An inverse image functor for Lie algebroids

Sophie Chemla

May 13, 2003

Abstract Extending material from the theory of D-modules to Lie algebroids (see also [C3]), we introduce an inverse image functor and show that it preserves coherence under appropriate circumstances. As in the case of D-modules, a suitable non characteristicity notion enables us to give a sufficient condition in order for the duality functor and the inverse image functor to commute. This generalizes a result for D-modules due to Kashiwara-Kawai-Sato but, even in the case of D-modules, our proof is different from theirs. In particular, we obtain a new duality formula for complexes of modules over an ordinary Lie algebra and, as a special case, we get a new adjunction formula for modules over Lie algebras. Moreover, our result will shed some light on the behaviour of Poisson cohomology (a notion introduced by Lichnerowitz) under a Poisson map (in the analytic case).

1 Introduction

We will follow the notation of [K-S-1] for sheaf theory. Let X be a complex manifold, O_{X} the sheaf of holomorphic functions on X, Θ_{X} the sheaf of holomorphic vector fields and D_{X} the sheaf of rings of differential operators on X. A complex Lie algebroid over X is a pair (L_{X}, ω_{L_{X}}) where

• L_{X} is endowed with the structure of a sheaf of C- Lie algebras and with that of a locally free O_{X}-module of constant finite rank.

• ω is an O_{X}-linear morphism of sheaves of C-Lie algebras from L_{X} to Θ_{X} (called the anchor map) such that the following compatibility relation holds

[ξ, fζ] = ω(ξ)(f)ζ + f[ξ, ζ]

for any (ξ, ζ) ∈ L_{X} and any f ∈ O_{X}.

The Lie algebroid L_{X} gives rise to the sheaf of algebras of generalized differential operators generated by O_{X} and L_{X}, D(L_{X}). If L_{X} = Θ_{X} and ω = id then D(L_{X}) is the sheaf of rings of differential operators over X. If X is a point, L_{X} is a Lie algebra and D(L_{X}) is its enveloping algebra. But there are many more Lie algebroids. In particular, Poisson manifolds give rise to Lie algebroids (if X is a Poisson manifold, then (X, Ω_{X}^{1}) is a Lie algebroid) and actions of groups over manifolds give rise to Lie algebroids. Denote by RHom_{D(L_{X})} (O_{X}, −) the right derived functor of Hom_{D(L_{X})} (O_{X}, −). It is interesting to notice that RHom_{D(L_{X})} (O_{X}, O_{X}) computes Lie algebra cohomology with values in O_{X} = C if X is a point, De Rham cohomology if L_{X} = Θ_{X} and Poisson cohomology if X is a Poisson manifold and L_{X} = Ω_{X}^{1} ([Li], [Hu]).

The construction of D(L_{X}) is analogous to the construction of D_{X}. This remark gave us the idea (see [C2], [C3]) of extending material from the theory of D-modules to Lie algebroids. Recall that basic concepts in the theory of D-modules are due to Bernstein and Kashiwara. We refer the reader to [Bj], [Bo], [Ho] and [S2] for an introduction to D-modules theory.

Let d_{L_{X}} be the rank of L_{X}. As in the D-modules case, D(L_{X}) ⊗_{O_{X}}Λ^{d_{L_{X}}} L_{X} is

endowed with a D(L_{X})⊗D(L_{X})-module structure and we have the following duality functor in the derived category of bounded complexes of left D(L_{X})- modules with coherent cohomology D_{coh}^{b}(D(L_{X})) (see [C3]) :

\underline{D}_{L_{X}}(N^{•}) = RHom_{D(L_{X})} (N^{•}, D(L_{X}) ⊗_{O_{X}}Λ^{d_{L_{X}}} L_{X}) [d_{L_{X}}]

which is defined for any N^{•} ∈ D_{coh}^{b} (D(L_{X})). A Lie algebroid morphism ([A-K], [C2], [C3]) Φ from (L_{X}, ω_{L_{X}}) to (L_{Y} , ω_{L_{Y}}) is a pair (f, F) where f is an analytic map from X to Y and F is an O_{X}-module morphism from L_{X} to f^{∗}L_{Y} = O_{X} ⊗_{f^{−1}O_{Y}} f^{−1}L_{Y} with some requirements (see section

2.3 for details). One of the requirements gives the existence of a transfer (D(L_{X}) ⊗ f^{−1}D(L_{Y})^{op}) -bimodule

D_{L_{X}→L_{Y}} = O_{X} ⊗_{f^{−1}O_{Y}} f^{−1}D(L_{Y})

(extending the transfer bimodule in the D-modules case, see [C3]). Denote

by ⊗^{L} the derived functor of the tensor product. We define ([C3]), as in the D-modules case an inverse image functor \underline{Φ}^{−1} : D^{b} (D(L_{Y})) → D^{b} (D(L_{X})) as follows.

\underline{Φ}^{−1}(R^{•}) = D_{L_{X}→L_{Y}} ⊗_{f^{−1}D(L_{Y} )}^{L}f^{−1}R^{•}.

In this article, we will study this inverse image functor. Generalizing the D-modules case (see [S2] for an exposition), we introduce the notion of non characteristicity and we prove the theorem (due to Kashiwara in the case of D-modules):

Theorem 3.4.4 Let (X, L_{X}) and (Y, L_{Y}) be Lie algebroids over the complex manifolds X and Y respectively. Let Φ = (f, F) be a Lie algebroid morphism from (X, L_{X}) to (Y, L_{Y}). Let R^{•} be an object of D_{coh}^{b} (D(L_{Y})). Assume that R^{•} is non characteristic with respect to Φ. Then \underline{Φ}^{−1}(R^{•}) is in D_{coh}^{b}(D(L_{X})).

