GAGA for DQ-algebroids

GAGA for DQ-Algebroids

HOU-YICHEN(*)

ABSTRACT- LetXbe a smooth complex projective variety with associated compact complex manifoldXan. IfAX is a DQ-algebroid onX, then there is an induced DQ-algebroidAXan onXan. We show that the natural functor from the derived category of bounded complexes ofAX-modules with coherent cohomologies to the derived category of bounded complexes of AXan-modules with coherent cohomologies is an equivalence.

Introduction.

Let X be a projective scheme over the complex number fieldC with associated complex analytic space X_an. Serre's GAGA paper [11] asserts that the category of coherent sheaves onXis equivalent to the category of coherentanalytic sheaves onX_an.

We consider the case whereXis a smooth algebraic variety overCor a complex manifold. In [8], the authors defined a DQ-algebraAX onXas a sheaf ofC^h:ˆC[[h]]-algebras locally isomorphic to (O_X[[h]]; ?) where?is a star product. The authors also defined a DQ-algebroid as aC^h-algebroid stack locally equivalent to the algebroid associated with a DQ-algebra. If AX is a DQ-algebroid onX, then we have the notion ofAX-modules. We denote by Mod(AX) the category of AX-modules and by D^b(AX) it s bounded derived category. We will recall these notions and their proper- ties from [8].

If (X;A_X) is a smooth variety overCendowed wit h a DQ-algebroid, then there is an induced DQ-algebroid AXan on the complex manifold

(*) Indirizzo dell'A.: Institut de MatheÂmatiques Universite Pierre etMarie Curie 175, rue du Chevaleret, 75013 Paris, France and Department of Mathe- matics, National Taiwan University, Taipei, 106, Taiwan.

E-mail address: [email protected] E-mail address: [email protected] Mathematics Subject Classification: 18D05, 32C38, 46L65, 53D55.

Xan. Then we const ruct a funct or f:D^b_coh(AX)!D^b_coh(AXan), where D^b_coh(AX) denotes the full triangulated subcategory of the bounded de- rived category D^b(A_X) with coherent cohomologies and similarly D^b_coh(A_Xan) denotes the full triangulated subcategory of the bounded derived category D^b(A_Xan) with coherent cohomologies. By using Lemma 1.2, Corollary 1.4 and some results in [8], we prove the following theorem:

MAIN THEOREM (See Theorem 4.12). Assume that X is projective.

Then the functor f:D^b_coh(A_X)!D^b_coh(A_Xan)is an equivalence.

This paper is organized as follows: In section 1, we review Serre's GAGA theorem and translate this theorem to the derived version. In section 2, we review some notions and results of DQ-modules from [8]. In particular, Remark 2.5 and Finiteness theorem (Theorem 2.13) are crucial for the paper. In section 3, we show how to induce an analytic DQ-algebroid from an algebraic DQ-algebroid on a smooth variety. In section 4, we prove the main theorem.

Throughout this paper, all varieties (or schemes) are over C if not otherwise specified.

1. Review on the GAGA Theorem.

LetXbe a scheme of finite type and letX_anbe the associated complex analytic space. We denote by Mod(OX) (resp. Mod(OXan)) the category of sheaves onX(resp.Xan). We also denote by Modcoh(OX) and Modcoh(OXan) the full subcategories of Mod(O_X) and Mod(O_Xan) consisting of coherent sheaves, respectively. There is a continuous map W:X_an!X of the un- derlying topological spaces and there is also a natural map of the structure sheaves W ¹OX !OXan: To F 2Mod(OX), one associates its complex analytic sheafF ^an:ˆOXan_W¹_OX W ¹F 2Mod(OXan). Hence we obtain a functor:

Y_X:Mod(O_X)!Mod(O_Xan):

()

IfF is a coherentsheaf, thenF ^anis also coherent.

The following theorem for a projective scheme is proved in Serre's famous paper GAGA (see [11]) which is generalized by Grothendieck for a proper scheme (see [6 XII]).

THEOREM 1.1. Let X be a projective scheme. Then the functor () induces an equivalence of categories

Mod_coh(O_X) ! Mod_coh(O_Xan):

Furthemore,for every coherent sheafF on X,the natural maps Hⁱ(X;F ) ! Hⁱ(Xan;F ^an)

are isomorphisms,for all i0. p

The following lemma is Theorem 2.2.8 in [1].

LEMMA 1.2. Let A⁰ and B⁰ be thick subcategories of abelian cate- goriesAandB,respectively,and letF:A ! Bbe an exact functor that takes A⁰ to B⁰. Assume furthemore that the following properties are satisfied:

1.AandBhave enough injectives,

2.Fis an equivalence of categories when restricted toA⁰! B⁰, 3.Finduces a natural isomorphism

Extⁱ_A(F;G)Extⁱ_B(F(F);F(G)) for any F;G2 A⁰and any i.

Then the natural functor Fe :D^b_A⁰(A) ! D^b_B⁰(B) induced by F is an equivalence of categories.

