HOLONOMIC DQ-MODULES FRANÇOIS PETIT
Abstract. We prove an analogue for holonomic DQ-modules of the codimension-three conjecture for microdifferential modules re-cently proved by Kashiwara and Vilonen. Our result states that any holonomic DQ-module having a lattice extends uniquely be-yond an analytic subset of codimension equal to or larger than three in a Lagrangian subvariety containing the support of the DQ-module.
1. Introduction
Problems of extension of analytic objects have been influential in complex analysis and complex analytic geometry. In 1966, Serre asked the following question in [20]. Let X be an analytic space and let S be a closed analytic subset of X.
Si codim S ≥ 3, est-il vrai que tout faisceau localement libre (ou même seulement réflexif) sur X − S est prolongeable? 1
This question stimulated intense activity around problems of extension of coherent analytic sheaves. One of the major results in this area is an extension theorem due to Trautmann, Frisch-Guenot, and Siu (see [4, 21, 22]) answering Serre’s question.
Theorem 1.1 (Frisch-Guenot, Trautmann, and Siu). Let X be a com-plex manifold, let S be a closed analytic subset of X and j : X \ S → X be the open embedding of X \ S into X. If F is a reflexive coherent
OX\S-module and codim S ≥ 3 then j∗F is a coherent OX-module.
In the 1970’s, Sato, Kashiwara and Kawai introduced and studied systematically the theory of microdifferential systems (see [18]). The codimension-three conjecture was formulated by Kashiwara at the end of the 1970’s and based on his study of holonomic systems with regular singularities. This conjecture is concerned with the extension of a holo-nomic microdifferential system over an analytic subset of the cotangent
The author was supported by the EPSRC grant EP/G007632/1.
1If codim S ≥ 3, is it true that any locally free (or even only reflexive) sheaf on
X − S is extendable ?
bundle of a complex manifold. It was recently proved by Kashiwara and Vilonen (see [14]).
Theorem 1.2 ([14, Theorem 1.2]). Let X be a complex manifold, U an open subset of T∗X, Λ a closed Lagrangian analytic subset of U , and Y a closed analytic subset of Λ such that codimΛY ≥ 3. Let EX
the sheaf of microdifferential operators on T∗X and M be a holonomic
(EX|U \Y)-module whose support is contained in Λ \ Y . Assume that
M possesses an (EX(0)|U \Y)-lattice. Then M extends uniquely to a
holonomic module defined on U whose support is contained in Λ.
The proof of the conjecture was made possible by the deep result of Kashiwara and Vilonen extending Theorem 1.1 to coherent sheaves over O~
X := OX[[~]].
Theorem 1.3 ([14, Theorem 1.6]). Let X be a complex manifold, let Y be a closed analytic subset of X and j : X \ Y → X the open embedding of X \ Y into X. If N is a coherent reflexive O~
X\Y-module
and codim Y ≥ 3 then j∗N is a coherent O~X-module.
The techniques involved in the proof of this result are inspired by ideas coming from deformation quantization (see [13]).
Deformation Quantization modules (DQ-modules for short) on sym-plectic manifolds provide a generalization of microdifferential modules to arbitrary symplectic manifolds (see [13]). It is therefore natural to try to extend the codimension-three conjecture to holonomic DQ-modules, and that is what we do here.
The case of DQ-modules is strictly more general than the case of mi-crodifferential modules. Indeed, in forthcoming work, we will explain how to recover the codimension-three conjecture for formal microdif-ferential operators from the statement for DQ-modules (the converse is not true). Although, our result is more general and does not depend on the codimension-three conjecture for microdifferential modules, it relies fundamentally on the tools elaborated by Kashiwara and Vilo-nen and our proof follows their general strategy. The main difference between the two approaches is that they are working in a conical set-ting whereas we are not. Hence, we have to implement their strategy differently.
DQ-modules in terms of perverse sheaves is not yet clear since the ana-logue of microlocal perverse sheaves is not known for DQ-modules. It is an interesting problem to define such objects.
Here are the precise statements of the results we are proving.
Theorem 1.4. Let X be a complex manifold endowed with a DQ-algebroid stack AX such that the associated Poisson structure is
sym-plectic. Let Λ be a closed Lagrangian analytic subset of X and Y a closed analytic subset of Λ such that codimΛY ≥ 3. Let M be a
holo-nomic (Aloc
X |X\Y)-module, whose support is contained in Λ \ Y . Assume
that M has an AX|X\Y-lattice. Then M extends uniquely to a
holo-nomic module defined on X whose support is contained in Λ.
and
Theorem 1.5. Let X be a complex manifold endowed with a DQ-algebroid stack AX such that the associated Poisson structure is
sym-plectic. Let Λ be a closed Lagrangian analytic subset of X and Y a closed analytic subset of Λ such that codimΛY ≥ 2. Let M be a
holo-nomic AlocX -module whose support is contained in Λ and let M1 be an
Aloc
X |X\Y-submodule of M|X\Y. Then M1 extends uniquely to a
holo-nomic AlocX -submodule of M.
We hope that the submodule version of the codimension-three conjec-ture for DQ-modules may have applications to the representation the-ory of Cherednik algebras, via the localization theorem due to Kashi-wara and Rouquier (see [10]).
Let us describe briefly the proof of the above statements. As we already remarked, we follow the general strategy of Kashiwara and Vilonen (see [14]). Keeping the notations of the theorems 1.4 and 1.5 and denoting by j : X \ Y → X the open embedding of X \ Y into
X, the problem essentially amount to show that j∗M and j∗M1 are
coherent. This is a local question. By adapting a standard technique in several complex variables in [18] to the case of DQ-modules, proving the coherence of a DQ-module amounts to proving the coherence of the pushforward of this module by a certain projection, the restriction of which to the support of the module is finite. Taking the pushforward by the aforementioned projection reduce the non-commutative problem to a commutative one. Then it is possible to use Theorem 1.3 to establish the coherence of the direct image of j∗M and j∗M1 by this specific
projection.
third section, we adapt the coherence criterion from [18] correspond-ing to Proposition 4.1 and 4.4 of [14] to the case of DQ-modules. In [14], it is indicated that the statements for microdifferential modules can be proved by using the Weierstrass preparation and division the-orems for microdifferential operators. In the case of DQ-modules, we follow a different route. We first recall the corresponding statement for coherent analytic sheaves and then lift these statements to the world of DQ-modules. In the fourth section, we prove the analogue of [14, Proposition 4.2] for DQ-modules. The proof is significantly more com-plicated than in the case of microdifferential modules since the duality theory for DQ-modules is entirely stated in terms of kernels and not of morphisms. Thus, one has to identify the actions of all the kernels involved in the duality with their corresponding operations on sheaves. In the short fifth section, we adapt the reduction technique of [18] to a non-conic setting and prove the version of the codimension-three con-jecture for holonomic DQ-modules in the sixth section. Finally, in the last section, we present a conjecture due to Pierre Schapira which ex-tends the codimension-three conjecture for holonomic DQ-modules to all Poisson manifolds.
Acknowledgments. I would like to express my gratitude to Masaki
Kashiwara and Pierre Schapira for many enlightening explanations. It is a pleasure to thanks Iain Gordon for his support and many useful discussions as well as Michael Wemyss, Thibault Lemanissier, Chris Dodd, Gwyn Bellamy and Evgeny Shinder for useful conversation and Will Donovan for proof-reading part of this paper.
2. DQ-modules
In this section, we recall and prove some results concerning the gen-eral theory of DQ-modules.
We denote by C~ := C[[~]] the ring of formal power series with
coefficients in C and by C~,loc:= C((~)) the field of Laurent series with
complex coefficients. In all this section, (X, OX) is a complex manifold.
