈ Ext2AeY(AY ,MY ) and assume thatα1(ρ(B))= 0, i.e., locallyB is a split

extension. Thus forU ∈ U we have

B(U)=MX(U)⊕AX(U)

with the multiplication

(m,a)(m′, a′)= (ma′ + am′, aa′)

and with the glueing given by algebra automorphisms

φij :B(Ui ∩Uj )∼−→ B(Ui ∩Uj ), φij (m,a)= (

m+ δij (a), a),

whereδij :AX(Ui ∩ Uj) → MX(Ui ∩ Uj) is a derivation. For an affine openU ⊂ X thespace

Ext1AX(U)⊗AoX(U)

(AX(U),MX(U)

)is the space of outer derivationsAX(U) → MX(U). The collection{δij } defines anelement inH 1(V,Ext1Ae

Y(AY ,MY )), which is equal toα2(ρ(B)).

Assume now thatα2(ρ(B))= 0. Then there exist elements

δi ∈ Ext1AX(Ui)⊗AX(Ui)0

(AX(Ui),MX(Ui)

)such thatδij = δi − δj . Changing the local trivializations ofB by the derivationsδi ’swe may assume thatδij ’s are inner derivations. Choosemij ∈ MX(Ui ∩ Uj) so thatδij (a)= [a,mij ]. The collection{mij } defines a 1-cochain inC(U,MX). Its coboundaryis a 2-cocycle which consists of central elementsmijk ∈MX(Ui ∩Uj ∩Uk). Thus it definesan element inH 2(V,HomAe

Y(AY ,MY )). It is equal toα3(ρ(B)).


8. Examples

Let X be a smooth complex quasiprojective variety. Letδ :X ↪→ Y = X × X bethe diagonal embedding,∆ = δ(X) be the diagonal, andp1,p2 :Y → X be the twoprojections.

8.1. Deformation of the structure sheaf

Let AX = MX = OX. ThenAeY = OY , AY = δ∗OX . Since theOX-bimoduleOX is

symmetric, we have the short exact sequence

0 → def(OX) → Ext2OY(δ∗OX, δ∗OX)→ H 2(X,OX)→ 0.

Assume thatX is projective. By the Hodge decomposition [GS,S]

Ext2OY(δ∗OX, δ∗OX)=H 0(X,

∧2TX

) ⊕H 1(X,TX)⊕H 2(X,OX).

The above short exact sequence identifies def(OX) with H 0(X,∧2

TX) ⊕ H 1(X,TX).The summandH 1(X,TX) corresponds to the first order deformations of the varietyX byKodaira–Spencer theory, i.e., to “commutative” deformations ofOX , while the summandH 0(X,

∧2TX) corresponds to “noncommutative” deformations.

8.2. Deformations of the sheaf of differential operators

LetAX =MX =DX be the sheaf of (algebraic) differential operators onX. LetωX bethe dualizing sheaf onX. Then

DoX = ωX ⊗OX

DX ⊗OXω−1X .

We haveDY = p∗1DX ⊗OY

p∗2DX , and hence

DeY = p∗

1ωX ⊗OYDY ⊗OY

p∗2ω

−1X .

The functorτ :M �→ p∗1ωX ⊗OY

M is an equivalence of categories

τ :DY -mod→ DeY -mod.

Denote byδ+ :DX-mod→ DY -mod the functor of direct image [Bo]. Then

DY = τ (δ+OX).

Let Xan denote the varietyX with the classical topology.

Proposition 8.1. There is a natural isomorphism

Ext•DeY

(DY , DY

) �H •(Xan,C).


Proof. By the above remarks

Ext•DeY

(DY , DY

) = Ext•DY(δ+OX, δ+OX).

Let Db∆(DY ) be the full subcategory ofDb(DY ) consisting of complexes with

cohomologies supported on∆. By Kashiwara’s theorem the direct image functor

δ+ :Db(DX)→ Db∆(DY )

is an equivalence of categories (see [Bo]). Thus, in particular,

Ext•DX(OX,OX)� Ext•DY

(δ+OX, δ+OX).

On the other hand, by (a special case of) the Riemann–Hilbert correspondence,

Ext•DX(OX,OX)�H •(Xan,C). ✷

Corollary 8.2. LetX be a smooth complex quasiprojective variety. Then we have an exactsequence

H 1(Xan,C)→ H 1(X,DX)→ def(DX)→ H 2(Xan,C)→ H 2(X,DX).

If X is D-affine( for exampleX is affine) then

def(DX)=H 2(Xan,C).