Then we establish the following duality theorem which generalizes a result of D-modules due to Sato, Kawai and Kashiwara ([SKK]). We do not know whether the proof of [SKK] can be generalized to Lie algebroids and, even in the D-modules case, our proof is different from theirs.

Theorem 4.1.1 Let Φ = (f, F) be a Lie algebroid morphism from (X, L_{X}) to (Y, L_{Y}). Let R^{•} be an object of D_{coh}^{b} (D(L_{Y})) which is supposed to be non characteristic with respect to Φ. Then there is a functorial isomorphism from \underline{D}_{L_{X}} ∘ \underline{Φ}^{−1}(R^{•}) to \underline{Φ}^{−1} ∘ \underline{D}_{L_{Y}} (R^{•}).

As a particular case, we get a duality formula for complexes of modules over Lie algebras.

Combining Theorem 4.1.1 with the results of [C3] concerning the direct image, we obtain adjunction formulas in the Lie algebroids setting (see [K-S 2] for the D-modules case). In particular, we get a new adjunction formula for modules over Lie algebras. Moreover, our result will shed some light on the behaviour of Poisson cohomology under a Poisson map (in the analytic case).

Acknowledgements : I would like to thank M.Kashiwara for indicating to me that the proofs of the main results could be reduced to the commutative case. I am grateful to A.D’Agnolo, M. Duflo, T. Levasseur, P. Schapira, J.P. Schneiders and A. Weinstein for fruitful discussions. I am grateful to M. Kashiwara for making comments on an earlier version of this article.

Notation

If A is a sheaf of rings on X, one denotes by D^{b} (A) (respectively D_{coh}^{b} (A)) the derived category of bounded complexes of left A-modules (respectively of left A-modules with coherent cohomology). In the sequel, we will take A = D(L_{X}) or A = D(L_{X}) ⊗ D(L_{X}). If M^{•} and M^{′•} are objects of D^{b} (D(L_{X})), one puts

Ext_{D(L_{X})}^{i} (M^{•}, M^{′•}) = H^{i} (RHom_{D(L_{X})} (M^{•}, M^{′•})) .

For the theory of filtered sheaves of rings, we refer the reader to [S2]. Let FA be a filtered sheaf of rings. One writes GrFA for the associated graded sheaf of rings. If FM is a filtered FA-module, one denotes by GrFM the associated graded GrFA-module. Let FM and FN be two FA-modules. A morphism of filtered FA-modules from FM to FN, Fu, is a morphism Fu : M → N of the underlying A-modules such that

Fu (F_{k}M) ⊂ F_{k}N .

The group of morphisms of FA-modules from FM to FN will be denoted by Hom_{FA} (FM, FN). With this notion of morphism, the category of FAmodules is an additive category. Let k ∈ ZZ. One denotes by FM(k) the sheaf of FA-modules endowed with the filtration F_{k+n}M. One defines also the sheaf of filtered groups FHom_{FA} (FM, FN) by setting

F_{k}Hom_{FA} (FM, FN) = Hom_{FA} (FM, FN (k)) .

To an element Fu of Hom_{FA} (FM, FN), one associates a morphism GrFu from GrFM to GrFN. One defines also the sheaf of groups Hom_{GFA} (GrFM, GrFN) and the sheaf of graded groups GHom_{GFA} (GrFM, GrFN) (see [S2]). In the category of FA-modules, the notion of strict morphism is defined. Let Fu be an element of Hom_{FA} (FM, FN). Denote by KerFu the sheaf kernel of Fu filtered with the family KerFu ⋂ F_{k}M. Similarly, one defines the sheaf of graded GrFAmodules KerGrFu. Recall that Fu is strict if and only if GrKerFu = KerGrFu.

An exact sequence of FA-modules is a sequence

FM ⟶^{Fu} FN ⟶^{Fv} FP

such that KerF_{k}v = ImF_{k}u.

A finite free FA-module is a FA-module of the form ⊕_{i=1}^{n}FA(r_{i}). If FM is a finite free FA-module, then the underlying sheaf of modules of FHom_{FA} (FM, FN) is Hom_{A} (M, N). A finite free resolution of the FAmodule FM is an exact sequence (of FA-modules) of the form

… ⟶ ⊕_{i=1}^{n_{1}}FA(r_{i,1}) ⟶ ⊕_{i=1}^{n_{0}}FA(r_{i,0}) ⟶ FM → 0.

2 Lie algebroids

2.1 Definitions

Let X be a complex analytic manifold and let O_{X} be the sheaf of holomorphic functions on X. Let Θ_{X} be the O_{X}-module of holomorphic vector fields on X.

Definition 2.1.1 A sheaf in Lie algebras L_{X} is a sheaf of C-vector spaces such that for any open subset U, L_{X}(U) is equipped with the structure of a Lie algebra and the restriction morphisms are Lie algebra homomorphisms.

A morphism between two sheaves of Lie algebras L_{X} and M_{X} is a C_{X}-module morphism which is a Lie algebra morphism on each open subset.

Definition 2.1.2 A complex Lie algebroid over X is a pair (L_{X}, ω) where

• L_{X} is a locally free O_{X}-module of finite constant rank,

• L_{X} is a sheaf of C-Lie algebras, • ω : L_{X} → Θ_{X} is an O_{X}-linear morphism of sheaves of C-Lie algebras

such that the following compatibility relation holds :

∀(ξ, ζ) ∈ L_{X}^{2}, ∀f ∈ O_{X}, [ξ, fζ] = ω(ξ)(f)ζ + f[ξ, ζ]

for any (ξ, ζ) in L_{X} and for any f in O_{X}.