2.1. DQ-algebras. In this subsection, we review some facts concern-ing DQ-algebras.
We define the following sheaf of C~-algebras
Definition 2.1. A star-product denoted ? on O~
X is a C~-bilinear
mul-tiplicative law satisfying
f ? g =X
i≥0
Pi(f, g)~i for every f, g ∈ OX,
where the Pi are bi-differential operators such that for every f, g ∈
OX, P0(f, g) = f g and Pi(1, f ) = Pi(f, 1) = 0 for i > 0. The pair
(O~
X, ?) is called a star-algebra.
Example 2.2. Let U be an open subset of T∗Cn endowed with a
symplectic coordinate system (x; u) with x = (x1, . . . , xn) and u =
(u1, . . . , un). Then there is a star-algebra (OU~, ?) on U given by
f ? g = X α∈Nn ~|α| α! (∂ α uf )(∂ α xg).
This star-product is called the Moyal-Weyl star-product.
Remark 2.3. On a symplectic manifold all the star-algebras are locally
isomorphic.
Definition 2.4. (i) A DQ-algebra AX on X is a C~-algebra locally
isomorphic to a star-algebra as a C~-algebra.
(ii) For a DQ-algebra AX, we set AlocX := C~,loc⊗ C~
AX.
There is a unique C-algebra isomorphism AX/~AX ∼
−→ OX. We
write σ0 : AX OX for the epimorphism of C-algebras defined by
AX → AX/~AX ∼
−→ OX.
This induces a Poisson bracket {·, ·} on OX defined as follows:
for every a, b ∈ AX, {σ0(a), σ0(b)} = σ0(~−1(ab − ba)).
Notation 2.5. If AX is a DQ-algebra, we denote by AXa the opposite algebra AopX. If X and Y are two manifolds endowed with DQ-algebras AX and AY, then X × Y is canonically endowed with the DQ-algebra
AX×Y := AXAY (see [13, §2.3]).
2.2. DQ-modules. We refer the reader to [13] for an in-depth study of DQ-modules. In this section, we briefly recall some general results that we will subsequently use.
Let (X, OX) be a complex manifold endowed with a DQ-algebroid
stack AX. We denote by Mod(AX) the Grothendieck category of
mod-ules over AX, by D(AX) its derived category, by Modcoh(AX) the
Abelian category of coherent modules over AX and by Dbcoh(AX) the
bounded derived category of DQ-modules with coherent cohomology and if Z is a closed analytic subset of X we denote by Db
full subcategory of Db
coh(AX) the objects of which are supported in Z.
We use similar notations for Aloc
X -modules.
If AX is a DQ-algebroid stack one can associate to it an algebroid
stack gr~AX (see [13, p.69]). If AX is a DQ-algebra, then gr~AX ' OX.
This provides a functor
gr~ : D(AX) → D(gr~AX), M 7→ C L
⊗
C~
M. We denote by f or the forgetful functor f or : Mod(Aloc
X ) → Mod(AX)
the left adjoint of which is given by
(·)loc : Mod(AX) → Mod(AlocX )
N 7→ Aloc
X ⊗
AX N .
Definition 2.6. Let M ∈ Mod(AX). We say that
(i) M has no ~-torsion if the map M→ M is a monomorphism,~ (ii) the module M is of uniform ~-torsion if there exists k ∈ N such
that ~kM = (0). The smallest such a k ∈ N is called the index of ~-torsion of M and is denoted tor~(M),
(iii) M is of ~-torsion if it is locally of uniform ~-torsion. Finally, we recall the following lemma.
Lemma 2.7. Let M ∈ Modcoh(AX). Then M is the extension of a
module without ~-torsion by a ~-torsion module.
2.2.1. Duality theory for DQ-modules. We briefly recall the main fea-tures of the duality theory for DQ-modules and refer the reader to [13, Ch. 6] for a detailed study.
Let X be a complex manifold endowed with a DQ-algebroid stack AX. We denote by δ : X ,→ X × X the diagonal embedding of X
into X × X and set CX := δ∗AX. The object CX is an AX×Xa-module simple along the diagonal.
We denote by ωX the dualizing complex for AX-modules. It is a
bi-invertible (AX ⊗ AXa)-module. Since the category of bi-invertible (AX ⊗ AXa)-modules is equivalent to the category of AX×Xa-modules simple along the diagonal we will regard ωX as an AX×Xa-module sim-ple along the diagonal and will still denote in by ωX. By [13, Theorem
6.2.4], we have
Lemma 2.8. Let X be a complex symplectic manifold. There is a canonical isomorphism
ωX ' CX[dX]
We set
D0AX :D(AX)
op → D(A X)
M 7→ RHomAX(M, AX). We use a similar notation for Aloc
X -modules.
Notation 2.9. (i) Consider a product of manifolds X1×X2×X3, we
write it X123. We denote by pi the i-th projection and by pij the
(i, j)-th projection (e.g., p13 is the projection from X1× X1a× X2
to X1× X2).
(ii) We write Ai and Aija instead of AX
i and AXi×Xja and similarly with other products. We use the same notations for CXi.
Recall Definitions 3.1.2 and 3.1.3 of [13].
Definition 2.10. Let Ki ∈ Db(Aija) (i = 1, 2, j = i + 1). One sets K1 L ⊗ A2 K2 = (K1 L K2) L ⊗ A22a CX2 = p−112K1 L ⊗ p−112A1a2 A123 L ⊗ p−123aA23a p−123K2, K1 ◦ X2 K2 = Rp13! K1 L ⊗ A2 K2 , K1 ∗ X2 K2 = Rp13∗ K1 L ⊗ A2 K2 .
Remark 2.11. (a) There is a morphism K1 L ⊗ A2 K2 → K1 L ⊗ A2 K2 which is an isomorphism if X1 = pt or X3 = pt.
(b) If the projection p13 : Supp(K1) ×X2Supp(K2) → X1× X3 is proper then, K1 ◦
X2
K2 ' K1 ∗ X2
K2.
We have the following duality results.
Theorem 2.12 ([13, Theorem 3.3.6]). Let (Xi, Ai) (i = 1, 2, 3) be a
complex manifold endowed with a DQ-algebroid Ai. Let Zi be a closed
subset of Xi× Xi+1 and assume that Z1×X2Z2 is proper over X1× X3. Set Z = p13(p−112Z1 ∩ p−123Z2). Let Ki ∈ Dbcoh,Zi(Ai(i+1)a) (i = 1, 2).
Then the object K1 ◦ X2
K2 belongs to Dbcoh,Z(A13a) and we have a natural
Definition 2.13. (i) An AX-module N is a submodule of an AlocX
-module M if there exist a monomorphism µ : N → f or(M) in Mod(AX).
(ii) Let N be an AX-submodule of an AlocX -module M. We say that
N generates M if µloc is an isomorphism.
(iii) A coherent AX-submodule of a coherent AlocX -module M is called
an AX-lattice of M if N generates M.
We recall the definition of good module and refer the reader to [13, p. 74] for a thorough treatment of this notion.
Definition 2.14. (i) A coherent Aloc
X -module is good if, for any
rel-atively compact open subset U of X, there exists an AX|U-lattice of
M|U.
(ii) One denotes by Modgd(AlocX ) the full subcategory of Modcoh(AlocX )
the objects of which are the good modules.
(iii) One denotes by Dbgd(AlocX ) the full subcategory of Dbcoh(AlocX ) the objects of which have good cohomology.
The Theorem 2.12 extends to Aloc
X -modules by replacing the category
Dbcoh(AX) with Dbgd(AlocX ). Hence, we have the following statement for
Aloc
X -modules.
Theorem 2.15. Let (Xi, Ai) (i = 1, 2, 3) be a complex manifold
en-dowed with a DQ-algebroid Ai. Let Zi be a closed subset of Xi× Xi+1
and assume that Z1×X2Z2 is proper over X1×X3. Set Z = p13(p
−1 12Z1∩
p−123Z2). Let Ki ∈ Dbgd,Zi(A loc
i(i+1)a) (i = 1, 2). Then the object K1X◦
2 K2
belongs to Dbgd,Z(Aloc13a) and we have a natural isomorphism D0Aloc 12a(K1) ◦X2a ωXloc 2a X◦ 2aD 0 Aloc 23a(K2) ∼ → D0Aloc 13a(K1X◦2 K2).