PROOF. (i) We prove that the functorFeis fully faithful, i.e., for anyF, G2D^b_A⁰(A),Fe induces an isomorphism

Hom_D^b

A0(A)(F;G)Hom_D^b

B0(B)(F(Fe );F(Ge )):

(1:1)

We'll use a technique known as devissageto prove it. The devissagetech- nique is the induction on the numbern(E) defined as

n(E)ˆmaxfj ijH^j(E)6ˆ0;Hⁱ(E)6ˆ0g:

Hence we shall prove (1.1) by induction onNˆn(F)‡n(G). IfNˆ 1, then one ofF orGis the zero complex, so there is nothing to prove. If Nˆ0, then there existF2 A⁰;G2 A⁰such thatFˆF[a] andGˆG[b]

for somea;b2Z. Then Hom_Db

A0(A)(F;G)ˆHom_Db

A0(A)(F[a];G[b])ˆExt^{b a}_A (F;G)

and

Hom_D^b

B0(B)(F(Fe );F(Ge ))ˆHom_D^b

B0(B)(F(F[a]);e F(G[b]))e

ˆExt^{b a}_B (F(F);F(G)):

Hence (1.1) follows from property 3 above.

Assume thatFe induces an isomorphism Hom_D^b

A0(A)(F;G)Hom_D^b

B0(B)(F(Fe );F(Ge ))

for allF;G2D^b_A⁰(A) withn(F)‡n(G)5N, and letF;Gbe objects of D^b_A⁰(A) withn(F)‡n(G)ˆN>0. We may assume thatn(G)ˆN>0 and thatGⁱˆ0 fori50, andH⁰(G)6ˆ0.

Let G⁰ be the complex with single non zero object H⁰(G) in degree zero. From the morphismG⁰!G, there exists a distinguished triangle G⁰⁰!G⁰!G!G⁰⁰[1]. By the long exact cohomology sequence, one deduces n(G⁰⁰)5n(G); also, from the assumption, n(G⁰)ˆ05n(G).

From the long exact sequence of Hom's; the five-lemma and the induction hypothesis, we conclude that

Hom_D^b

A0(A)(F;G)Hom_D^b

B0(B)(F(Fe );F(Ge ))

which is what we needed to prove thatFe is fully faithful. (The case when n(G)ˆ0 butn(F)>0 follows in a similar way.)

(ii) Next, we shall prove that the functorFeis essentially surjective, i.e., any objectG of D^b_B⁰(B) is isomorphic to an object of the formF(Fe ) for some F2D^b_A⁰(A). We prove this by induction on nˆn(G). The case nˆ 1is trivial, andnˆ0 follows from property 2.

So assume n>0, and as before construct a distinguished triangle G⁰⁰!G⁰!G!G⁰⁰[1] whereG⁰ˆH⁰(G)6ˆ0 and we assume thatG is zero in degrees50. SinceFis an equivalence of categories betweenA⁰ and B⁰, we can find an F⁰2D^b_A⁰(A) such thatF(F⁰)G⁰. Also, by in- duction hypothesis, we can find anF⁰⁰2D^b_A⁰(A) such thatF(Fe ⁰⁰)G⁰⁰. Since we proved thatFeis fully faithful, we can find a mapF⁰⁰!F⁰whose image by Fe is the side of the distinguished triangle constructed before.

Again, we have the distinguished triangle F⁰⁰!F⁰!F!F⁰⁰[1].

SinceFeis a derived functor, it is a triangulated functor. Hence, we see that

F(Fe ) is isomorphic toG, as required. p

The following proposition is well known (see, e.g., [9]).

PROPOSITION1.3. LetRbe a sheaf of rings on a topological space X.

Then the categoryMod(R)ofR-modules is a Grothendieck category. In particular,Mod(R)has enough injectives.

LetXbe a scheme of finit e t ype. Set A:ˆMod(O_X),B:ˆMod(O_Xan), A⁰:ˆModcoh(OX) andB⁰:ˆModcoh(OXan). We also setD^b_coh(X):ˆD^b_A⁰(A) and D^b_coh(Xan):ˆD^b_B⁰(B). Clearly,A⁰andB⁰are full thick subcategories ofA andB, respect ively.

As an application of Lemma 1.2, we have the following.

COROLLARY1.4. Let X be a projective scheme. Then the functorY_X of () induces an equivalence (we keep the same notation)

Y_X :D^b_coh(X) ! D^b_coh(X_an):

PROOF. We shall apply Lemma 1.2. First, note thatOXan;xis a flatOX;x- module for eachx2X. Hence the functorY_X :A ! Bis an exactfunctor.

BothAandBhave enough injectives by Proposition 1.3. Hence condition 1 is satisfied and condition 2 is Theorem 1.1. To check condition 3, it is suf- ficient to prove that

RHom(F ;G)'RHom(F ^an;G^an) forF ;G 2 A⁰: (1:2)

Since RHom(F ;G)'RG(X;F ^L

OX

G) and RHom(F ^an;G^an)'RG(X_an; (F ^an) ^L

OXan

G^an), whereF ˆRHomOX(F ;O_X) and (F ^an)ˆRHomOXan

(F ^an;O_Xan), we reduce (1.2) to the following isomorphism RG(X;F )'RG(X_an;F ^an); whereF 2D^b_coh(X):

(1:3)

Now, there is a morphism RG(X;F )!RG(Xan;F ^an) forF 2D^b_coh(X) and itis an isomorphism by Theorem 1.1. Hence the resultfollows. p

2. Review on DQ-modules (after K-S).

In this section, we recall some notions and results from [8].