Proof. The first part follows immediately from Proposition 8.1 and Corollary 5.4. IfX isD-affine, thenHi(X,DX)= 0 for i > 0. An affine variety isD-affine sinceDX is a quasi-coherent sheaf of algebras. This implies the last assertion.✷Example 8.3. Let X = Cn. Then def(DX) =H 2(X,C)= 0. SinceX is affine, def(DX)=def(DX(X)), whereDX(X) is the Weyl algebra. It is well known that the Hochschildcohomology of the Weyl algebra is trivial.

9. Deformation of differential operators

9.1. Induced deformations of differential operators

Let S be a commutative ring andC be anS-algebra with a finite filtration

0= C−1 ⊂ C0 ⊂ C1 ⊂ · · · ⊂ Cn = C,

such that the associated gradedgrC is commutative. Then it makes sense to define the ringDS(C) =D(C) of (S-linear) differential operators onC in the usual way. More generally,


given two leftC-modulesM, N define the space of differential operators of order� m

fromM to N as follows.

Dm(M,N)= {d ∈ HomS(M,N)

∣∣ [fm, . . . ,

[f1, [f0, d]] . . . ] = 0

for all f0, . . . , fm ∈ C}.

ThenD(M,N) := ⋃mDm(M,N) and in particular we obtain a filtered (by the order

of differential operator) ringD(C) = D(C,C). Note thatC ⊂ D(C) acting by leftmultiplication. Sometimes we will be more explicit and will writeD(CM,C N) forD(M,N). If the algebraC is commutative then eachk-subspaceDm(M,N) ⊂ D(M,N)

is also a (left and right)C-submodule.

Lemma 9.1. Denote bySn the ringS[t]/(tn+1). Then

DSn(C ⊗S Sn) �DS(C)⊗S Sn

canonically. In particular, for a commutativek-algebraA we have

Dkn(A⊗k kn)�Dk(A)⊗k kn.

Proof. Indeed, everyf ∈ EndSn(C ⊗S Sn) = HomS(C,C ⊗S Sn) can be uniquely decom-posed as

f =n⊕

i=0

fi ⊗ t i ,

wherefi ∈ EndS(C,C). Now the inclusionf ∈ DmSn(C ⊗S Sn) is equivalent to inclusions

fi ∈ DmS (C) for all i. Whence the assertion of the lemma.✷

For the rest of this section we will consider onlyk[t]-algebras, and all differentialoperators will bek[t]-linear, so we will omit the corresponding subscript. We denote asbeforekn = k[t]/(tn+1).

Let A be a commutativek-algebra andB be a kn-algebra with an isomorphismgrB � A ⊗k kn, i.e., B defines an element in defn(A). Consider the inclusion of ringsD(B) ⊂ Endkn(B). Both these rings are filtered the powers oft , hence we obtain a naturalhomomorphism (of degree 0 of graded algebras)

α : grD(B) → grEndkn(B).

Note thatα may not be injective. On the other hand, we have a natural homomorphism ofgraded algebras

δ : grEndkn(B) → Endkn(grB),

which is, in fact, an isomorphism.


We denote the composition of the two maps again byγ : grD(B) → Endkn(grB).

Lemma 9.2.

(i) The homomorphismγ mapsgrD(B) to D(grB).(ii) The following are equivalent:

(a) The mapγ : grD(B) → D(grB) is injective.(b) The mapγ : grD(B) → D(grB) is surjective.

Proof. (i) Since everything iskn-linear, it suffices to prove thatγ (D(B)/tD(B)) ⊂D(B/tB). Letd ∈ Dm(B) and denote byd ∈ D(B)/tD(B) its residue. Letb0, . . . , bm ∈B

with the corresponding residuesb0, . . . , bm ∈ B/tB. We have

[b0, . . . , [bm,d] . . .] = 0, hence

[b0, . . . ,

[bm, γ (d)

]. . .

] = 0.

Thusγ (d) ∈ Dm(B/tB).(ii) The injectivity of γ : grD(B) → D(grB) is equivalent to the injectivity of the

natural mapα : grD(B) → grEndkn(B). Consider the subspaceD(B/tB) �D(B, tnB) ⊂D(B,B). The injectivity ofα is equivalent to the assertion that everyd ∈ D(B, tnB) isequal totnd1 for somed1 ∈ D(B). But this last assertion is equivalent to the surjectivityof the mapD(B)/tD(B) → D(B/tB) and hence to the surjectivity ifγ : grD(B) →D(grB). ✷Definition 9.3. Assume that the mapγ : grD(B) → D(grB) is an isomorphism. Then bythe Lemma 7.1 the algebraD(B) defines an element in defn(D(A)). We callD(B) theinduced(byB) deformation ofD(A). We also say thatB inducesa deformation ofD(A).