One calls ω the anchor map. When there is no ambiguity, we will drop the anchor map in the notation of the Lie algebroid. If (L_{X}, ω) is a Lie algebroid over X, then (L_{X|U}, ω_{|U} ) is a Lie algebroid over U which will be denoted by L_{U}.

A Lie algebroid (L_{X}, ω) gives rise to the sheaf of generalized differential operators generated by O_{X} and L_{X} which is denoted by D(L_{X}).

Definition 2.1.3 D(L_{X}) is the sheaf associated with the presheaf:

U ↦ T_{C}^{+} (O_{X}(U) ⊕ L_{X}(U)) /J_{U}

where J_{U} is the two sided ideal generated by the relations

∀(f, g) ∈ O_{X}(U), ∀(ξ, ζ) ∈ L_{X}(U)^{2}
1)f ⊗ g = fg
2)f ⊗ ξ = fξ
3)ξ ⊗ ζ − ζ ⊗ ξ = [ξ, ζ]
4)ξ ⊗ f − f ⊗ ξ = ω(ξ)(f)

D(L_{X}) is endowed with the filtration (F_{n}D(L_{X}))_{n∈IN} = (D(L_{X})_{n})_{n∈IN} defined as follows:

\begin{matrix}D(L_{X})_{0} = O_{X} \\ D(L_{X})_{n} = D(L_{X})_{n−1} · L_{X} + D(L_{X})_{n−1} \end{matrix}

The anchor map ω_{L_{X}} : L_{X} → Θ_{X} induces a sheaf of rings morphism from D(L_{X}) to D_{X}.

As L_{X} is a locally free O_{X}-module of finite rank, we have the following theorem due to Rinehart [R].

Theorem 2.1.4 The sheaves of O_{X}-algebras S_{O_{X}}(L_{X}) and GrFD(L_{X}) are isomorphic.

Proof of theorem 2.1.4:

The proof of this theorem is done in the affine case (that is to say the case of a Lie-Rinehart algebra) in [R]. The case of Lie algebroids follows.

Proposition 2.1.5 The sheaf D(L_{X}) is coherent, noetherian and has finite global homological dimension. Moreover, there exists an integer p such that, locally, every coherent D(L_{X})-module M has a free resolution of less than p or equal to p. In other words, any point x has an open neighborhood V such that there exists a resolution of the type

0 → D(L_{V})^{l_{p}} → … → D(L_{V})^{l_{0}} → M_{|V} → 0.

The proof of proposition 2.1.5 in the D-modules case extends to the setting of Lie algebroids ([S2] p.14).

A sheaf of rings (respectively graded rings) is syzygic if it is coherent, has finite global homological dimension and syzygic fibers (i.e any finite type S(L_{X,x})-module has a finitely free resolution of finite length).

Proposition 2.1.6 The sheaf S(L_{X}) is syzygic. Hence, locally, any coherent FD(L_{X})-module FM has a finite free resolution. In other words, any point x has an open neighborhood V such that there exists a resolution of the type

⋯ → ⊕_{i=1}^{n_{1}}FD(L_{V})(r_{i,1}) → ⊕_{i=1}^{n_{0}}FD(L_{V})(r_{i,0}) → FM_{|V} → 0.

See [S2] for details.

Proposition 2.1.7 Let M^{•} be a complex of FD(L_{X})-modules. If H^{j} (GrM^{•}) is a coherent S(L_{X})-module, then H^{j}(M^{•}) is a coherent D(L_{X})- module.

Proof of proposition 2.1.7:

We put

M^{•} : 0 → M_{0} ⟶^{d_{0}} M_{1} ⟶ …

For each i, endow Kerd_{i} and Imd_{i−1} with the induced filtration. We have the following strict short exact sequence (i.e the arrows are strict morphisms)

0 → Imd_{i−1} → Kerd_{i} → H^{i}(M^{•}) → 0.

Consequently, we get the short exact sequence

0 → GrImd_{i−1} → GrKerd_{i} → GrH^{i}(M^{•}) → 0.

And, as GrKerd_{i} ⊂ KerGrd_{i} and ImGrd_{i} ⊂ GrImd_{i}, we have the following inclusion

GrH^{i}(M^{•}) ⊂ H^{i}(GrM^{•})

from which the proposition follows.

2.2 Resolution of O_{X} as a left D (L_{X})-module.

Let (L_{X}, ω_{X}) be a Lie algebroid over X. Set L_{X}^{∗} = Hom_{O_{X}} (L_{X}, O_{X}). Consider the graded left D(L_{X})-module D(L_{X}) ⊗_{O_{X}} ⋀^{•}(L_{X}) = ⊕_{n} D(L_{X}) ⊗_{O_{X}}Λ^{n}L_{X}

where D(L_{X}) acts by left multiplication. It is known ([R] p 200) that the endomorphism of degree -1, d, defined below is a differential on D(L_{X}) ⊗_{O_{X}} ⋀^{•}(L_{X}).

∀v ∈ D(L_{X}), ∀ξ_{i} ∈ L_{X},

d(v ⊗ ξ_{1} ∧ ... ∧ ξ_{n}) = ∑_{i=1}^{n}(−1)^{i−1}vξ_{i} ⊗ ξ_{1} ∧ ... ∧ ξ̂_{i} ∧ ... ∧ ξ_{n}

+ ∑_{k<i}(−1)^{i+k}v ⊗ [ξ_{k}, ξ_{i}] ∧ ξ_{1} ∧ ... ∧ ξ̂_{k} ∧ ... ∧ ξ̂_{i} ∧ ... ∧ ξ_{n}

d(v ⊗ 1) = ω_{X}(v)(1)

where the notation x̂ means that x is omitted.