2.3. Holonomic DQ-modules. In this subsection we will be con-cerned with DQ-modules on complex symplectic manifolds. Thus, we first recall some notions of symplectic geometry.
Definition 2.16. Let (X, ω) be a complex symplectic manifold.
(i) A locally closed complex analytic subvariety Λ of X is isotropic if there exists a dense open complex manifold Λ0 ⊂ Λ such that
ω|Λ0 = 0.
(ii) A subvariety Λ in X is co-isotropic if for every function f and g such that f |Λ = g|Λ = 0 then {f, g} = 0 on Λ where {·, ·} is the
Poisson bracket associated to the symplectic form ω.
Following [13], we review the notion of holonomic DQ-modules. In this section, X is a complex manifold of dimension dX endowed with
a DQ-algebroid AX such that the associated Poisson structure is
sym-plectic and we set n = dX 2 .
Definition 2.17. (i) An Aloc
X -module M is holonomic if it is
coher-ent and if its support is a Lagrangian subvariety of X.
(ii) An AX-module N is holonomic if it is coherent, without ~-torsion
and Nloc is a holonomic Aloc
X -module.
Recall the following lemma.
Lemma 2.18 ([10, Lemma 2.1]). Let r be an integer and M be a coherent AlocX -module so that E xtjAloc
X
(M, Aloc
X ) = 0 for any j > r.
Then HjY(M) = 0 for any closed analytic subset Y of X and any
j < codim Y − r.
Holonomic DQ-modules enjoy the following properties.
Lemma 2.19 ([13, Lemma 7.2.2]). Let M be a holonomic Aloc
X -module.
Then D0Aloc X
(M)[dX/2] is concentrated in degree zero and is holonomic.
Lemma 2.20. Let M be a holonomic AlocX -module supported in a La-grangian subvariety Λ of X and Y a closed analytic subset of Λ. Then
HjY(M) = 0 for 0 ≤ j < codimΛY .
Proof. Recall that n = dX
2 . It follows from Lemma 2.19 that
ExtjAloc X
(M, AlocX ) = 0 for any j > n.
Then by Lemma 2.18, HjY(M) = 0 for 0 ≤ j < codimΛY .
Lemma 2.21. Let n = dX
2 and N be a holonomic AX-module. Then ExtjA
X(N , AX) = 0 for j < n and E xt n
AX(N , AX) has no ~-torsion.
Proof. Since N has no ~-torsion, we have Supp(Nloc) = Supp(N ).
Thus, the analytic set Supp(gr~N ) = Supp(Nloc) is Lagrangian. It
follows that
codim Supp(gr~N ) = n. This implies that
Corollary 2.22. Let N be an holonomic AX-module. Then the module
ExtAX(N , AX) is holonomic.
Proof. It is immediate that the module E xtAX(N , AX) is coherent. By Lemma 2.21, it has no ~-torsion. Moreover, [ExtAX(N , AX)]
loc is
holo-nomic.
Proposition 2.23. Let M be a holonomic AlocX -module such that it has a lattice N . Then E xtn
AX(E xt n
AX(N , AX), AX) is a lattice of M.
Proof. We set F = E xtn
AX(N , AX). By functoriality, there is a mor-phism
RHomAX(F , AX) → RHomAlocX (A loc X ⊗ AX F , Aloc X ) and Aloc X ⊗ AX F = Aloc X ⊗ AX Extn AX(N , AX) ' Ext n Aloc X (M, A loc X ).
Thus, we have a morphism
RHomAX(F , AX) → RHomAlocX (E xt n Aloc X (M, A loc X ), A loc X ).
Since M is a holonomic module RHomAloc X (E xt n Aloc X (M, A loc X ), A loc X ) ' M[−n].
Taking the degree n cohomology, we obtain a morphism
γ : E xtnAX(E xtnAX(N , AX), AX) → M such that γloc : [E xtnA X(E xt n AX(N , AX), AX)] loc→ Mloc' M is an isomorphism.
By Lemma 2.21 and Corollary 2.22, E xtn AX(E xt
n
AX(N , AX), AX) has no ~-torsion. Thus, Ker γ has no ~-torsion. Hence, there is a monomor-phism Ker γ ,→ (Ker γ)loc. Since, Aloc
X ⊗
AX
is an exact functor, we have (Ker γ)loc ' Ker(γloc) and Ker(γloc) = (0). It follows that Ker γ =
(0).
3. A coherence criterion
Definition 3.1. Let X and Y be two Hausdorff spaces and let p : X → Y be a continuous map. Let F be a sheaf of Abelian groups. The
sheaf F is p-finite if the restriction of p to the support of F is a finite morphism.
3.1. A coherence criterion: the case of OX-modules. We prove
an equivalence between certain categories of OX-modules and also a
coherence criterion for OX-modules. We will use these results to
de-duce their analogue for DQ-modules. The statements of this subsection should be compared with [7].
In all this subsection, we let D be an open subset of some Cp, we set
X = D × Cn and
p : D × Cn → D the projection D × Cn 3 (y, z) 7→ y.
We write Modcoh(OX) for the category of coherent OX-modules on X
and we denote by Modp−fincoh (OX) the full sub-category of Modcoh(OX)
the objects of which are the p-finite modules.
We set OD[t] := OD[t1, . . . , tn] and recall that the sheaf OD[t] is a
Noetherian sheaf of ring (cf. [8, Theorem A.31]).
The projection p : X → D clearly induces a morphism of ringed spaces
(p, p†) : (X, OX) → (D, OD[t]).
We consider the following functors
p∗ : Mod(OX) → Mod(OD[t]) F 7→ p∗F p] : Mod(OD[t]) → Mod(OX) G 7→ OX ⊗ p−1O D[t] p−1G.
Proposition 3.2. The functors p] is left adjoint to p∗.
Proof. Clear.
Proposition 3.3. The functor p] is exact.
Proof. This follows from the fact that OX is flat over p−1OD[t] (cf. [15,
Example 4.11]).
Notation 3.4. We denote by ModOD−coh(OD[t]) the full subcategory of the category Modcoh(OD[t]) the objects of which are also OD-coherent
modules.
Proposition 3.5. If F ∈ Modp−fincoh (OX), then p∗F ∈ ModOD−coh(OD[t]).
Proof. The sheaf p∗F is clearly an OD[t]-module and it is OD-coherent
since p|Supp(F ) is finite.
Proposition 3.6. If G ∈ ModOD−coh(OD[t]), then p
Proof. Since G is a coherent OD-module, it is a coherent OD[t]-module.
Since OX and OD[t] are coherent sheaves of rings, it implies that p]G
is a coherent OX-module.
Let y ∈ D. Consider AnnOD[t](G) the annihilator of G in OD[t]. Since G is coherent, we have
AnnOD[t](G)y ' AnnOD,y[t](Gy).
The ti, 1 ≤ i ≤ n define OD,y-linear endomorphisms Ti : Gy →
Gy, m 7→ ti · m, 1 ≤ i ≤ n and Gy is a finitely generated module over
OD,y. Thus, their exist monic polynomials ωi,y ∈ OD,y[ti] such that
ωi,y(Ti) = 0. This implies that ωi,y ∈ AnnOD,y[t](Gy). It is well known that
Supp(p]G) = {x ∈ X| AnnOX(p ]G)
x ⊂ mx}
where mx is the maximal ideal of OX,x. Since p] is exact and G is
coherent, AnnOX(p ]G) ' p]Ann OD[t](G). It follows that Supp(p]G) = {x ∈ X|OX,x ⊗ OD,y[t] AnnOD,y[t](Gy) ⊂ mx}.
