Fourier-Mukai transform in the quantized setting

We prove that a coherent DQ-kernel induces an equivalence between the derived categories of DQ-modules with coherent cohomology if and only if the graded commutative kernel associated to it induces an equivalence between the derived categories of coherent sheaves.


Introduction
Fourier-Mukai transform has been extensively studied in algebraic geometry and is still an active area of research (see [1] and [4]). In the past years, several works have extended to the framework of deformation quantization of complex varieties some important aspects of the theory of integral transforms. In [6], Kashiwara and Schapira have developed the necessary formalism to study integral transforms in the framework of DQ-modules and some classical results have been extended to the quantized setting. In particular, in [9], Ben-Bassat, Block and Pantev have quantized the Poincaré bundle and shown it induces an equivalence between certain derived categories of coherent DQ-modules.
Our paper grew out of an attempt to understand which properties the integral transforms associated to the quantization of a coherent kernel would enjoy.
The main result of this paper is Theorem 3.16 which states that a coherent DQ-kernel induces an equivalence between the derived categories of DQ-modules with coherent cohomology if and only if the graded commutative kernel associated to it induces an equivalence between the derived categories of coherent sheaves. Whereas the second part of the proof relies on technique of cohomological completion, the first part builds upon the results of [10]. Indeed, as explained in section 2 there is a pair of adjoint functors between the categories of qcc objects and the derived category of quasi-coherent sheaves. Both of these functors preserve compact generators. Then, roughly speaking, to show that a certain property of the quantized integral transform implies a similar properties at the commutative level it is sufficient to check that the category of objects satisfying this properties is thick and that this property hold at the quantized level for a compact generator of the triangulated category of qcc objects. This paper is organized as follow. In the second section we review some material about DQ-modules, cohomological completeness, compactly generated categories, thick subcategories and qcc modules. In the third section, we study integral transforms in the quantized setting. We start by extending the framework of convolutions of kernels of [6] to the case of qcc objects and prove that an integral transform of qcc objects preserving compact objects has a coherent kernel (Thm. 3.12). Then, we concentrate our attention to the case of integral transforms with coherent kernel. We start by extending to DQ-modules some classical adjunction results and then establish the main theorem of this paper. Finally, in an appendix we show that the cohomological dimension of a certain functor is finite.
Aknowledgement: I would like to thank Oren Ben-Bassat, Andrei Cǎldǎraru, Carlo Rossi, Pierre Schapira, Nicolò Sibilla, Geordie Williamson for many useful discussions and Damien Calaque and Michel Vaquié for their careful reading of early version of the manuscript and numerous suggestions which have allowed substantial improvements.
2. Some recollections on DQ-modules 2.1. DQ-modules. We refer the reader to [6] for an in-depth study of DQ-modules. Let us briefly fix some notations. Let (X, O X ) be a smooth complex algebraic variety endowed with DQ-algebroid A X . It is possible to define a quotient algebroid stack A X / A X . It comes with a canonical morphism of algebroid stack A X → A X / A X . On a smooth complex algebraic variety the stack A X / A X is equivalent to the algebroid stack associated to O X . Thus, there is a natural morphism A X → O X of C-algebroid stacks which induces a functor The functor ι g is exact and fully faithful and induces a functor Finally, we have the following proposition.

Cohomologically complete modules.
We briefly present the notion of cohomologically complete module and state the few results that we need. Again, we refer the reader to [6, §1.5] for a detailed study of this notion. We denote by C the ring of formal power series with coefficient in C. Let R be a C -algebroid stack without -torsion. We set R 0 = R/ R and R loc = C ,loc ⊗ C R where C ,loc is the field of formal Laurent's series.  We call this functor the functor of cohomological completion.
The name of functor of cohomological completion is also justified by the fact that (·) cc • (·) cc ≃ (·) cc .
There is a natural transformation It enjoys the following property. is an isomorphism in D(R 0 ).

Compactly generated categories and thick subcategories.
In this subsection, we review a few facts about compactly generated categories and thick subcategories. These facts play an essential role in the proof of Theorems 3.12 and 3.16. A classical reference is [8]. We also refer to [2, § 2].
Definition 2.8. Let T be a triangulated category. Let G = {G i } i∈I be a set of objects of T . One says that G generates T if the following condition is satisfied.
If F ∈ T is such that for every G i ∈ G and n ∈ Z Hom T (G i [n], F ) = 0 then F ≃ 0.

Definition 2.9.
Assume that T is a cocomplete triangulated category.
(i) An object L in T is compact if the functor Hom T (L, ·) commutes with coproducts. We write T c for the full subcategory of T whose objects are the compact objects. (ii) The category T is compactly generated if it is generated by a set of compact objects. The next result is probably well known. We include a proof for the sake of completeness. Proposition 2.12. Let F, G : T → S be two functors of triangulated categories and α : F ⇒ G a natural transformation between them. Then the full subcategory T α of T whose objects are the X such that Proof. The category T α is triangulated and is closed under isomorphism. Let X be an object of T α and Y and Z two objects of T such that X ≃ Y ⊕ Z. By definition of the direct sum there is a map Since α is a natural transformation we have the following commutative diagram.
Thus, Y belongs to T α . It follows that T α is a thick subcategory of T .