Theorem 2.2.1 Let (L_{X}, ω_{X}) be a Lie algebroid. The complex K_{L_{X}}^{•} defined by

∀n ∈ ZZ, K_{L_{X}}^{−n} = D(L_{X}) ⊗_{O_{X}}Λ^{n}(L_{X})

and the differential above is a resolution of O_{X} by locally free left D(L_{X})- modules. Filter K_{L_{X}}^{i} by F_{k}K_{L_{X}}^{i} = F_{k}D(L_{X}) ⊗_{O_{X}}Λ^{i}L_{X}. Then

0 → K_{L_{X}}^{d_{L_{X}}} → K_{L_{X}}^{d_{L_{X}} −1}(1) → ⋯ → K_{L_{X}}^{1}(d_{L_{X}} − 1) → K_{L_{X}}^{0}(d_{L_{X}}) → O_{X} → 0

is a resolution of O_{X} by locally free left FD(L_{X})-modules.

Proof of theorem 2.2.1 : See [R] p 202.

We call K_{L_{X}}^{•} the Koszul resolution of the left D(L_{X})-module O_{X}.

2.3 Lie algebroid morphisms

Definition 2.3.1 Let (L_{X}, ω_{X}) and (L_{Y} , ω_{Y}) be Lie algebroids over X and Y respectively. A morphism Φ from (L_{X}, ω_{X}) to (L_{Y} , ω_{Y}) is a pair (f, F) such that

• f : X → Y is a holomorphic map

• F : L_{X} → f^{∗}L_{Y} = O_{X} ⊗_{f^{−1}O_{Y}} f^{−1}L_{Y} is an O_{X}-module morphism such

that the two following conditions are satisfied:

1) The diagram

[figure: Θ_{X}; f^{∗}Θ_{Y}; T f; L_{X}; f^{∗}L_{Y}; F; ω_{X}; f^{∗}ω_{Y}]

commutes (where T f is the differential of f).

2) Let ξ and η be two sections of L_{X}. Put F (ξ) = ∑_{i=1}^{m}a_{i} ⊗ ξ_{i} and

F (η) = ∑_{j=1}^{m}b_{j} ⊗ η_{j} with a_{i}, b_{j} ∈ O_{X} and ξ_{i}, η_{j} ∈ f^{−1}L_{Y}. Then

F ([ξ, η]) = ∑_{j=1}^{n}ω_{X}(ξ)(b_{j}) ⊗ η_{j} − ∑_{i=1}^{n}ω_{X}(η)(a_{i}) ⊗ ξ_{i} + ∑_{i,j}a_{i}b_{j} ⊗ [ξ_{i}, η_{j}].

The condition 2) is equivalent to the following property : O_{X} ⊗_{f^{−1}O_{Y}} f^{−1}D(L_{Y})

endowed with the two operations below is a left D(L_{X})-module.

∀(a, b) ∈ O_{X}^{2}, ∀ξ ∈ L_{X}, ∀v ∈ f^{−1}D_{Y}
a · (b ⊗ v) = ab ⊗ v
ξ · (b ⊗ v) = ω_{X}(ξ)(b) ⊗ v + ∑_{i} ba_{i} ⊗ ξ_{i}v

(where F (ξ) = ∑_{i}a_{i} ⊗ ξ_{i} with a_{i} ∈ O_{X} and ξ_{i} ∈ f^{−1}L_{Y}).

Our definition ([C2], [C3]) coincides with that of Almeida and Kumpera ([A-K]).

Notation : O_{X} ⊗_{f^{−1}O_{Y}} f^{−1}D(L_{Y}) considered as a D(L_{X}) ⊗ f^{−1}D(L_{Y})^{op}-module will be de-

noted D_{L_{X}→L_{Y}} (as in [C3]).

Remark : Let (L_{X}, ω_{L_{X}}) be a Lie algebroid. Then Kerω_{L_{X}} endowed with the operations below is a D(L_{X})-module which is coherent as an O_{X}-module: for any f ∈ O_{X}, any D ∈ L_{X} and any Δ ∈ Kerω_{L_{X}},

\begin{matrix}D · Δ = [D, Δ] \\ f · Δ = fΔ \end{matrix}

The composition of two Lie algebroid morphisms is a Lie algebroid morphism.

2.4 Examples

1) The Lie algebroid (X, Θ_{X}, id) gives rise to the usual ring of differential operators. Moreover, if f : X → Y is a morphism between complex analytic manifolds, then its differential T f : Θ_{X} → O_{X} ⊗_{f^{−1}O_{Y}} f^{−1}Θ_{Y} defines a Lie

algebroid morphism (f, T f) from (X, Θ_{X}) to (Y, Θ_{Y}).

2) Let 𝔤 be a finite dimensional Lie algebra. It is a Lie algebroid over a point with trivial anchor map. The ring of differential operators in this case is the universal enveloping algebra of 𝔤. Lie algebroid morphisms generalize Lie algebra morphisms.

3) Let 𝔤 be a Lie algebra. Assume that there is a Lie algebra morphism σ : 𝔤 → Θ_{X}. Then O_{X} ⊗ 𝔤 has a natural Lie algebroid structure with anchor map ω defined by

∀f ∈ O_{X}, ∀ξ ∈ 𝔤, ω(f ⊗ ξ) = fσ(ξ).

The Lie algebra bracket on O_{X} ⊗ 𝔤 is given

[f ⊗ ξ, g ⊗ η] = fσ(ξ)(g) ⊗ η − gσ(η)(f) ⊗ ξ + fg ⊗ [ξ, η].