This means that if an element x0 = (y0, z0) of X belongs to Supp(p]G)
then, for every 1 ≤ i ≤ n, ωi,y0(z) = 0. The ωi,y0(ti) being monic polynomials, they have a finite number of roots. This implies that
p−1(y) is finite.
Let us prove that p|Supp(p]G)is closed. Since being closed is a property local on the base, it is sufficient to check that p|Λ : Λ → Ui is closed
for a covering (Ui)i∈I of D.
Let y0 ∈ D and consider the polynomials ωi,y0(ti) ∈ OD,y0[ti], 1 ≤ i ≤ n previously defined. There exists an open neighbourhood U of y0 such that the wi(ti) ∈ OD[ti]|U are well defined and belong to
AnnOD[t](G)|U. It follows immediately that they induce Weierstrass polynomials ωi(y, zi) ∈ AnnOX(p
]G)| U ×Cn. We set Z = {(y, z) ∈ U × Cn|ω
1(y, z1) = . . . = ωn(y, zn) = 0} and
Λ = Supp(p]G) ∩ U × Cn. We have the following commutative diagram
Λ iΛ p|Λ // U Z p|Z ??
The map iΛis a closed immersion. Thus, we are reduced to show that
p|Z is a closed map. This follow immediately by repeated applications
Lemma 3.7. Let (T, OT) be a closed analytic subspace of (X, OX) such
that OT is p-finite.
(i) If x ∈ X and y = p(x), then for every xi ∈ p−1(y), such that
xi 6= x, OX,x ⊗ OD,y[t]
OT ,xi = 0.
(ii) p]p∗OT ' OT.
Proof. (i) It follows from Propositions 3.5 and 3.6 that p]p
∗OT is a
p-finite coherent OX-module. Thus,
(p]p∗OT)x ' M xi∈p−1(y) OX,x ⊗ OD,y[t] OT ,xi
is finitely generated OX,x-module. Moreover, OX,x is a Noetherian
ring. This implies that the module OX,x ⊗ OD,y[t]
OT ,xi is also a finitely generated OX,x-module. Since xi 6= x, there exists a polynomial P ∈
OD,y[t] such that P (x) = 0 and P (xi) 6= 0. Denoting by (P ) the
ideal generated by P in OX,x, we have that (P )(OX,x ⊗ OD,y[t]
OT,xi) = OX,x ⊗
OD,y[t]
OT,xi and (P ) is included in the maximal ideal of OX,x. Hence, Nakayama’s lemma implies that OX,x ⊗
OD,y[t]
OT ,xi = 0.
(ii) It is a direct consequence of (i).
Theorem 3.8 below is well-known to experts and should be compared with [7, Section 3]. We give a proof for the convenience of the reader.
Theorem 3.8. The functors
Modp−fincoh (OX) p∗ //
ModOD−coh(OD[t]) p]
oo
are equivalences of categories inverse to each other. Proof. Consider the morphism
(3.1) p]p∗F → F .
By lemma 3.7, OX,x ⊗ OD,y[t]
OT ,xi = 0 if x 6= xi. It follows that (p ]p
∗F )x '
Fx. Then the morphism (3.1) is an isomorphism.
Consider the map
(3.2) G → p∗p]G.
We prove it is an isomorphism.
Let y ∈ D. Then the map (3.2) induces a morphism
(3.3) Gy → (p∗p]G)y.
We construct the inverse of the morphism (3.3). Since p|Supp(p]G) is finite, we have (p∗p]G)y ' M xi∈p−1(y) OX,xi ⊗ OD,y[t] Gy ' ( M xi∈p−1(y) OX,xi) ⊗ OD,y[t] Gy.
As already explained, the ti, 1 ≤ i ≤ n define OD,y-linear
endo-morphisms Ti : Gy → Gy, v 7→ ti · v, 1 ≤ i ≤ n and Gy is a finitely
generated module over OD,y. Thus, there exists monic polynomials
ωi,y ∈ OD,y[ti] such that ωi,y(Ti) = 0. Shrinking D if necessary (and X
since X = D ×Cn), we set Q = (O
X/(ω1, . . . , ωn)OX) and denote by m
the cardinal of p−1(y). By the generalized Weierstrass division theorem ([5, Ch. 2 §4]) for any choice of m germsfj ∈ Qxj, xj ∈ p
−1(y), i ≤ j ≤
m, there exists a uniquely determined polynomial r ∈ OD,y[t] whose
degree in ti is less than deg ωi, 1 ≤ i ≤ n and such that the residue
epimorphism OX,xj → Qxj maps r onto fj for all j = 1, . . . , m. We define the morphism
φ : ( M xi∈p−1(y) OX,xi) ⊗ OD,y[t] Gy → Gy (fxi)xi∈p−1(y)⊗ n 7→ rn
where r is the polynomial obtained via the generalized Weierstrass division Theorem. It is immediate that φ is the inverse of the morphism (3.3). This implies that morphism (3.2) is an isomorphism.
We now prove the following coherence criterion.
Proposition 3.9. Let F be a OX-module and assume that F is a
p-finite OX-module. Then F is a coherent OX-module if and only if p∗F
is a coherent OD-module.
Proof. (i) Assume that F is coherent. Since p|Supp(F ) is finite, p∗F is
coherent.
(ii) Assume that p∗F is coherent. The functor p] and p∗ are adjoint.
We denote by ε : p]p
η : id → p∗p] the unit of the adjunction. Recall that the natural transformation p∗ ηp∗ → p∗p]p∗ p∗ε → p∗
is the identity. Thus p∗ε is an epimorphism of sheaves.
Let Q be the cokernel of εF : p]p∗F → F . By Propositions 3.5 and
3.6, p]p
∗F is a p-finite coherent OX-module and Q is also a p-finite
sheaf since Supp(Q) ⊂ Supp(F ). It follows that we have a right exact sequence of p-finite sheaves
(3.4) p]p∗F
εF
→ F → Q → 0.
Since all the sheaves in the exact sequence (3.4) are p-finite, we obtain by applying the functor p∗ the right exact sequence
p∗p]p∗F p∗εF
→ p∗F → p∗Q → 0.
Since p∗εF is an epimorphism of sheaves, it follows that p∗Q ' 0 which
implies that Q ' 0. Thus, εF is an epimorphism. It results that F
is a locally finitely generated OX-module and that AnnOX(p ]p
∗F ) ⊂
AnnOX(F ). Setting OT = OX/ AnnOX(p ]p
∗F ), it follows from the
in-clusion AnnOX(p ]p
∗F ) ⊂ AnnOX(F ) that F is a OT-module. Moreover, OX,x ⊗ OD,y[t] Fxi ' OX,x ⊗ OD,y[t] OT ,xi ⊗ OT ,xiFxi, and by Lemma 3.7, OX,x ⊗ OD,y[t] OT,xi = 0 if xi 6= x. Then we have the following commutative diagram.
(p]p ∗F )x o εF ,x // Fx o L xi∈p−1(y) OX,x ⊗ OD,y[t] Fxi // o Fx o Fx ∼ // Fx. Hence, p]p
∗F ' F which implies that F is coherent.
Let M be an open subset of Cn. We set X = T∗M and denote by
ρ : T∗M → M the projection on the base. We denote by AX the
Moyal-Weyl star-algebra on X and define the algebra BM =OM~ ⊗ C C[~∂ 1, · · · , ~∂n]. Then Bloc M 'OM~,loc⊗ CC[∂ 1, · · · , ∂n].
Definition 3.10. We say that a BM-submodule L of a coherent BMloc
-module N is a finiteness BM-lattice of N if L is a BM-lattice of N and
it is coherent as a O~
M-module.
We denote by Modfgd(BlocM) the Abelian full-subcategory of Mod(BMloc)
the objects of which are the Bloc
M-modules admitting locally a finiteness
lattice.
Let (x, u) be a symplectic coordinate system on X = T∗M . Then ρ : X → M
(x, u) 7→ x.