Algebroid.

In this subsection, we denote byXa topological space and byKa com- mutative unital ring. IfAis a ring, anA-module means a leftA-module. Recall

that the notion algebroid was first introduced by Kontsevich [10], see also [2]

and [7]. AK-algebroidAonXis aK-linear stack locally non empty and such that for any open subsetUofX, two objects ofA(U) are locally isomorphic.

Let U = fU_ig_i2I be an open covering of X. In the sequel, we set U_ij:ˆU_i\U_j,U_ijk:ˆU_i\U_j\U_k, et c.

Consider the data of

aK-algebroidA onX

si2A(U) and isomorphismsW_ij:sjj_Uij!sij_Uij: (

(2:1)

To these data, we associate:

A_iˆEnd_K(s_i),

fij:Ajj_Uij !Aij_Uij theK-algebra isomorphisma7!W_ijaW_ij¹, a_ijk, the invertible element ofA_i(U_ijk) given byW_ijW_jkW_ik¹. Then:

f_ijf_jkˆAd(a_ijk)f_ik aijkaiklˆfij(ajkl)aijl:

(2:2)

(Recall that Ad(a)(b) =aba ¹).

Conversely, let A_i be sheaves of K-algebras on U_i (i2I), let f_ij:A_jj_Uij !A_ij_Uij (i;j2I) be K-algebra isomorphisms, and let a_ijk (i;j;k2I) be invertible sections ofA_i(U_ij) satisfying (2.2). One calls:

(fAig_i2I;ffijg_i;j2I;faijkg_i;j;k2I) (2:3)

a gluing datum forK-algebroids onU.

THEOREM 2.1 ([5]). Assume that the topological space X is para- compact. Considering a gluing datum (2.3) on U. Then there exist an algebroidAon X andfsi;W_ijgi;j2Ias in(2.1)to which this gluing datum is associated. Moreover,the data(A;si;W_ij)are unique up to an equivalence of stacks,this equivalence being unique up to a unique isomorphism.

In general, if a topological spaceXis notparacompact, for example for algebraic varieties, then Theorem 2.1 may be false. Hence we need another local description of such algebraic algebroids.

DEFINITION 2.2. Let A and A⁰ be two sheaves of K-algebras. An AA⁰-moduleL is calledbi-invertibleif there exists locally a sectionv ofL such thatA3a7!(a1)v2L andA⁰3a⁰7!(a⁰1)v2L give isomorphism ofA-modules andA⁰-modules, respectively.

LetU=fUig_i2Ibe an open covering ofX. Consider the data of aK-algebroidA onX

s_i2A(U):

(2:4)

To these data, we associate:

A_iˆEndK(s_i),

L_ij:ˆHom_AijUij(s_jj_Uij;s_ij_Uij), henceL_ijis a bi-invertibleA_iA^op_j - module onU_ij.

the natural isomorphismsa_ijk:L_ijAjL_jk !L_ik: Hence we obtain:

(fA_ig_i2I;fL_ijg_i;j2I;fa_ijkg_i;j;k2I) (2:5)

an algebraic gluing datum forK-algebroids onU.

THEOREM 2.3 ([8] Proposition 2.1.13). Consider an algebraic gluing datum(2.5)onU. Then there exist an algebroidAon X andfsi;W_ijg_i;j2I as in (2.1)to which this gluing datum is associated. Moreover,the data (A;s_i;W_ij) are unique up to an equivalence of stacks,this equivalence being unique up to a unique isomorphism.

For an algebroid A, one defines the Grothendieck K-linear abelian category Mod(A), whose objects are calledA-modules, by setting:

Mod(A):ˆFct_K(A;Mod(K_X)):

HereMod(K_X) is t heK-linear stack of sheaves ofK-modules onX, and FctKis the category ofK-linear functors of stacks.

We have the well defined notion of tensor product for twoK-algebroids C and C⁰, say C KC ⁰. For a K-algebroid A, Mod(AKA^op) has a canonical objectgiven by

A_KA^op3(s;s^0op)7!Hom_A(s⁰;s)2Mod(K_X):

We denote this object by the same letterA.

ForK-algebroidsA_i(iˆ1;2;3), we have functors:

A2:Mod(A_1KA^op₂ )Mod(A_2KA^op₃ )!Mod(A_1KA^op₃ ) and

HomA₁(;):Mod(A1KA^op₂ )^opMod(A1KA^op₃ )!Mod(A2KA^op₃ ):

In particular, we have

A:Mod(A^op)Mod(A)!Mod(K_X)

and

HomA(;):Mod(A)^opMod(A)!Mod(KX):

LetYbe another topological space. Letf :X7!Y be a continuous map and letAbe aK-algebroid onY. We denote byf ¹AtheK-linear stack associated with the prestackSgiven by

S(U)ˆ f(s;V); V is an open subsetofY such thatf(U)V and s2A(V)gfor any open subset U of X;

Hom_S(U)((s;V);(s⁰;V⁰))ˆG(U;f ¹HomA(s;s⁰)):

Thenf ¹Ais aK-algebroid onX.