Let G and G′ be two complex Lie groups with Lie algebras 𝔤 and 𝔤′ and let χ : G → G′ be a Lie group morphism. We will denote by dχ the differential of χ at the unity. Let X (respectively X′) be an analytic manifold with action of G (respectively G′). Let f : X → X′ be an equivariant map in the sense that

∀g ∈ G, ∀x ∈ X, f(g · x) = χ(g) · f(x).

Let us define F : O_{X} ⊗ 𝔤 → O_{X} ⊗_{f^{−1}O_{X′}}f^{−1} (O_{X′} ⊗ 𝔤′) by

F (f ⊗ ξ) = f ⊗ 1 ⊗ dχ(ξ)

Then (f, F) is a Lie algebroid morphism from O_{X} ⊗ 𝔤 to O_{X′} ⊗ 𝔤′.

4) Take the same notation as in the example 5. Assume that X = V is a finite dimensional vector space and that G is a connected algebraic group acting on V. Put LieG = 𝔤 and L_{V} = ω(O_{V} ⊗𝔤). If max_{v∈V} dimG·v = dimG, then L_{V} is a locally free O_{V} -module ([P] p. 186) and (V, L_{V}) (with the natural embedding as an anchor) is a Lie algebroid.

5) Let X be an analytic Poisson manifold. The Poisson bracket on O_{X} is denoted by { , }. The O_{X}-module of differential forms of degree 1, Ω_{X}^{1}, is endowed with a natural Lie algebroid structure (see [Hu]) with anchor map

\begin{matrix}Ω_{X}^{1} & → & Θ_{X} \\ fdg & ↦ & f\{g, •\}. \end{matrix}

Recall that the Lie bracket on Ω_{X}^{1} is given by

[f ⊗ da, g ⊗ db] = fg ⊗ d{a, b} + f{a, g} ⊗ db − g{b, f} ⊗ da.

Let Y be another Poisson analytic manifold and let f : X → Y be a Poisson map. We endow the O_{X}-module O_{X} ⊗_{f^{−1}O_{Y}} f^{−1}Ω_{Y}^{1} with a Lie algebroid

structure as follows : We define the Lie algebra bracket by

∀(a, a′) ∈ O_{X}, ∀(b, b′, v, v′) ∈ O_{Y} ; \begin{aligned}[a ⊗ dv, a′ ⊗ dv′] & = & a\{v ∘ f, a′\} ⊗ dv′ − a′\{v′ ∘ f, a\} ⊗ dv \\ & + & aa′ ⊗ d\{v, v′\} \end{aligned}

and the anchor map by

\begin{matrix}O_{X} ⊗_{f^{−1}O_{Y}} f^{−1}Ω_{Y}^{1} & → & Θ_{X} \\ a ⊗ dv & ↦ & a\{v ∘ f, .\}. \end{matrix}

The couple (f, id) is a Lie algebroid morphism from (X, O_{X} ⊗_{f^{−1}O_{Y}} f^{−1}Ω_{Y}^{1} ) to

(Y, Ω_{Y}^{1} ).

We construct the following correspondence :

\xymatrix{ & O_{X} ⊗_{f^{−1}O_{Y}} f^{−1}Ω_{Y}^{1} \ar@{-->}[dl]_{(id,F)} \ar@{-->}[dr]^{(f,id)} \\ (X, Ω_{X}^{1}) &  & (Y, Ω_{Y}^{1} ),}

where F is defined as follows : for all α in O_{X} and all β in O_{Y},

\begin{matrix}O_{X} ⊗_{f^{−1}O_{Y}} f^{−1}Ω_{Y}^{1} & → & Ω_{X}^{1} \\ α ⊗ βdq & ↦ & α(β ∘ f)d(q ∘ f). \end{matrix}

6) Let (X, L_{X}) be a Lie algebroid. We associate to it the abelian Lie algebroid (X, L_{X}^{ab}) defined by

• L_{X}^{ab} = L_{X}

• The Lie bracket on L_{X}^{ab} is zero

• The anchor map on L_{X}^{ab} is 0.

Let Φ = (f, F) be a Lie algebroid morphism from (X, L_{X}) to (Y, L_{Y}). It induces a Lie algebroid morphism Φ^{ab} = (f, F^{ab}) from (X, L_{X}^{ab}) to (Y, L_{Y}^{ab}).

For other examples of Lie algebroids, see [M] and [C3].

2.5 The characteristic variety

Let (X, L_{X}) be a Lie algebroid over X. Let L_{X} be the vector bundle associated with L_{X} and let π be the projection from L_{X}^{∗} to X. As in the D-modules case ([S2]), one can define the notion of good filtration for a coherent D(L_{X})-module (see [Bj] p. 24). Locally, each coherent D(L_{X})- module admits a good filtration. Let N be a coherent D(L_{X})-module and let U be an open subset on which N_{|U} admits a good filtration. The subset

Supp (O_{L_{U}^{∗}} ⊗_{π^{−1}S(L_{U})}π^{−1}GrN_{|U} ) does not depend on the good filtration. The

characteristic variety, char(N), is the closed conic subset of L_{X}^{∗} defined by

Supp (O_{L_{U}^{∗}} ⊗_{π^{−1}S(L_{U})}π^{−1}GrN ) = char(N) ⋂ L_{U}^{∗}.

If N^{•} is an element of D^{b} (D(L_{X})), set

char(N^{•}) = ⋃_{j∈Z}charH^{j} (N^{•}) .

Then charN^{•} = charN^{•}[1]. As in the D-modules case, one can show the following property: If N^{′•} → N^{•} → N^{′′•} →^{+1} is a distinguished triangle in D_{coh}^{b} (D(L_{X})), then charN^{•} ⊂ charN^{′•} ∪ charN^{′′•}.