We denote by Modρ−fincoh (AX) (resp. Modρ−fincoh (AlocX )) the full sub-category
of the category Modcoh(AX) (resp. Modcoh(AlocX )) the objects of which
are the ρ-finite modules.
Remark 3.11. A coherent ρ-finite AlocX -module M is holonomic. We have a morphism of C~-algebras
ρ† : ρ−1O~
M → AX, ρ†(xi) = xi for 1 ≤ i ≤ n.
There is also a monomorphism of C~-algebras from ρ−1B
M to AX de-fined as follows. ψ : ρ−1BM → AX, xi 7→ xi, ~∂i 7→ ui. (3.5) We set ρ[ : Mod(BM) → Mod(AX) N 7→ AX ⊗ ρ−1B M ρ−1N
ρ\: Mod(BlocM) → Mod(AlocX ) N 7→ Aloc
X ⊗
ρ−1Bloc M
ρ−1N .
Proposition 3.12. The functors ρ[ and ρ
∗ (resp. ρ\ and ρ∗) form the
adjoint pair (ρ[, ρ∗) (resp. (ρ\, ρ∗)).
Proof. Clear.
Proof. We need to prove that for every BM-module N Hi(AX L ⊗ ρ−1B M ρ−1N ) = 0 f or i < 0.
which is equivalent to prove that for every z ∈ X and every N ∈ Mod(BM) Hi(AX,z L ⊗ ρ−1B M,z ρ−1Nz) = 0 f or any i < 0.
For that purpose, we adapt the proof of Theorem 1.6.6 of [13]. Let
z ∈ X. We set A = AX,z, B = BM,ρ(z), N = Nρ(z), Q = OX,z and
R = OM,z[t1, . . . , tn] and we recall that Q is flat over R (see [15, example
4.11]). The problem reduces to show that ToriB(A, N ) = 0 for all i < 0. Since, Tor commute with filtrant colimits, we can further assume that
N is a B-module of finite type. Furthermore, every finitely generated B-module is an extension of a B-module without ~-torsion by a module
of torsion. So, we just need to treat the case of modules without ~-torsion and the case of ~-~-torsion modules.
(i) We assume that N has no ~-torsion, then gr~(A⊗L
B
N ) ' Q ⊗
RN/~N.
Moreover, N is a finitely generated B-module. Hence, A⊗L
B
N belongs
to Dbf(A). It follows from Theorem 1.6.1 of [13] that A⊗L
BN is
cohomo-logically complete. Hence, Proposition 1.5.8 of [13] implies that A⊗L
BN
is concentrated in degree zero.
(ii) We assume that N is of ~-torsion. Clearly N = S
k∈N∗Nk where Nk = {n ∈ N |~kn = 0} and A⊗L B N1 ' Q L ⊗ R N1 ' Q ⊗ R N1. Thus, A⊗L
BN1 is concentrated in degree zero. By recursion, we
ex-tend this to all the Nk. Since TorB commutes with filtrant colimits
ToriB(A, N ) = 0 for i 6= 0. It follows that for any finitely generated module N , A⊗L
B
N is concentrated in degree zero.
The result for ρ\ follows immediately since ρ\ is the composition of
ρ[ with the exact functor C~,loc
X ⊗
C~X
Proposition 3.14. If M ∈ Modρ−fincoh (AX), then ρ∗M is an object of
ModO~
M−coh(BM).
Proof. Let M ∈ Modcohρ−fin(AX). The sheaf ρ∗M is clearly a BM-module
and by Grauert’s direct image theorem for DQ-modules (cf. [13, Thm
3.2.1 ]) ρ∗M is O~M-coherent.
Proposition 3.15. If N ∈ ModO~
M−coh(BM), then ρ
[N ∈ Modρ−fin coh (AX).
Proof. The coherence of ρ[N is clear. Now, assume that N has no
~-torsion. Then
Supp(ρ[N ) = Supp(gr~ρ[N ) = Supp(ρ](N /~N )). It follows that ρ[N ∈ Modρ−fin
coh (AX).
Let N be a module of ~-torsion with tor~(N ) = 1. We have the
following isomorphism ρ[N ' AX ⊗ ρ−1B M ρ−1N ' OX ⊗ ρ−1O M[t] ρ−1N ' ρ]N . Thus, ρ[N is a ρ-finite A X-module.
Let N be a module of ~-torsion with tor~(N ) = k, k ≥ 2. Every
BM-submodule P of N with tor~(P) = 1 is ρ-finite. Assume that ρ [P
is ρ-finite for each BM-submodule P of N which is of uniform ~-torsion
with tor~(P) = k − 1. Let Q be a BM-submodule of N which is of
~-torsion with tor~(Q) = k. Then we have the following short exact
sequence
0 → ~Q → Q → Q/~Q → 0. Moreover, ρ[ is an exact functor, then
0 → ρ[(~Q) → ρ[Q → ρ[(Q/~Q) → 0.
So, Supp(ρ[Q) ⊂ Supp(ρ[(~Q)) ∪ Supp(ρ[(Q/~Q)) which implies that
ρ[Q is ρ-finite. In particular, ρ[N is ρ-finite.
Assume that N is of ~-torsion. Thus, there exists an open covering (Ui)i∈I of M such that tor~(N |Ui) = ki with ki ∈ N. We set ρi :=
ρ|Ui ρ−1(U
i). It follows from the previous cases that ρ [
iN |Ui is ρi-finite. Since being finite for a morphism is a property local on the base, the sheaf ρ[N is ρ-finite.
Let N ∈ ModO~
M−coh(BM). Then N is the extension by an object of ModO~
The following proposition is the analogue for DQ-modules of [14, Lemma 4.3].
Proposition 3.16. The functors
Modρ−fincoh (AX) ρ∗ //
ModO~
M−coh(BM) ρ[
oo
are equivalences of categories and inverse to each other.
Proof. Once again, we will use the fact that a coherent DQ-module is
the extension of a coherent module without ~-torsion by a DQ-module of ~-torsion. Let M ∈ Modρ−fincoh (AX). The adjunction between
ρ∗ and ρ[ provides the map
(3.6) ρ[ρ∗M → M.
(i) Assume that M has no ~-torsion. Applying the gr~ functor to (3.6),
we get
(3.7) ρ]ρ∗(M/~M) → (M/~M).
The module (M/~M) belongs to Modρ−fincoh (OX), thus by the Theorem
3.8, it follows that the morphism (3.7) is an isomorphism. Since M and ρ[ρ
∗M are complete and have no ~-torsion, it follows that the
morphism (3.6) is an isomorphism.
(ii) Assume that M is of ~-torsion with tor~(M) = 1. Then we have
the following commutative diagram
ρ[ρ∗M // o M o ρ]ρ∗M ∼ // M
which implies that the map (3.6) is an isomorphism.
(iii) If M is a module of ~-torsion with tor~(M) = k, the fact that the
morphism (3.6) is an isomorphism is proved by recursion on the index of ~-torsion.
(iv) Assume that M is of ~-torsion. Let z = (x, u) a point of X = T∗M .
Consider the stalk of morphism (3.6) at z
(3.8) (ρ[ρ∗M)z → Mz.
If ρ−1(x) ∩ Supp(M) = ∅, it is clear that (3.8) is an isomorphism.
Assume that ρ−1(x) ∩ Supp(M) = {z1, . . . , zs}. Since M is
coher-ent we can find open pairwise disjoint neighbourhoods U10, . . . , Us0 of
z1, . . . , zs such that Ui0∩ Uj0 = ∅ for i 6= j and M|U0
i is of ~-torsion with tor~(M|U0
open subset V of M such that ρ−1(V ) is an open neighbourhood of
z1, . . . , zs and tor~(M|T∗V) = k with k = max1≤i≤s{ki}.
Using the fact that the morphism (3.6) is an isomorphism for ~-torsion modules with index of ~-~-torsion equal to k, it follows that
(ρ[ρ∗M)|ρ−1(V ) → M|ρ−1(V ). is an isomorphism which proves the claim.