Notations: For the rest of this section, we denote by X a complex manifold or a smooth variety and byC^h:ˆC[[h]] the power series algebra.

InvertibleO_X-algebroids.

DEFINITION 2.4. AC-algebroidP onX is called an invertible OX- algebroid if for any open subset U of X and any s2P(U), there is a C-algebra isomorphism End_P(s)'O_U:

We shall state some properties for invertibleO_X-algebroids.

Let P be an invertible OX-algebroid. Then for any s;s⁰2P(U), Hom(s;s⁰) is an invertibleOU-module.

For two invertible O_X-algebroids P1 and P2. We denot e by P_1OXP₂theC-linear stack associated with the prestack whose objects over an open set U is P₁(U)P₂(U), and Hom((s₁;s₂);(s⁰₁;s⁰₂))

ˆHom(s1;s⁰₁)OX Hom(s2;s⁰₂). Then P1OXP2 is an invertible OX- algebroid. Note that the set of equivalence classes of invertible OX-al- gebroids has structure of an additive group by the operation OX , and this group is isomorphic toH²(X;O_X) ([4], [9]).

The following remark is due to Prof. Joseph Oesterle and is crucial for the paper.

REMARK2.5. For a smooth algebraic varietyXas Zariski topology over C, the groupH²(X;O_X) is trivial. Hence any invertibleO_X-algebroidP is equivalenttoO_X.

We sketch the proof of it. LetK be the field of rational functions onX

and letK_Xbe the constant sheaf with stalk the abelian groupK. Denote by X1ˆ fx2XjdimOX;xˆ1g(or the set of closed irreducible hypersurfaces ofX). Letx2X₁, sinceXis a variety, the ringO_X;xis a DVR with valuation y_xand quotient fieldK. LetZ_xˆ(i_x)(Z) wherei_x:x!Xand letUX be an open set, thenZ_x(U)ˆ0 ifx2= UandZ_x(U)ˆZifx2U. Consider the sheaf L

x2X1

Zx, then L

x2X1

Zx

(U)ˆ L

x2X1

Zx(U)ˆZ^U\X1. Hence we can define a morphism of sheaves

y:K_X!M

x2X1

Zx

by: y(f)ˆ(y_x(f))_x2X1\U where U is a nonempty open subset of X and f 2K_X(U)ˆK. Then one has an exactsequence

0!O_X !^u K_X !^y M

x2X1

Z_x!0

whereuis the natural morphism. SinceK_X is constant, it is a flabby sheaf for the Zariski topology. On the other hand, the sheaf L

x2X1

Zxis also flabby.

Itfollows thatH^j(X;O_X) is zero forj>1.

Letf :X!Ybe a morphism of complex manifolds or smooth varieties.

For an invertibleO_Y-algebroidP_Y, we denote byfP_YtheC-linear stack onXassociated with the prestack whose objects on Uare the objects of (f ¹PY)(U) andHom(s;s⁰)ˆOX_f ¹_OYHom_f ¹_PY(s;s⁰). ThenfPY is an invertibleO_X-algebroid.

Star-products.

Let X be a complex manifold (or a smooth variety). We denote by d_X :X,!XXthe diagonal embedding and we set4_X ˆd_X(X). We de- note byO_Xthe structure sheaf onX, byV_Xthe sheaf of differential forms of maximal degree and byUXthe sheaf of vector fields. As usual, we denote byDX the sheaf of rings of differential operators onX. Recall that a bi- differential operator P onX is a C-bilinear morphism O_XO_X !O_X which is obtained as the compositiond_X¹P~ whereP~ is a differential op- erator onXXdefined on a neighborhood of the diagonal andd ¹is the restriction to the diagonal:

P(f;g)(x)ˆ(P(x~ 1;x2;@x1; @x2)(f(x1)g(x2))j_x1ˆx2ˆx:

Hence the sheaf of bi-differential operators is isomorphic toD_XOXD_X,

where the both DX are regarded as OX-modules by the left multi- plications.

DEFINITION 2.6. A star algebra on O_X[[h]] is a C^h-bilinear sheaf morphism

?:O_X[[h]]O_X[[h]]!O_X[[h]]

satisfying the following conditions:

(i) the star product makesO_X[[h]] into a sheaf of associated unitalC^h- algebra with unit 12O_X:

(ii) there is a sequence P_i:OX OX !OX of bi-differential opera- tors, such that for any two local sectionsf;g2O_X one has

f?gˆfg‡X¹

iˆ1

P_i(f;g)hⁱ:

Note that f?gfg mod h, and P_i(f;1)ˆP_i(1;f)ˆ0 for all f and i>0. We call (O_X[[h]]; ?) a star algebra.