3 Some operations for modules on Lie algebroids

In this section, we generalize some basic notions of D-modules theory due to Bernstein and Kashiwara. We refer the reader to [Bj], [Bo], [Ho] and [S2] for an exposition.

3.1 Left and right-modules

The following proposition is classical for D-modules and is easy to generalize to Lie algebroids ([C2], [C3]).

Proposition 3.1.1 a) If N and N′ are left D(L_{X})-modules, then N ⊗_{O_{X}}N′

endowed with the two following operations :

∀a ∈ O_{X}, ∀n ∈ N , ∀n′ ∈ N′, ∀D ∈ L_{X}
a · (n ⊗ n′) = a · n ⊗ n′ = n ⊗ a · n′
D · (n ⊗ n′) = D · n ⊗ n′ + n ⊗ D · n′

is a left D(L_{X})-module.

b) If M (respectively N) is a right (respectively a left) D(L_{X})-module, then M ⊗_{O_{X}}N endowed with the two following operations :

∀a ∈ O_{X}, ∀m ∈ M, ∀n ∈ N , ∀D ∈ L_{X}
(m ⊗ n) · a = m ⊗ a · n = m · a ⊗ n
(m ⊗ n) · D = m · D ⊗ n − m ⊗ D · n

is a right D(L_{X})-module.

c) If M and M′ are two right D(L_{X})-modules, then Hom_{O_{X}} (M, M′) endowed with the two following operations

∀ϕ ∈ Hom_{O_{X}} (M, M′) , ∀m ∈ M, ∀a ∈ O_{X}, ∀D ∈ L_{X}
(a · ϕ) (m) = ϕ(m) · a
(D · ϕ) (m) = −ϕ(m) · D + ϕ(m · D)

is a left D(L_{X})-module.

d) If N and N′ are two left D(L_{X})-modules, then Hom_{O_{X}} (N , N′) endowed with the two following operations

∀ϕ ∈ Hom_{O_{X}} (N , N′) , ∀m ∈ M, ∀a ∈ O_{X}, ∀D ∈ L_{X}
(a · ϕ) (m) = a · ϕ(m)
(D · ϕ) (m) = D · ϕ(m) − ϕ(D · m)

is a left D(L_{X})-module.

The following theorem is now a consequence of the previous proposition.

Theorem 3.1.2 Let E be a right D(L_{X})-module which is a locally free O_{X}- module of rank one. The functor N^{•} ↦ E ⊗_{O_{X}}N^{•} establishes an equivalence of

categories between complexes of left and complexes of right D(L_{X})-modules. Its inverse functor is given by M^{•} ↦ Hom_{O_{X}} (E, M^{•}).

It is a well known fact that Ω_{X}^{dimX} (the sheaf of differential forms of maximal degree) is endowed with a right D_{X}-module structure (see [S2] p.9, [Bo] p. 226 ). By the morphism D(L_{X}) → D_{X}, Ω_{X}^{dimX} has a structure of right D(L_{X})-module. Hence theorem 3.1.2 applies in particular if E = Ω_{X}^{dimX}. Put

L_{X}^{∗} = Hom_{O_{X}} (L_{X}, O_{X})

and let d_{L_{X}} be the rank of L_{X}. Then one may take E = Λ^{d_{L_{X}}} (L_{X}^{∗}). Indeed, L_{X} acts on L_{X}^{∗} as follows :

< D, Δ · λ >=< [D, Δ], λ > +Δ(< D, λ >)

for any (D, Δ) ∈ L_{X}^{2} and any λ ∈ L_{X}^{∗}. Hence L_{X} acts on Λ(L_{X}^{∗}). The action of an element D of L_{X} on Λ(L_{X}^{∗}) is called the Lie derivative of D and is denoted L_{D}. Put det(L^{∗}) = Λ^{d_{L_{X}}} (L_{X}^{∗}). Then det(L_{X}^{∗}), endowed with the following two operations,

∀σ ∈ det(L_{X}^{∗}), ∀D ∈ L_{X}, ∀a ∈ O_{X}
σ · a = a · σ
σ · D = −L_{D}(σ)

is a right D(L_{X})-module (see [C1]).

Consider

\begin{aligned}H_{L_{X}} & = & Hom_{O_{X}} (Λ^{d_{L_{X}}} (L_{X}^{∗}), D(L_{X})) \\ & = & D(L_{X}) ⊗_{O_{X}}Λ^{d_{L_{X}}} (L_{X}) \end{aligned}

and

\begin{aligned}\overline{H}_{L_{X}} & = & Hom_{O_{X}} (Ω_{X}, D(L_{X})) \\ & = & D(L_{X}) ⊗_{O_{X}}Ω_{X}^{−1} \end{aligned}

They are endowed with a natural D(L_{X})⊗D(L_{X})-left module structure (the first left D(L_{X})-module structure is given by left multiplication, the second one is obtained by proposition 3.1.1 c) .

3.2 Duality functor

If N^{•} is an element of D_{coh}^{b} (D(L_{X})), we set

\underline{D}_{L_{X}} (N^{•}) = RHom_{D(L_{X})} (N^{•}, H_{L_{X}}) [d_{L_{X}}].

Since the natural arrow N^{•} ↦ \underline{D}_{L_{X}} (\underline{D}_{L_{X}} (N^{•})) is an isomorphism (see [C3]), one calls \underline{D}_{L_{X}} a duality functor. Similarly, one sets

\underline{Δ}_{L_{X}} (N^{•}) = RHom_{D(L_{X})} (N^{•}, \overline{H}_{L_{X}} ) [dimX].

\underline{Δ}_{L_{X}} is a duality functor.