(v) Let M be a ρ-finite AX-module. Then M is the extension of a
ρ-finite module without ~-torsion by a ρ-ρ-finite module of ~-torsion. Using the preceding cases the result follows immediately.
The proof that the unit id → ρ∗ρ[ is an isomorphism is simpler and
follows a similar path.
Lemma 3.17. Let M ∈ Modρ−fincoh (Aloc
X ) then, M is locally good on the
base i.e. there exists an open covering (Ui)i∈I of M such that for every
i ∈ I, M|ρ−1(U
i) is good.
Proof. If ρ−1(x) ∩ Supp(M) = ∅, then by Lemma 8.3, there exists a neighbourhood W of x such that M|W ×Cn = 0 and 0 is a lattice of M on W × Cn. Assume that ρ−1(x) ∩ Supp(M) = {z
1, . . . , zs}.
Since M is coherent, we can find open pairwise disjoint neighbourhoods
U10, . . . , Us0 of z1, . . . , zssuch that Ui0∩Uj0 = ∅ and such that M|U0
i admits a lattice Mi for 1 ≤ i ≤ s. It follows from Proposition 8.5, that there
exists pairwise disjoint neighbourhoods U1, . . . , Us of z1, . . . , zs such
that Ui ⊂ Ui0 and such that if we write γi : Ui ,→ V × Cn for the
inclusion of Ui into V × Cn we have the isomorphism
M|V ×Cn ∼ → s Y i=1 γi∗γi−1(M|V ×Cn). It follows immediately that Qs
i=1γi∗γi−1Mi is a lattice of M|V ×Cn. The following proposition is the analogue for DQ-modules of [14, Proposition 4.4].
Proposition 3.18. The functors
Modρ−fincoh (Aloc X )
ρ∗ //
Modfgd(BlocM) ρ\
oo
are equivalences of categories and inverse to each other.
Proof. Using Lemma 3.17 the proof of the proposition reduces to
Propo-sition 3.16.
Theorem 3.19. Let (T, OT) be a complex manifold endowed with a
DQ-algebra AT. Let M be a locally finitely generated AT-module. Then
M is coherent if and only if ~nM/~n+1M is a coherent O
T-module for
any n ≥ 0.
Proposition 3.20. (i) Let M be a ρ-finite AX-module. Then M is a
coherent AX-module if and only if ρ∗M is a coherent OM~ -module.
(ii) Let M be a ρ-finite AlocX -module.
(a) If M is a coherent AlocX -module, then ρ∗M is a coherent O~,locM
-module.
(b) If ρ∗M is a coherent OM~,loc-module and has a coherent O~M-lattice
L such that L is a BM-module, then M is a coherent AlocX -module.
Proof. (i) The first direction is Proposition 3.14. Let us prove the converse. Let M be a ρ-finite module and assume that ρ∗M is coherent.
We denote by ε : ρ[ρ
∗ → id the co-unit and by η : id → ρ∗ρ[ the unit
of the adjunction (ρ[, ρ
∗). Recall that the natural transformation
ρ∗ ηρ∗
→ ρ∗ρ[ρ∗ ρ∗ε → ρ∗
is the identity. Thus, ρ∗ε is an epimorphism of sheaves. By an argument
similar to the one of the proof of Proposition 3.9, we show that εM is
an epimorphism of sheaves which implies that M is locally finitely generated.
The restriction of ρ∗ to the category Modρ−fin(AX) is exact and the
sheaves M, ~nM and ~nM/~n+1M are ρ-finite. This implies that
~nρ∗M/~n+1ρ∗M ' ρ∗(~nM)/ρ∗(~n+1M) ' ρ∗(~nM/~n+1M).
This means that the sheaf ρ∗(~nM/~n+1M) is a coherent O~M-module
and hence, a coherent OM-module since tor~(ρ∗(~ n
M/~n+1M)) = 1.
By Proposition 3.9, ~nM/~n+1M is a coherent O
X-module. It follows
from Theorem 3.19 that M is a coherent AX-module.
(ii) The point (a) is clear. Let us prove (b). Let M be a ρ-finite module such that ρ∗M is a coherent OM~,loc-module and has a coherent
O~
M-lattice L such that L is a BM-module. This implies that ρ[L ∈
Modρ−fincoh (AX).
By an argument similar to the one of the proof of Proposition 3.9, we obtain that the co-unit
(3.9) ρ\ρ∗M → M
is an epimorphism. Moreover, there is a canonical map
(3.10) ρ[L → ρ\ρ
Composing the maps (3.9) and (3.10), we get
(3.11) θ : ρ[L → ρ\ρ
∗M → M.
Moreover, θloc: (ρ[L)loc→ Mloc ' M is an epimorphism. This implies
that (Im θ)loc ' M. The A
X-module Im θ is ρ-finite and the morphism
(3.11) induces the epimorphism ρ[L Im θ. Thus, ρ
∗Im θ is a locally
finitely generated O~
M-submodule of the coherent OM~,loc-module ρ∗M.
It follows by [13, Lemma 2.3.13] that ρ∗Im θ is a coherent O~M-module.
Then point (i) implies that Im θ is a coherent AX-module. Hence, M
is a coherent Aloc
X -module.
Remark 3.21. If U is an open subset of X, the above criterion can
also be used to prove the coherence of a ρ|U-finite AX|U-module M.
Indeed, if i : U ,→ ρ(U ) × Cndenotes the inclusion of U into ρ(U ) × Cn
then, by Corollary 8.2 i∗M is a ρ|ρ(U )×Cn-finite Aρ(U )×Cn-module the coherence of which is equivalent to the coherence of M.
4. A duality result
In this section, we prove the analogue for DQ-modules of [14, Propo-sitions 4.2 & 4.6 ]. We let X be an open subset of T∗M and
shrink-ing M is necessary we assume that ρ(X) = M . We denote by AX
the Moyal-Weil star-algebra on X and, on M , we consider the triv-ial star-algebra AM := OM~ . We endow M × X with the star-algebra
AM ×Xa = OM~ AXa. Following page 61 of [13] the star-product on AM ×Xa is given by
f ? g =X
i≥0
~i(m Pi)(f, g) for every f, g ∈ OM ×X
where the Pi are the bi-differential operators defining the star-product
on AXa and m is the multiplication of function on O~M. Consider the projection ρ : X → M and define
γρ: X × X → M × X
(x, s) 7→ (ρ(x); s)
δρ: X → M × X
x 7→ (ρ(x); x).
Notice that δρ= γρ◦ δ. We define the morphism
γρ†:γρ−1AM ×Xa → AX×Xa
f 7→ f ◦ γρ.
It follows from the formula defining the star-product on AM ×Xa and AX×Xa that γ†ρ is a morphism of C~-algebras.
induces on γρ∗δ∗AX ' δρ∗AX a structure of AM ×Xa-module. Since δρ is finite, δρ∗AX is a coherent AM ×Xa-module.
We denote by Dbρ−fin(AX) the full subcategory of Db(AX) the objects
of which have ρ-finite cohomology and define the functor
ρ∗ : Mod(AMa) → Mod(AXa) N 7→ ρ−1N ⊗
ρ−1A
M AX
Lemma 4.1. Let M ∈ Dbρ−fin(AX) and N ∈ Db(AMa). Then
δρ∗AX ◦ AX M ' R ρ∗M, N ◦ AM δρ∗AX ' Lρ∗N .
Proof. These formulas are direct consequences of the projection
for-mula.