Example 9.4. It follows from Lemma 7.1 that the trivial deformation ofA induces a de-formation ofD(A), which is also trivial.

Remark 9.5. It would be interesting to see which deformations ofA induce deformationsof D(A).

9.2. Two lemmas about induced deformations

Assume thatA andB are as above andB induces a deformation ofD(A). Denote theresidue mapτ :D(B) → D(A). Moreover, assume thatD(B) is a split extension ofD(A)

with a splitting homomorphism (ofk-algebras)s :D(A) → D(B). SinceA ⊂ D(A), themap s defines, in particular, a structure of a leftA ⊗k kn-module onB. The next twolemmas will be used in what follows.

Lemma 9.6. (i) The residue mapβ :B → A is a homomorphism of leftA-modules.(ii) B is a freeA⊗k kn-module of rank1.


Proof. (i) Givena ∈A, b ∈ B we need to show thatβ(s(a)b)= aβ(b). This follows fromthe identityτs(a)= a and the commutativity of the diagram

D(B) ×B(τ,β)

D(A)×A

Bβ

A,

where the vertical arrows are the action morphisms.(ii) The A-module mapβ :B → A has a splittingα :A → B, which induces an

isomorphismα ⊗ 1 :A⊗k kn →B of left A⊗k kn-modules. ✷Lemma 9.7. Assume that thek-algebraA is finitely generated. ConsiderB with thestructure of a leftA⊗k kn-module defined above. ThenD(BB) = D(A⊗kknB) as subringsof Endkn(B).

Proof. DenoteA = A ⊗k kn. SinceD(B) is a deformation ofD(A) the graded ringgrD(B) coincides with the subringD(grB) ⊂ Endkn(grB). The isomorphism ofA-mod-ulesAB � A defines an isomorphism of rings

D(AB)� D(A)=D(grB).

Hence, in particular, grD(AB) is a graded submodule of Endkn(grB) and as suchcoincides withD(grB). We conclude that the graded subrings of Endkn(grB), grD(B) andgrD(AB) coincide(=D(grB)). So it suffices to prove the inclusionD(BB)⊂D(AB).

We will prove by descending induction onp that

D(BB,Bt

pB) ⊂D

(AB, At

pB).

It follows from Lemma 7.6(i) that theA- andB-module structure onB coincide modulot .More precisely, ifb ∈ B anda = β(b) ∈A, then

s(a)− b : t•B → t•+1B.

This implies that

D(BB,Bt

nB) =D

(AB, At

nB).

Suppose that we proved the inclusionD(BB,Btp+1B) ⊂ D(AB, At

p+1B). Let a1,

. . . , al be a set of generators of the algebraA. Choosed ∈ Dm(BB,BtpB). Then the

operators

di0...im := [s(ai0), . . . ,

[s(aim), d

]. . .

], ij ∈ {1, . . . , l},


mapB to tp+1B. Sinces(ai) ∈D(BB), also

di0...im ∈ D(BB,Bt

p+1B) ⊂D

(AB, At

p+1B).

Thus there existsN such that everydi0...im ∈ DN(AB, Atp+1B). SinceA is commutative,

this implies that for anyc1, . . . , cm ∈ A

[s(c1), . . . ,

[s(cm), d

]. . .

] ∈ DN(AB, At

p+1B).

But thend ∈ DN+m(AB, AtpB). HenceD(BB,Bt

pB) ⊂D(AB, AtpB), which completes

the induction step and proves the lemma.✷9.3. Sheafification

Let Y be a scheme overk, B is a sheaf ofkn-algebras onY with an isomorphism ofsheaves ofkn-algebras grB � OY ⊗k kn, i.e., B defines an element in defn(OY ). Thenusing the commutator definition as in Section 9.1 above we define the sheafD(B) ofkn-linear differential operators onB. Thus, in particular,D(B) is a subsheaf ofEndkn(B).In this section all the differential operators will bek[t]-linear, so we omit the correspondingsubscript.

As in the ring case we obtain a natural homomorphism of sheaves of gradedkn-algebras(which, probably, is neither injective, nor surjective in general)

γ : grD(B)→ Endkn(grB).

The following two lemmas are the sheaf versions of Lemmas 9.1 and 9.2 which will beused later. The proofs are the same.

Lemma 9.8. D(OY ⊗k kn)=D(OY )⊗k kn (=DY ⊗k kn).

Lemma 9.9. The homomorphismγ mapsgrD(B) to D(grB).

Definition 9.10. Assume thatγ : grD(B) → D(grB) is an isomorphism. Then byLemma 9.8 the sheafD(B) defines an element in defn(DY ). We callD(B) the induced(byB) deformation ofDY and say thatB inducesthis deformation.