Let N and N′ be two D(L_{X})-modules. We endow RHom_{O_{X}} (N , N′) with a left D(L_{X})-module structure as follows. Let I^{•} be an injective resolution of N′ in the category of left D(L_{X})-modules. Then, as D(L_{X}) is a flat O_{X}- module, I^{•} is an injective resolution of N′ in the category of O_{X}-modules. Hence, one has

RHom_{O_{X}} (N , N′) ≃ Hom_{O_{X}} (N , I^{•}) .

As, by proposition 3.1.1, the right-hand side is endowed with a left D(L_{X})- module structure, so is the left-hand side. In other words, deriving the functor Hom_{O_{X}} (N , •) in the category of left D(L_{X})-modules is the same as deriving it in the category of O_{X}-modules.

Lemma 3.2.1 Let N and N′ be two left D(L_{X})-modules. The objects RHom_{D(L_{X})} (N , N′) and RHom_{D(L_{X})} (O_{X}, RHom_{O_{X}}(N , N′)) are isomorphic in D^{b}(C_{X})

Proof of lemma 3.2.1 :

Let K_{L_{X}}^{•} be the Koszul resolution of O_{X} and let N′ → I^{•} be an injective resolution of the D(L_{X})-module N′. One has the following sequence of isomorphisms :

RHom_{D(L_{X})} (O_{X}, RHom_{O_{X}}(N , N′)) ≃ Hom_{D(L_{X})} (K_{L_{X}}^{•}, Hom_{O_{X}}(N , I^{•}))
≃ Hom_{D(L_{X})} (K_{L_{X}}^{•} ⊗_{O_{X}}N , I^{•}))
≃ RHom_{D(L_{X})} (N , N′)

Proposition 3.2.2 If N is a left D(L_{X})-module which is coherent as an O_{X}- module, then \underline{D}_{L_{X}} (N) and RHom_{O_{X}} (N , O_{X}) are isomorphic in D^{b}(O_{X}).

This proposition is well known for D-modules (see [Ho] p. 93). Note that in the D-module case, the O_{X}-module N is necessarily locally free of finite rank.

Proof of proposition 3.2.2 : If N is a locally free O_{X}-module of finite rank, the proposition was already proved in [C3].

Assume now that N is only a coherent O_{X}-module, then one has the following sequence of left D(L_{X})-modules.

\underline{D}_{L_{X}}(N) ≃ RHom_{D(L_{X})} (O_{X}, RHom_{O_{X}} (N , H_{L_{X}})) [d_{L_{X}}]
≃ RHom_{D(L_{X})} (O_{X}, RHom_{O_{X}} (N , O_{X}) ⊗_{O_{X}}H_{L_{X}} ) [d_{L_{X}}]
≃ \underline{D}_{L_{X}}(O_{X}) ⊗_{O_{X}}^{L}RHom_{O_{X}} (N , O_{X})
≃ RHom_{O_{X}} (N , O_{X}) .

The first isomorphism follows from lemma 3.2.1, the second one follows from the fact that N is a coherent O_{X}-module, the third one follows from the fact that O_{X} is a coherent D(L_{X})-module and the last isomorphism follows from \underline{D}_{L_{X}}(O_{X}) ≃ O_{X}.

Corollary 3.2.3 Let M^{•} and N^{•} be two elements of D_{coh}^{b}(D(L_{X})). There is an isomorphism from RHom_{D(L_{X})} (M^{•}, N^{•}) to

RHom_{D(L_{X})} (O_{X}, \underline{D}_{L_{X}}(M^{•}) ⊗_{O_{X}}^{L}N^{•})

Proof of corollary 3.2.3 :

Let K_{L_{X}}^{•} the Koszul resolution of O_{X} and let P^{•} → M_{|V}^{•} be a bounded locally free resolution of M_{|V}^{•}. The morphism we are looking for, χ_{L_{X}} (P^{•}, N^{•}) or χ_{L_{X}} for short, can be made explicite as follows : It is a morphism from

Hom_{D(L_{X})}^{•} (P^{•}, N^{•}) to Hom_{D(L_{X})}^{•} (K_{L_{X}}^{•}, Hom_{D(L_{X})}^{•} (P^{•}, H_{L_{X}}) ⊗_{O_{X}}N^{•}) [d_{L_{X}}],

and if ϕ_{p} is in Hom_{D(L_{X})}^{p} (P^{•}, N^{•}) then χ(ϕ_{p}) is defined by

\begin{matrix}∀α ∈ D(L_{X}) ⊗_{O_{X}} Λ_{O_{X}}^{q}(L_{X}), χ(ϕ_{p})(α) = 0 if q ≠ d_{L_{X}} \\ ∀ω ∈ ⋀_{O_{X}}^{d_{L_{X}}} (L_{X}), χ(ϕ_{p})(1 ⊗ ω)(e_{i,r}) = (1 ⊗ ω) ⊗ ϕ_{p}(e_{i,r}) \end{matrix}

where (e_{i,r})_{i∈[1,n_{r}]} is a basis of the free module P^{r}. It is easy to check that χ is a morphism of complexes

Remark : Note that if M^{•} and N^{•} are bounded complexes of filtered FD(L_{X})-modules, we could take a resolution P^{•} of M_{|V}^{•} by finite free FD(L_{V})-modules.

Then FHom_{FD(L_{X})}^{•} (K_{L_{X}}^{•}, FHom_{FD(L_{X})}^{•} (P^{•}, H_{L_{X}}) ⊗_{O_{X}}N^{•}) [d_{L_{X}}] and

FHom_{FD(L_{X})}^{•} (P^{•}, N^{•}) are complexes of filtered FD(L_{V})-modules. The morphism χ_{L_{X}} (P^{•}, N^{•}) is a morphism of filtered complexes and

(#) Grχ_{L_{X}} (P^{•}, N^{•}) = χ_{L_{X}^{ab}} (GrP^{•}, GrN^{•}) .