DQ-algebras.

DEFINITION 2.7. A DQ-algebra A on X is a C^h-algebra locally iso- morphic to a star-algebra (OX[[h]]; ?) as aC^h-algebra.

Clearly, a DQ-algebra is a sheaf ofh-adically complete flatC^h-algebra onXsatisfyingA=hA 'O_X. Note also that for an algebraic varietyX, a DQ-algebraA is calleddeformation quantizationofO_X in [3] and [12].

REMARK 2.8. For a smooth projective varietyX, there exists a DQ- algebraA_X onX. For details, one refers to [3].

DQ-algebroids.

DEFINITION2.9. A DQ-algebroidA onXis aC^h-algebroid such that for each open setUXand eachs2A(U), theC^h-algebraHom_A(s;s) is a DQ-algebra onU.

Let AX be a DQ-algebroid on X. For an AX-module M, the local notions of being coherent or locally free, etc. make sense.

The category Mod(AX) is a Grothendieck category and we denote by D(AX) its derived category and by D^b(AX) its bounded derived category.

We also denote by D^b_coh(A_X) the full triangulated subcategory of D^b(A_X) consisting of objects with coherent cohomologies.

Graded modules.

Let A_X be a DQ-algebroid onX. Letus denote by gr(A_X) t heC-al- gebroid associated with the prestackSgiven by

Ob(S(U))ˆOb(AX(U)) for an open subsetU of X;

Hom_S(s;s⁰)ˆHom_AX(s;s⁰)=hHom_AX(s;s⁰) fors;s⁰2A_X(U):

Then it is easy to see that gr(AX) is an invertibleOX-algebroid and the left derived functor of the right exact functor Mod(A_X)!Mod(gr (A_X)) given byM !M=hMis denoted by gr:D^b(A_X)!D^b(gr (A_X)).

The functor gr induces a functor (we keep the same notation):

gr:D^b_coh(AX)!D^b_coh(gr (AX)):

(2:6)

The following lemma is in [8].

LEMMA2.10. The functorgrof(2.6)is conservative (i.e.,a morphism in D^b_coh(AX) is an isomorphism as soon as its image by gr is an iso- morphism inD^b_coh(gr (AX)).

Denote by D^b_f(C^h):ˆD^b_coh(C^h) and D^b_f(C):ˆD^b_coh(C) the full triangu- lated subcategories of D^b(C^h) and D^b(C) consisting of objects with finitely generated cohomologies, respectively.

Hence we have a well defined functorC^L_C^h :D^b_f(C^h)!D^b_f(C). As an application of Lemma 2.10, we get the following.

COROLLARY 2.11. The functor C^L_Ch :D^b_f(C^h)!D^b_f(C). is con- servative.

PROOF. Applying the functor gr in Lemma 2.10 toXˆ fptg. p The following proposition is in [8] which will be used in Theorem 4.2.

PROPOSITION 2.12. Let (X_i;A_Xi) be complex manifolds or smooth varieties endowed withDQ-algebroidsA_Xi (iˆ1;2;3).

(i) LetKi2D^b(A_XiX^a_i‡1) (iˆ1;2). Then

gr (K₁^L_A2K₂)'gr (K₁)^L_gr(A2)gr (K₂):

(ii) LetK_i2D^b(A_XiXi‡1) (iˆ1;2). Then

gr RHom_A2(K₁;K₂)'RHom_{gr (A2)}(gr (K₁);gr (K₂)):

Finiteness for DQ-kernels.

Recall that we have the following Finiteness theorem.

FINITENESS THEOREM 2.13 ([8]). Let (X;AX)be a compact complex manifold or a smooth projective variety endowed with a DQ-algebroid A_X. Let M and N be two objects of D^b_coh(A_X). Then the object RHom_AX(M;N )belongs toD^b_f(C^h).

3. Analytization of a DQ-algebroid.

In this section, we denote byXa smooth algebraic variety. LetXanbe the corresponding complex analytic manifold of X with continuous map f :X_an!X.

Let A_X be a DQ-algebroid on X and letUˆ fU_ig_iˆ1;...;n be a finite affine open covering ofX. Consider the data:

aC-algebroidA_X onX si2AX(Ui):

(

Then by Theorem 2.3, we have the following gluing data:

A_i:ˆEnd_A(s_i)ˆ(O_Ui[[h]]; ?_i),

f_ij:A_jj_Uij !A_ij_Uij theC^h-algebra isomorphism, aijk: invertible elements ofAi(Uijk)

which satisfies:

f_ijf_jkˆAd(aijk)fik

a_ijka_iklˆf_ij(a_jkl)a_ijl: (

SinceA_iˆ(O_Ui[[h]]; ?_i) is a star algebra for eachiˆ1; ;n, by definition,

we have

f_i?_ig_iˆf_ig_i‡X¹

jˆ1

b_j(f_i;g_i)h^j

for f_i;g_i2C_i:ˆG(U_i;O_X)and b_j :O_UiO_Ui!O_Ui is a bi-differential operators for eachj.