Lemma 4.2. Let M ∈ Db
coh,ρ−fin(AXa) (resp. in Db
gd,ρ−fin(AlocXa)). Then
we have the following isomorphisms
(ωMa ◦ AM aD 0 AM ×Xa(δρ∗AX)) ◦A Xa M→ ρ∼ ∗M in Dbcoh(AMa), (4.1) (ωMloca ◦ Aloc M a D0Aloc M ×Xa (δρ∗AlocXa)) ◦ Aloc Xa M→ ρ∼ ∗M in Dbgd(A loc Ma). (4.2)
Proof. We prove formula (4.1). Let N ∈ Db
Applying RΓ(M ; ·), we get RΓ(M ; R pM ∗(p−1MD 0 AM a(N ) L ⊗ p−1MAM a (D0AM ×Xa(ωMaA◦ M a δρ∗AX)) L ⊗ p−1XaAXa p−1XaM)) ' ' RΓ(M × X; p−1MD0AM a(N ) L ⊗ p−1MAM a (ωMa ◦ AM a D0AM ×Xa(δρ∗AX)) L ⊗ p−1XaAXa p−1XaM) ' RΓ(X; R pX∗(p−1MD 0 AM a(N ) L ⊗ p−1MAM a (ωMa ◦ AM aD 0 AM ×Xa(δρ∗AX)) L ⊗ p−1XaAXa p−1XaM)) ' RΓ(X; (D0AM a(N ) L ⊗ p−1MAM a (ωMa ◦ AM aD 0 AM ×Xa(δρ∗AX)) L ⊗ p−1XaAXa p−1XaM)) ' RΓ(X; [D0AM a(N )) ◦A M a ωMa ◦ AM aD 0 AM ×Xa(δρ∗AX)] L ⊗ AXa M)).
Applying Theorem 2.12, we obtain RΓ(X; [D0AM a(N )) ◦A M a ωMa ◦ AM aD 0 AM ×Xa(δρ∗AX)] L ⊗ AXaM)) ' ' RΓ(X; D0AXa(N ◦A M a δρ∗AX)) L ⊗ AXa M)) ' RHomAXa(Lρ ∗N , M) ' RHomAM a(N , R ρ∗M) It follows from the Yoneda Lemma that
(ωMa ◦ AM aD 0 AM ×Xa(δρ∗AX)) ◦A Xa M ' R ρ∗M.
The construction of the morphism (4.2) is similar. First, one as-sumes that N ∈ Db
gd(AlocX ) and instead of using Theorem 2.12, one uses
Theorem 2.15 and get the desired isomorphism.
Finally, we get the following duality result which is the analogue for DQ-modules of [14, Proposition 4.2].
Proposition 4.3. Let M ∈ Modρ−fincoh (AX) (resp. in Mod ρ−fin gd (AlocX )). Then we have ρ∗ExtkAX(M, AX) ' Ext k−n O~ M (ρ∗M, OM~ ), (4.3) ρ∗ExtkAloc X (M, A loc X ) ' Ext k−n O~,loc M (ρ∗M, O~,locM ). (4.4) where n = dX/2 = dM.
Proof. We only prove formula (4.3), the proof of formula (4.4) being
similar. Applying Theorem 2.12 with X1 = M , X2 = X and X3 = pt,
get that D0M(ρ∗M) ' D0M(δρ∗AX ◦ XM) ' D0M ×Xa(δρ∗AX) ◦ XaωXaX◦aD 0 X(M) ' ωM−1a ◦ AM a ωMa ◦ AM a D0M ×Xa(δρ∗AX) ◦ XaD 0 X(M)[dX] ' ωMa ◦ AM aD 0 M ×Xa(δρ∗AX) ◦ XaD 0 X(M)[dX − dM] ' ρ∗D0X(M)[dX − dM].
Taking the cohomology and using the fact that ρ : Supp(M) → M is finite, we get that
ρ∗ExtkAX(M, AX)) ' Ext k−n O~ M (ρ∗M, OM~ ). We will need the following result (see [3], [17]).
Theorem 4.4 (Popescu, Bhatwadekar, Rao). Let R be a regular local ring, containing a field, with maximal ideal m and t ∈ m \ m2. Then every finitely generated projective module over the localized ring Rt =
R[t−1] is free.
As a consequences of Proposition 4.3 and Theorem 4.4, we get the analogue for DQ-modules of [14, Proposition 4.6].
Proposition 4.5. Let M ∈ Modρ−fincoh (Aloc
X ). The sheaf ρ∗M is locally
free over O~,loc M .
Proof. The question is local on M and ρ-finite coherent modules are
locally good on the base. Then there exists an open covering (Ui)i∈I of
M such that for every i ∈ I, M|ρ−1(U
i)is good. Once, one has restricted M to such an Ui the proof is exactly the same as in [14, Proposition
4.6]. We include it for the sake of completeness. By Lemma 3.11, M is holonomic. Then
ExtkAloc X (M, A loc X ) = 0 for k 6= dX 2 . It follows from Proposition 4.3 that
Extk O~,loc
M
(ρ∗M, OM~,loc) = 0 for k 6= 0.
By taking the stalk at m ∈ M , we get that (4.5) ExtkO~,loc
M,m
The ring O~,loc
M,m is a Noetherian ring with finite global dimension.
This combined with equation (4.5) imply that ρ∗Mm is a finitely
gen-erated projective module.
It follows from Theorem 4.4 that ρ∗Mm is a finitely generated free
module. Since ρ∗M is a coherent O~,locM -module, this implies that ρ∗M
is a locally free sheaf.
5. Geometric preliminary
Let X be a complex manifold and Z be a closed analytic subset of
X and p ∈ X. We denote by Cp(Z) the tangent cone of Z in X at p.
We refer the reader to [24] and [11] for a thorough study of the notion of tangent cone and to [16] for a gentle introduction.
The following result can be found in [19, Ch. III Proposition 1.1.3].
Proposition 5.1. Let X be a complex manifold, let Z be a closed analytic subset of X and let p be a point of X. The set Cp(Z) is a
closed analytic (even algebraic) subset of TpX, conic for the action of
C∗ on TpX.
Lemma 5.2 ([24, Lemma 8.11]). Let X be a complex manifold, let Z be a closed analytic subset of X and p ∈ X. Then dim Cp(Z) = dimpZ. If
Z is of constant dimension near p then, Cp(Z) is of constant dimension.
The next proposition is a special case of [6, Proposition 3.2.3].
Proposition 5.3. Let (X, p) be a germ of analytic space of dimen-sion dX, (Y, q) be a germ of complex manifold of dimension dX and
f : (X, p) → (Y, q) be a germ of analytic map. If dpf : (Cp(X), 0) →
(TqY, 0) is finite, then f : (X, p) → (Y, q) is finite.
We will need the following result to prove Proposition 5.5.
Lemma 5.4 ([12, Lemma 7.3]). Let X be a complex manifold, Y be a closed complex subvariety of X and f : X → C be a holomorphic function. Set Z := f−1(0), Y0 := (Y \ Z) ∩ Z. Consider a p-form η, a (p − 1)-form θ on X and set
ω = df ∧ θ + f η.
Assume that ω|Y = 0. Then θ|Y0 = 0 and η|Y0 = 0.
Proposition 5.5. Let (X, ω) be a symplectic complex manifold, Λ be a closed isotropic analytic subset of X and p ∈ Λ. The tangent cone Cp(Λ) of Λ at p is an isotropic subset of the symplectic vector space
Proof. We adapt the proof of [12, Proposition 7.1] to our needs. Let
(x; u) be a local symplectic coordinate system on X such that
p = (0; 0), ωp = n
X
i=1
dui∧ dxi.
Consider the deformation to the normal cone ˜Xp of X along p. We
have the following diagram
X X˜p q oo t // C p? OO q−1(p). q oo ? OO
Let (x; ξ, t) be the coordinate system on ˜Xp. Then
q(x; ξ, t) = (x; tξ),
TpX ' {(x; ξ, t) ∈ ˜Xp|t = 0},
Cp(Λ) ' q−1(Λ) \ t−1(0) ∩ t−1(0).
The last isomorphism is a direct consequence of the sequential descrip-tion of the tangent cone.