9.4. Deformations of differential operators on a flag variety

Theorem 9.11. Let G be a complex linear simple simply connected algebraic group,B ⊂ G—a Borel subgroup,X = G/B—the corresponding flag variety. Then any induceddeformation ofDX is trivial.

Remark 9.12. SinceH 1(X,TX) = 0 (the varietyX is rigid) the only deformations ofOX

are “purely noncommutative,” i.e., they correspond to elements ofH 0(X,∧2TX). In this


respect one may ask the following question: SupposeY is a smooth projective variety,B isa purely noncommutative deformation ofOY . Assume thatB induces a deformationD(B)of DY . IsD(B) a trivial deformation ofDY ?

Proof. Assume that a sheaf ofkn-algebrasB, which represents an element in defn(OX),induces a deformation (of ordern) D(B) of DX . Then for anym> 0 the sheafB/tm+1Binduces a deformation of orderm, D(B/tm+1B), of DX . By induction we may assumethatD(B/tnB)�DX ⊗k kn−1, i.e.,D(B) represents an element in defn

0 (DX). (Recall thatdefn0 (DX)� def(DX).) We need to prove thatD(B) is the trivial element in defn(DX). Forsimplicity of notation we assume thatn = 1 (the proof in the general case is the same).

It is well known thatX has an open coveringX = ⋃w∈W Uw, whereW is the Weyl

group ofG andUw � Cd , d = dim(X). Denote the coveringU = {Uw}. It follows fromExample 8.3 thatD(B)Uw is the trivial deformation ofDUw for eachw ∈W .

The varietyX is D-affine [BB], thus def(DX) � H 2(Xan,C) (Corollary 8.2). ButH 2(Xan,C)=H 1,1(Xan,C)= Pic(X)⊗Z C. Let us describe the isomorphismσ : Pic(X)⊗C → def(DX) directly. LetL be a line bundle onX. ThenL|Uw � OUw for all w ∈ W .HenceL is defined by aCech 1-cocycle{fij ∈ O∗

Uwi∩Uwj

}. Define derivations

δij :DUwi∩Uwj

→ DUwi∩Uwj

by δij (d)= [d, log(fij )

].

Note that though log(fij ) is a multivalued analytic function,[· , log(fij )] is a well-definedderivation of the ring of differential operators and it preserves the algebraic operators. Soδij is well defined. Using these derivations, we define the glueing overUw1 ∩ Uw2 of thesheavesDUwi

⊗ C[t]/(t2) andDUwj⊗ C[t]/(t2). We denote the corresponding global

sheafσ(L). The mapσ : Pic(X) → def(DX) is a group homomorphism which extends toan isomorphism

σ : Pic(X)⊗ C∼−→ def(DX).

Let us get back toD(B) ∈ def(DX). By the above isomorphism,D(B)= σ(L) for someL ∈ Pic(X) ⊗ C. We haveD(B)Uw = DUw ⊗ C[t]/(t2), so thatBUw has a structure of aDUw -module and, in particular, of anOUw -module. By (a sheaf version of) Lemma 9.6(ii)BUw � OUw ⊗ C[t]/(t2) as anOUw -module. Since the glueing of differentD(B)Uw ’s isby means of derivations[· , log(fij )], it follows that the localOUw -module structure onBagree on the intersectionsUwi ∩ Uwj . HenceB is anOX-module, which fits in the shortexact sequence ofOX-modules

0 →OX → B →OX → 0.

Since Ext1OX(OX,OX) = 0, B = OX ⊗ C[t]/(t2). Thus D(OX

B) = DX ⊗ C[t]/(t2).But by (a sheaf version of) Lemma 9.7D(OX

B) = D(BB) (= D(B)), which proves thetheorem. ✷


Acknowledgments

It is my pleasure to thank Paul Bressler for his references to the literature on thedeformation theory and Michael Larsen for helpful discussions of the subject. I also thankthe referee for useful comments.

References

[BB] A. Beilinson, J. Bernstein, Localisation desg-modules, C. R. Acad. Sci. 292 (1981) 15–18.[Bo] A. Borel, et al., AlgebraicD-Modules, Academic Press, Boston, 1987.[GS] M. Gerstenhaber, S.D. Schack, Algebraic cohomology and deformation theory, in: Deformation Theory of

Algebras and Structures and Applications, in: NATO Adv. Sci. Inst. Ser. C, Vol. 247, 1986, pp. 11–264.[LR] V.A. Lunts, A.L. Rosenberg, Localization for quantum groups, II, in preparation.[S] R.G. Swan, Hochschild cohomology of quasiprojective schemes, J. Pure Appl. Algebra 110 (1996) 57–80.

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