Qcc modules.
We review some facts about qcc modules. They may be considered as a substitute to quasi-coherent sheaves in the quantized setting. For a more detailed study one refers to [10]. In this subsection, (X, O X ) is a smooth complex algebraic variety endowed with a DQ-algebroid A X . We denote by D qcoh (O X ) the derived category of sheaves with quasi-coherent cohomology and by D b coh (A X )) the derived category of bounded complexes of O X -modules (resp. A X -modules) with coherent cohomology. One easily shows that the category D qcc (A X ) is a triangulated subcategory of D(A X ).
The functors gr and ι g induce the following functors.
We have the following proposition. (ii) Let us prove the claim for the functor gr .
is fully faithful and exact, we have H i (M) ≃ 0. It follows that M ≃ 0. Moreover, gr G is coherent and on a smooth algebraic variety coherent sheaves are compact.
Remark 2.16. We know by [2] that for a complex algebraic variety the category D qcoh (O X ) is compactly generated by a single compact object i.e by a perfect complex. As shown in [10], this implies in particular that D qcc (A X ) is compactly generated by a single compact object.
Corollary 2.17. Let X be a smooth complex algebraic variety endowed with a DQ-algebroid.
(ii) On a complex smooth algebraic variety the category of compact objects is equivalent to D b coh (O X ). Hence the results follows from Theorem 2.11.
Finally, let us recall the following result from [10].

Fourier-Mukai functors in the quantized setting
The aim of this section is to study integral transforms in the framework of DQ-modules. In the first subsection, we review some results, from [6], concerning the convolution of DQ-kernels. In the second one, we adapt to qcc modules the framework for integral transforms develloped in [6]. We prove that an integral transform preserving the compact objects of the qcc has a coherent kernel. In the last subsection, we focus our attention on integral transforms of coherent DQ-modules on projective smooth varieties. We first extend some classical adjunction results and finally prove that a coherent DQ-kernel induced an equivalence between the derived categories of DQ-modules with coherent cohomology if and only if the graded commutative kernel associated to it induces an equivalence between the derived categories of coherent sheaves.
All along this section we use the following notations. Notation 3.1. (i) If X is a smooth complex variety endowed with a DQ-algebroid A X , we denote by X a the same variety endowed with the opposite DQ-algebroid A op X and we write A X a for this algebroid.
(ii) Consider a product of smooth complex varieties X 1 × X 2 × X 3 , we write it X 123 . We denote by p i the i-th projection and by p ij the (i, j)-th projection (e.g., p 13 is the projection from X 1 × X a 1 × X 2 to X 1 × X 2 ). (iii) We write A i and A ij a instead of A X i and A X i ×X a j and similarly with other products.

Convolution of DQ-kernel.
We review some results, from [6], concerning the convolution of DQ-kernels.
3.1.1. Tensor product and convolution of DQ-kernels. The tensor product of DQ-modules is given by The composition of kernels is given by  (A i(i+1) a ). Assume that X 2 is proper. Then the object Let X i (i = 1, 2, 3) be a smooth projective complex variety endowed with the Zariski topology and let A i be a DQ-algebroid on X i . We recall some duality results for DQ-modules from [6,Chap. 3]. First, we need the following result.
We denote by ω i the dualizing complexe for A i . It is a bi-invertible (A i ⊗ A i a )-module. Since the category of bi-invertible (A i ⊗ A i a )modules is equivalent to the category of coherent A ii a -modules simple along the diagonal, we will regard ω i as an A ii a -module supported by the diagonal and we will still denote it by ω i .

Integral transforms for qcc modules.
In this section, we adapt to qcc objects the framework of convolutions of kernels of [6]. In view of Definitions 3.2 and 3.3, it is easy, using the functor of cohomological completion (see Definition 2.6), to define a tensor product and a composition for cohomologically complete modules.
Proof. Using morphism (2.1), we get a map It induces a morphism By Proposition 1.5.12 of [6] the direct image of a cohomologically complete module is cohomologically complete. Then, This gives us a map Using the fact that the functor gr commutes with direct image and Proposition 2.7, we get the following commutative diagram.
It follows that the morphism gr (( is an isomorphism. Applying Proposition 2.5, we obtain that the morphism (3.1) is an isomorphism.
The second formula is proved similarly.
From now on all the varieties considered are smooth complex algebraic varieties endowed with the Zariski topology Let K ∈ D qcc (A 12 a ). The above corollary implies that the functor (3.2) is well-defined.
Before proving Theorem 3.12, we need to establish the following result.
. It follows immediately from [6,Prop. 1.5.8], that M ∈ D + (A X ). Then to establish that M ∈ D b (A X ), it is sufficient to prove that there exists a number q such that τ ≥q M ∈ D b (A X ). For that purpose, we essentially follow the proof of Proposition 1.5.8 of [6]. Since gr M ∈ D b coh (O X ), there exists p ∈ Z such that for every i ≥ p, H i (gr M) = 0. We deduce from the exact sequence we get the distinguished triangle The module M is cohomologically complete. Hence, we have the iso-