Taking the pull-back of ω by q, we get
q∗ω = n X i=1 d(tξi) ∧ dxi = dt ∧ ( n X i=1 ξi ∧ dxi) + t n X i=1 dξi∧ dxi.
The form q∗ω vanishes on (q−1(Λ))reg. Hence, by Lemma 5.4 the
formPn
i=1dξi∧ dxi vanishes on Cp(Λ) which proves the claim.
Proposition 5.6 ([9, Proposition 1.6.1]). Let (E, ω) be a symplectic vector space and let Λ be an isotropic homogeneous analytic subset of E. Then there is a Lagrangian linear subspace λ such that Λ ∩ λ ⊂ {0}. Corollary 5.7. Let (E, ω) be a symplectic complex vector space of finite dimension. Let Λ be Lagrangian analytic subset of E and let p be a point of Λ. Then there exists a linear Lagrangian subspace λ of E such that λ ∩ Cp(Λ) ⊂ {0}.
Proposition 5.8. Let (E, ω) be a complex symplectic vector space of finite dimension, let U be an open subset of E, let Λ be a closed grangian analytic subset of U and let p ∈ Λ. There exists a linear La-grangian subspace λ of E, an open neighbourhood V of p and an open neighbourhood M of πλ(p) the image of p by the canonical projection
πλ : E → E/λ, such that the map
ρ := πλ|MV : V → M
Proof. Since U is an open subset of a E, we identify TpU and E. It
follows from Proposition 5.1 that Cp(Λ) is an algebraic subset of E conic
for the action of C∗ on E. By Corollary 5.7, there exist a Lagrangian linear subspace λ of E such that Cp(Λ) ∩ λ ⊂ {0}. The quotient
map πλ : E → E/λ induces a germ of analytic map ρ : (Λ, p) →
(E/λ, πλ(p)). We consider the differential dpπλ : (Cp(Λ), 0) → (E/λ, 0).
Since dpρ = πλ, we have
(dpρ)−1(0) = πλ−1(0) ∩ Cp(Λ) ⊂ {0}.
Thus, the germ of map dpρ : (Cp(Λ), 0) → (E/λ, 0) is finite and
dimpΛ = dim E/λ which implies that the germ of analytic map ρ :
(Λ, p) → (E/λ, πλ(p)) is finite.
6. The codimension-three conjecture
As indicated in the introduction the strategy to prove the codimension-three conjecture for DQ-modules is to reduce the non-commutative statement to a commutative problem where it is possible to apply The-orem 1.3 due to Kashiwara and Vilonen.
6.1. Proof of the Theorem 1.4. The aim of this subsection is to prove the codimension-three conjecture for DQ-modules. Let us recall Theorem 1.4.
Theorem. Let X be a complex manifold endowed with a DQ-algebroid stack AX such that the associated Poisson structure is symplectic. Let Λ
be a closed Lagrangian analytic subset of X, and Y be a closed analytic subset of Λ such that codimΛY ≥ 3. Let M be a holonomic (AlocX |X\Y
)-module, whose support is contained in Λ \ Y . Assume that M has an AX|X\Y-lattice. Then M extends uniquely to a holonomic module
defined on X whose support is contained in Λ.
Let X, Λ, Y and M as in Theorem 1.4. We write j : X \ Y → X for the open embedding of X \ Y into X.
6.1.1. Unicity of the extension. We prove the unicity of the extensions of M to X. As in [14], we have
Proposition 6.1. Let M0 be an extension of M. Then M0 is isomor-phic to j∗M.
Proof. Let M0 be an extension of M. We have the following exact sequence.
0 → H0Y(M0) → M0 → j∗j−1M0 → H1Y(M 0
By definition of an extension, j−1M0 ' M. The above exact sequence
becomes
0 → H0Y(M0) → M0 → j∗M → H1Y(M 0
) → 0.
Since, codimΛY = 3, it follows by Lemma 2.20 that HiY(M0) ' 0 for
i = 0, 1. This implies that M0 ' j∗M.
6.1.2. The support of the extension is Lagrangian. As in [14], we have
Proposition 6.2. The support of j∗M is a closed Lagrangian analytic
subset of X.
Proof. The module M is holonomic. Thus, Supp(M) is a complex
analytic Lagrangian subset of X \ Y and Supp(j∗M) = Supp(M).
By the Remmert-Stein theorem the set Supp(M) is a closed analytic subset of X of dimension dX/2 and is isotropic since the closure of an
isotropic subset is isotropic.
6.1.3. Proof of the coherence. The only thing that remain to prove is that the Aloc
X -module j∗M is coherent. We will need the following
proposition.
Proposition 6.3. Let X be a complex manifold and Y be a closed analytic subset of X and j : X \ Y → X be the inclusion of X \ Y into X. If N is a coherent reflexive O~
X\Y-module. Then
(j∗N )loc ' j∗(Nloc).
Proof. By adjunction between j−1 and j∗, we have
j−1j∗N → N . Then j−1(C~,loc X\Y ⊗ C~X\Y j∗N ) → C~,locX ⊗ C~X N . Finally we get a canonical map φ : (j∗N )loc → j∗(Nloc).
Since N is reflexive, it has no torsion and in particular no ~-torsion. Thus, the morphism φ is a monomorphism.
The morphism φ is an epimorphism. Indeed, let U be a connected open subset of X and u ∈ (j∗(Nloc))(U ). Locally on U \ Y , there exist
n ∈ N such that ~nu ∈ N . Let us show that this is true globally.
Let φn : Nloc → Nloc/~−nN with n ∈ N. Assume that there exist
x ∈ U \ Y such that φn(u)x = 0. Set v = φn(u). Let us show that
supp(v) is open in U \ Y . There is the following isomorphism Nloc
/~−nN ' [
m∈N
Thus, locally v belongs to some ~−m
N /~−n
N . Moreover, ~−m
N /~−nN
are OX\Y-modules without OX\Y-torsion this implies that the support
of their sections is open. Thus, the support of v is a union of open sets which implies that supp(v) is open. The set Y is a closed analytic subset of U different from U . Thus, U \ Y is connected. The support of
v is open and closed and is different form U \ Y by assumption. Hence,
we have supp(v) = ∅. This implies that u ∈ ~−nN (U \ Y ). It follows
that φ is an epimorphism.
We now address the question of the coherence of j∗M. Let us briefly
recall the setting. We consider a complex symplectic manifold X, Λ is a closed Lagrangian analytic subset of X and Y is a closed analytic subset of Λ such that codimΛY ≥ 3. The problem of showing that a
certain DQ-module is coherent is a local problem. Thus, we just need to work in a neighbourhood U of a point p ∈ Y . As pointed out in [14], we can assume, by working inductively on the dimension of the singular locus of Y that p is a smooth point of Y .
Shrinking U if necessary, we can assume by Darboux’s theorem that
U is an open subset of a symplectic vector space (E, ω). Then by
Proposition 5.8, we know that there exists a Lagrangian linear subspace
λ of E, an open neighbourhood V of p in U and an open neigbhourhood M of πλ(p) in E/λ such that the map ρ : V → M is finite when
restricted to Λ ∩ V . By shrinking U , we can assume that U = V and that Y is smooth. As ρ is an open map we can also assume that
M = ρ(U ). We set N = ρ(Y ), then the map ρ|Y : Y → N
is an isomorphism and codim N ≥ 3. We denote by ρ0 the restriction of ρ to U \ ρ−1(N ), that is
ρ0 : U \ ρ−1(N ) → M \ N.
Writing i : U \ ρ−1(N ) ,→ U \ Y for the inclusion, we have the following Cartesian square
U \ ρ−1(N ) j◦i // ρ0 U ρ M \ N j0 //M.
Since λ is a linear Lagrangian subspace of E, we can find a symplectic basis (e1, . . . , en; f1, . . . , fn) of E such that fi ∈ λ for 1 ≤ i ≤ n.
Let (x, u) be the coordinates system associated to this basis. Then