From the inclusionC_i,!C^an_i :ˆG(U_i;O_Xan), we can define a star pro- duct?^an_i on the analytic sheafO_Ui t o be

f_i^an?^an_i g^an_i ˆf_i^ang^an_i ‡X¹

jˆ1

b^an_j (f_i^an;g^an_i )h^j forf_i^an;g^an_i 2C^an_i ;

whereb^an_j :O_UiO_Ui!O_Uiis a bi-differential operators on the analytic sheaf OUi for each j. Hence, we obtain the (analytic) star algebra A^an_i ˆ(OUi[[h]]; ?^an_i ) for eachi.

Therefore, we get the corresponding descent data onX_an: A^an_i ˆ(OUi[[h]]; ?^an_i ),

f_ij^an:A^an_j j_Uij!A^an_i j_Uij theC^h-algebra isomorphism, a^an_ijk, the invertible element ofA^an_i (U_ijk)

and we obtain the DQ-algebroidA_XanonX_anby Theorem 2.1 (note thatX_an is paracompact).

Hence for a DQ-algebroidAX onX, we have the induced analytic DQ- algebroidAXan onXan.

Furthemore,

AXan 2Mod(f ¹AX_C^hA^op_Xan):

Hence for a DQ-algebroid A_X on a smooth variety X, we have the functor f :ˆA_Xanf ¹_(AX)f ¹():Mod(A_X)!Mod(A_Xan) which sends MtoA_Xanf 1(AX)f ¹(M). Denote by Mod_coh(A_X) and Mod_coh(A_Xan) t he categories consisting of coherent A_X-modules and A_Xan-modules, re- spectively. IfM 2Modcoh(AX), thenf(M)2Modcoh(AXan).

4. The main theorem.

In this section, we prove the main theorem of this paper. LetA_X be a DQ-algebroid on a smooth algebraic varietyX.

Flatness.

Let Xan be the corresponding complex analytic manifold of X with continuous map f :X_an!X. First, we need the following lemma. The following lemma over one pointas a corollary of Theorem 1.6.5 of [8].

LEMMA 4.1. The functor f:Mod(AX)!Mod(AXan) constructed above is exact.

PROOF. We may assume thatA_XandA_Xanare DQ-algebras. We need to show thatB:ˆA_Xan;xis flatoverR:ˆA_X;xfor eachx2X. Note that:

(a)Bhas noh-torsion,

(b)B0:ˆB=hBˆOXan;xis a flatR0:ˆR=hRˆOX;x-module, (c)B'lim

n

B=hⁿB.

Hence applying Theorem 1.6.5 of [8] toXˆ fptg, one gets the result. p From Lemma 4.1, one can see that the functor f:Mod(A_X)!

!Mod(AXan) induces a functor (we keep the same notation):

f:D^b_coh(A_X) ! D^b_coh(A_Xan):

Fully faithfulness.

Now we can prove the following theorem.

THEOREM4.2. Let X be a smooth projective variety,then the functor f: D^b_coh(AX)!D^b_coh(AXan)is fully faithful.

PROOF. For any M;N 2D^b_coh(A_X), we need to show that the morphism

Hom_D^b

coh(AX)(M;N ) ! Hom_D^b

coh(AXan)(f(M);f(N )) (4:1)

is a bijection. In order to show that the morphism of (4.1) is a bijection, it is sufficient to show that the morphism

RHom_Db

coh(AX)(M;N )!RHom_Db

coh(AXan)(f(M);f(N )) (4:2)

is an isomorphism. Since X is projective, by Theorem 2.13, the com-

plexes RHom_D^b

coh(AX)(M;N ) and RHom_D^b

coh(AXan)(f(M);f(N )) belong t o D^b_f(C^h). Moreover, since X is a smooth variety and gr (A_X) is an invertible O_X-algebroid, gr (A_X) is equivalentto O_X by Remark 2.5 and hence gr (A_Xan) is equivalentto O_Xan. Thus, the equivalence D^b_coh(OX) ! D^b_coh(OXan) (Corollary 1.4) implies that the following morphism

(4:3) RHom_D^b_{coh(gr (AX))}(grM;grN )!RHom_D^b_{coh(gr (AXan))}(f(grM);f(grN )) is an isomorphism in D^b_f(C). Applying the functorC^L_C^h to (4.2) and using Proposition 2.12, we get (4.3). Since the functorC^L_C^h is conservative by Corollary 2.11, the morphism of (4.2) is an isomorphism and the result

follows. p

COROLLARY4.3. Let X be a smooth projective variety,then the natural functor f :Mod_coh(AX)!Mod_coh(AXan)is exact and fully faithful.

For eachn>0, we denote by Mod(AX=hⁿAX) (resp. Mod(AXan=hⁿAXan)) the full subcategory of Mod(AX) (resp. Mod(AXan)) consisting of objectsM such thathⁿ:M !Mis the zero morphism.

Similarly, we denote by Modcoh(AX=hⁿAX) (resp. Modcoh(AX_an=hⁿAX_an)) the full subcategory of Modcoh(AX) (resp. Modcoh(AXan)) consisting of objects M such thathⁿ :M!M is the zero morphism for each n>0.