Corollary 4.4 implies that RHom
We now restrict our attention to the case of smooth proper algebraic varieties. The next result is inspired by [12,Thm. 8.15]. Recall that the objects of D b coh (A X ) are not necessarily compact in D qcc (A X ) (see Theorem 2.18). Theorem 3.12. Let X 1 (resp. X 2 ) be a smooth complex algebraic variety endowed with a DQ-algebroid A 1 (resp. A 2 ). Let K ∈ D qcc (A 12 a ). Assume that the functor Φ K : D qcc (A 2 ) → D qcc (A 1 ) preserves compact objects. Then, K belongs to D b coh (A 12 a ). Proof. The kernel gr K induces an integral transform Let G be a compact generator of D qcoh (O 2 ). Then, by Proposition 2.15 ι g (G) is a compact generator of D qcc (A 2 ). By hypothesis, Φ K (ι g (G)) is a compact object of D qcc (A 1 ). It follows that the object Φ gr K (gr ι g (G)) belongs to D b coh (O 1 ) and thus is a compact object of . Applying Theorem 8.15 of [12], we get that gr K is an object of D b coh (O 12 ). Applying Proposition 3.11, we get that K ∈ D b (A 12 a ). Now, Theorem 1.6.4 of [6] implies that K ∈ D b coh (A 12 a ). 3.3. Integral transforms of coherent DQ-modules. In this section we study integral transforms of coherent DQ-modules. Recall that all the varieties considered are smooth complex projective varieties endowed with the Zariski topology.
Proof. It is a direct consequence of Proposition 3.2.4 of [6].
We extend to DQ-modules some classical adjunctions results. They are usually established using Grothendieck duality which does not seem possible to do here. Our proof relies on Theorem 3.6. Definition 3.14. For any object K ∈ D b coh (A 12 a ), we set Proof. We have Applying Theorem 3.6 and the projection formula, we get Taking the global section and applying again the projection formula, we get N ) which proves the claim. The proof is similar for K L .
Finally, we have the following theorem. Theorem 3.16. Let X 1 (resp. X 2 ) be a smooth complex projective variety endowed with a DQ-algebroid A 1 (resp. A 2 ). Let K ∈ D b coh (A 12 a ). The following conditions are equivalent is fully faithful (resp. an equivalence of triangulated categories).
is fully faithful (resp. an equivalence of triangulated categories).
Proof. We recall the following fact. Let F and G be two functors and assume that F is right adjoint to G. Then, there are two natural morphisms (1) (i) ⇒ (ii). Proposition 3.15 is also true for O-modules since the proof works in the commutative case without any changes. Moreover, the functor gr commutes with the composition of kernels. Hence, we have gr (K R ) ≃ (gr K) R . Therefore, the functor Φ gr K R is a right adjoint of the functor Φ gr K . Thus, there are morphisms of functors is an isomorphism. It follows from Proposition 2.12 that T 2 is a thick subcategory of D b coh (O 2 ). Let G be a compact generator of D qcoh (O 2 ). By Corollary 2.17, D b coh (O 2 ) = G . Since Φ K is a fully faithful we have the isomorphism Applying the functor gr , we get that gr ι g (G) belongs to T 2 and by Corollary 2.17, gr ι g (G) is a classical generator of D b coh (O 2 ). Hence, . Thus, the morphism (3.7) is an isomorphism of functors. A similar argument shows that if Φ gr K is an equivalence the morphism (3.6) is also an isomorphism which proves the claim.
Since Φ K and Φ K R are adjoint functors we have natural morphisms of functors Applying the functor gr , we get (3.9) gr M → Φ gr K R • Φ gr K (gr M).
If Φ gr K is fully faithful, then the morphism (3.9) is an isomorphism. The objects Φ K R • Φ K (M) and M are cohomologically complete since they belongs to D b coh (A 2 ). Thus the morphism (3.8) is an isomorphism that is to say It follows that Φ K is fully faithful.
Similarly, one shows that if Φ gr K is an equivalence then in addition It follows that Φ K is an equivalence.
Remark 3.17. The implication (ii) ⇒ (i) of Theorem 3.16 and Proposition 3.15 still hold if one replaces smooth projective varieties by complex compact manifolds. This result implies immediatly that the quantization of the Poincaré bundle constructed in [9] induces an equivalence.

Appendix
In this appendix we show that the cohomological dimension of the functor RHom C X (C ,loc X , M) is finite. We refer to [5] for a detailed account of pro-objects. Recall that to an abelian category C one associates the abelian category Pro(C) of its pro-objects. Then, there is a natural fully faithful functor i C : C → Pro(C). The functor i C is exact. For any small filtrant category I the functor " lim ← − " : Fct(I op , C) → Pro(C) is exact. If C admits small projective limits the functor i C admits a right adjoint denoted π.
If C is a Grothendieck category, then π has a right derived functor R π : D(Pro(C)) → D(C).
Let us recall Lemma 1.5.11 of [6].  Then by Proposition 6.1.9 of [5] adapted to the case of pro-objects, we have It follows from Corollary 13.3.16 from [5] that ∀i > 1, R i π(" lim ← −