Therefore, we have a functorf_n:ˆfj_Modcoh(AX=hⁿ_AX):Modcoh(AX=hⁿAX)!

!Modcoh(AX_an=hⁿAX_an) for eachn>0.

Note that fornˆ1, the category Modcoh(AX=h¹AX)'Modcoh(OX) is equivalent to the category Mod_coh(A_Xan=h¹A_Xan)'Mod_coh(O_Xan) by Theorem 1.1.

COROLLARY4.4. Let X be a smooth projective variety,then the functor f_n:Mod_coh(A_X=hⁿA_X)!Mod_coh(A_Xan=hⁿA_Xan)is exact and fully faith- ful for each n>0.

Essential surjectivity.

Denote byXa smooth projective variety. Next, we shall prove that the functorf: D^b_coh(AX)!D^b_coh(AXan) is essentially surjective.

We first prove that the functor f_n : Mod_coh(AX=;hⁿAX)!

!Modcoh(AXan=hⁿAXan) is essentially surjective for each n>0. We need the following lemma.

LEMMA4.5. LetA⁰andB⁰be thick subcategories of abelian categories AandB,respectively. LetF:A ! Bbe an exact functor which takesA⁰to B⁰ and such that the natural functor (we keep the same notation) F:D^b_A⁰(A) ! D^b_B⁰(B) induced by F is fully faithful. Consider an exact sequence inB

0!F(M⁰)!N!F(M⁰⁰)!0;

()

with M⁰;M⁰⁰2 A⁰and N2 B⁰.

Then there exists a commutative diagram

for some M2 A⁰(note that the middle arrow is an isomorphism).

PROOF. Since () is an exactsequence, we getthe morphism v:F(M⁰⁰)!F(M⁰)[1]ˆF(M⁰[1]) in D^b_B⁰(B). SinceFis fully faithful, there exists a morphismu:M⁰⁰!M⁰[1] in D^b_A⁰(A) such thatvˆF(u). Consider the distinguished triangle

M⁰⁰ !^u M⁰[1] !L ^‡1!

in D^b_A⁰(A) induced byuwithL2D^b_A⁰(A). Then from the long exact sequence !Hⁱ(M⁰⁰) !Hⁱ(M⁰[1]) !Hⁱ(L) !Hⁱ(M⁰⁰[1]) ! ; we get Hⁱ(L)ˆ0 for i6ˆ 1. Hence L[ 1] is isomorphic to H⁰(L[ 1])2 A⁰ in D^b_A⁰(A). Denote by MˆH⁰(L[ 1]), then from the morphism of distinguished triangles

we obtainN'F(M) and the result follows. p

Set A :ˆ Mod (AX=hⁿAX), A⁰ :ˆ Modcoh(AX=hⁿAX), B:ˆ

:ˆMod(AXan=hⁿAXan) and B⁰ˆModcoh(AXan=hⁿAXan). We shall apply Lemma 4.5.

THEOREM 4.6. The functor

f_n : Modcoh(AX=hⁿAX) !Modcoh(AXan=hⁿAXan) is essentially surjective for each n>0.

PROOF. We shall prove this by induction.

Whennˆ1, itis Theorem 1.1.

We shall prove the theorem forn>1.

For anyM^an2Modcoh(AXan=hⁿAXan), consider the exact sequence 0!hM^an!M^an!M^an=hM^an!0

wherehM^an2Modcoh(AXan=h^{n 1}AXan) andM^an=hM^an2Modcoh(OXan).

Denote by M^an₁ ˆhM^an and by M^an₂ ˆM^an=hM^an. By induction hypothesis, there existsM12Mod_coh(A_X=h^{n 1}A_X) such thatf_{n 1}(M1)' 'M^an₁ . On the other hand, by Theorem 1.1, there existsM₂2Mod_coh(O_X) such thatf₁(M₂)'M^an₂ . Sincefj_A :A ! Bis exactby Lemma 4.1 and the functor D^b_A⁰(A)!D^b_B⁰(B) induced by fj_A is fully faithful by the proof of Theorem 4.2, applying Lemma 4.5, we obtain the following commutative diagram

for someM2 A⁰. Hencef_nis essentially surjective. p From Corollary 4.4 and Theorem 4.6, we obtain the following.

THEOREM4.7. The functor

f_n:Modcoh(AX=hⁿAX)!Modcoh(AXan=hⁿAXan) is an equivalence for each n>0.

In order to prove that the functor f: D^b_coh(AX) !D^b_coh(AXan) is es- sentially surjective, we need the notion of projective limit in the 2-category Cat. For its definition, we refer to [9] Definition 19.1.6.

Recall that a presiteXis nothing but a category which we denote byCX. IfSis a prestack onX, then we have the morphismu:U1!U2inCX, and the functorru:S(U₂)!S(U₁) for anyU₁;U₂ 2 C_X.

Denote byNthe set of positive integers, viewed as a category defined by Ob(N)ˆN

HomN(i;j)ˆ fptg if ij;

; otherwise.

(

We define prestacksSandSan onNas follows:

S(n):ˆModcoh(AX=hⁿAX) for anyn2N,

r_u:S(j)!S(i) is the functor for anyijandu2Hom_N(i;j), and

San(n):ˆModcoh(AXan=hⁿAXan) for anyn2N,

ru:San(j)!San(i) is the functor for anyijandu2HomN(i;j).

The following lemma shows that coherentA-modules areh-complete.

LEMMA 4.8 ([8]). Let (X;A_X) be a complex manifold or a smooth variety endowed with aDQ-algebroidAX. LetfMng_n0 be a projective system of coherent AX-modules. Assume that h^n‡1Mn ˆ0 and the induced morphism Mn‡1=h^n‡1Mn‡1!Mn is an isomorphism for any n0. Then M:ˆlim

n

M_n is a coherent A_X-module and M=h^n‡1M !M_n is an isomorphism for any n0.

We need the following theorem.

THEOREM4.9. We have the following equivalences:

(1) lim

n2N ƒƒS(n) ! Mod_coh(A_X), (2) lim

n2N ƒƒSan(n) !Mod_coh(A_Xan):

PROOF. We only need to prove (1) and (2) can be proved similarly. Let M 2Mod_coh(AX), then we obtain the familyf(Fn;W_u)gwhere:

(i) F_n :ˆM=hⁿM2S(n) for anyn2N,

(ii) W_u :r_uF_j !F_i for any ij andu2Hom_N(i;j) andr_u:S(j)!

!S(i) is defined by sendingM toM=hⁱM.

It is easy to check thatf(Fn;W_u)gsatisfies the cocycle condition (a) of [9]

19.1.6 and hencef(Fn;W_u)g 2 lim

n2N ƒƒS(n).

Let M;M⁰2Mod_coh(AX), then these define two objects Fˆ

ˆ f(F_n;W_u)g and F⁰ˆ f(F⁰_n;W⁰_u)g in lim

n2N ƒƒS(n). Let f :M!M⁰2 2Mod_coh(A_X), then we have the set of families ff_ng_n2N where fn:M=hⁿM !M⁰=hⁿM⁰2Hom_S(n)(M=hⁿM;M⁰=hⁿM⁰):Itis easy to check that ffng satisfies the commutative diagram of definition (b) of [9]

19.1.6 and hence ffng_n2N2Hom_lim

ƒn2N ^S(n)(F;F⁰). Hence we can define a functor F:Mod_coh(A_X)!lim

n2N ƒƒS(n) by sending M to f(F_n;W_u)g and f 2Hom_AX(M;M⁰) t off_ng_n2N.

On the other hand, iff(F_n;W_u)g 2 lim

n2N ƒƒS(n), then by definition (a) of [9]

19.1.6, we have

(i) Fn2S(n) for anyn2N,

(ii) W_u:ruFj ! Fi for any ij and u2HomN(i;j) and ru :S(j)!

!S(i).

Hence the system fF_ng_n2N is a projective system and lim

n2N ƒƒF_n2 2Mod_coh(A_X) by Lemma 4.8.

For two objects Fˆ f(Fn;W_u)g and F⁰ˆ f(F_n⁰;W⁰_u)g in lim

n2N ƒƒS(n). By definition (b) of [9] 19.1.6, Hom_lim

n2Nƒ^S(n)(F;F⁰) is the set of families f ˆ ffng_n2N such that fn2Hom_S(n)(Fn;F_n⁰) and the following diagram commutes for anyu:i!jandij,

Hence the systemf ˆ ffng_n2N is a projective system and we get that the morphism lim

n2Nƒƒ f_n belongs to Mod_coh(A_X). Therefore, we can define a functor C: lim

n2N ƒƒS(n) !Mod_coh(A_X) by C(f(Fn;W_u)g)ˆlim

n2N ƒƒFn and C(f ˆ ffng_n2N)ˆ lim

n2N ƒƒfn. Now it is easy to check thatFC'id_lim ƒn2N ^S(n)

andCF'idModcoh(AX). Therefore, the result follows. p

COROLLARY4.10. The functor f:Modcoh(AX)!Modcoh(AXan) is an equivalence.

PROOF. This follows from Theorem 4.7 and Theorem 4.9. p From Theorem 4.2, Corollary 4.10 and the proof of Lemma 1.2 for es- sentially surjectivity, we obtain what we want mentioned above.

COROLLARY4.11. The natural functor f: D^b_coh(A_X)!D^b_coh(A_Xan)is essentially surjective.

Equivalence.

Therefore, we obtain the main theorem of this paper.

MAINTHEOREM4.12. The natural functor f:D^b_coh(A_X)!D^b_coh(A_Xan) is an equivalence.

PROOF. This follows from Theorem 4.2 and Corollary 4.11. p Acknowledgments. I would like to thank Pierre Schapira for getting me interested in this problem and helping me with suggestions. Also, I like to express my thanks to Jungkai Alfred Chen for teaching me algebraic geometry and supporting me when I studied in Taiwan. Finally, I thank the support of FIT and NTU when I studied in France.

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Manoscritto pervenuto in redazione il 18 maggio 2009.