Corollary 3.2.4 Let M^{•} and N^{•} be two elements of D_{coh}^{b}(D(L_{X})). The

groups Hom_{D^{b}(D(L_{X}))} (M^{•}, N^{•}) and Hom_{D^{b}(D(L_{X}))} (O_{X}, \underline{D}_{L_{X}}(M^{•}) ⊗_{O_{X}}^{L}N^{•}) are

isomorphic.

3.3 Direct images

In this paragraph we recall results of [C3].

Let Φ = (f, F) be a Lie algebroid morphism from (L_{X}, ω_{X}) to (L_{Y} , ω_{Y}). Let M^{•} be an object of D^{b} (D(L_{X})^{op}). In [C3], the direct image functor is defined by

\underline{Φ}_{!}(M^{•}) = Rf_{!} (M^{•} ⊗_{D(L_{X})}^{L}D_{L_{X}→L_{Y}} ) .

Then \underline{Φ}_{!}(M^{•}) is in D^{b} (D(L_{Y})^{op}). If Φ = (f, T f), we recover the D-module construction (see [S2] for example). Then D_{Θ_{X}→Θ_{Y}} is denoted by D_{X→Y} and \underline{Φ}_{!} is denoted by \underline{f}_{!}.

Proposition 3.3.1 Let Φ and Ψ be Lie algebroids morphism from (L_{X}, ω_{X}) to (L_{Y} , ω_{Y}) and from (L_{Y} , ω_{Y}) to (L_{Z}, ω_{Z}) respectively, then

\underline{Ψ}_{!} ∘ \underline{Φ}_{!} = (\underline{Ψ ∘ Φ})_{!} .

The proof of proposition 3.3.1 is similar to the D-modules case (see [Bo] p. 251).

We recall here a definition due to Kashiwara (see [S-S]).

Definition 3.3.2 A right coherent D(L_{X})-module is good if, for any compact subset K of X, there exists an open neighborhood U of K such that M_{|U} has a filtration (M_{k})_{k∈[1,n]} by coherent right D(L_{U})-submodules such that each quotient M_{k}/M_{k−1} is generated by a coherent O_{U}-module.

Note that if X is a smooth algebraic variety, all the coherent D(L_{X})- modules are good. Good D(L_{X})-modules form a thick subcategory of the category of coherent D(L_{X})-modules. The associated full subcategory of D^{b} (D(L_{X})^{op}) consisting of objects with good cohomology is denoted by D_{good}^{b} (D(L_{Y})^{op})

Theorem 3.3.3 Assume that M^{•} is in D_{good}^{b} (D(L_{X})^{op}) and that f is proper on Supp(M), then \underline{Φ}_{!}(M) is in D_{good}^{b} (D(L_{Y})^{op}).

The proof of Schneiders ([S2] p. 38) in the case of D-modules extends without any change to our situation. The particular case where f is projective and M has a global good filtration was treated in [Ka].

Theorem 3.3.4 Let X and Y be two complex manifolds. Let (L_{X}, ω_{X}) and (L_{Y} , ω_{Y}) be Lie algebroids over X and Y respectively. Let Φ = (f, F) be a Lie algebroid morphism from (L_{X}, ω_{X}) to (L_{Y} , ω_{Y}). Let M^{•} be an element of D_{good}^{b} (D(L_{X})^{op}) such that f is proper on the support of M^{•}. Then there is a functorial isomorphism from \underline{Φ}_{!}Δ_{L_{X}} (M^{•}) to Δ_{L_{Y}} \underline{Φ}_{!} (M^{•}) in D_{good}^{b} (D(L_{Y})^{op}) .

Theorem 3.3.4 generalizes a result in Schneiders’ thesis [S1] (see also the work of Schapira-Schneiders [S-S]) where the case of relative differential operators is treated. The algebraic smooth case had been previously treated by Bernstein ([Be], [Bo], [Ho]) (in the D-modules context ) for a proper morphism. Moreover Mebkhout had treated the absolute case (i.e Y consists of a single point, see corollary 4.3.6 in [Me1], [Me2]).

3.4 Inverse image

Let Φ = (f, F) be a Lie algebroid morphism from (L_{X}, ω_{X}) to (L_{Y} , ω_{Y}). Let R^{•} be an object of D^{b} (D(L_{Y})). Set

\underline{Φ}^{−1}(R^{•}) = D_{L_{X}→L_{Y}} ⊗_{f^{−1}D(L_{Y} )}^{L}f^{−1}R^{•}.

Then \underline{Φ}^{−1}(R^{•}) is in D^{b} (D(L_{X})). We call it the inverse image of R^{•} by Φ. If Φ = (f, T f), we recover the D-module construction (see [S2] for example). Then D_{Θ_{X}→Θ_{Y}} is denoted by D_{X→Y} and \underline{Φ}^{−1} is denoted by \underline{f}^{−1}.

Remark : If X = Y and f = id, then \underline{Φ^{−1}}(R^{•}) is nothing but R^{•} considered as an element of D^{b} (D(L_{X})). We will write \underline{Φ^{−1}}(R^{•}) = R_{|L_{X}}^{•}.

To define the inverse image of an object of D^{b} (D(L_{Y})^{op}), as in the Dmodules case, one uses the (f^{−1}D(L_{Y}) ⊗ D(L_{X})^{op})-bimodule D_{L_{Y} ←L_{X}} defined by

D_{L_{Y} ←L_{X}} = Λ^{d_{L_{X}}} (L_{X}^{∗}) ⊗_{O_{X}}D_{L_{X}→L_{Y}} ⊗_{f^{−1}O_{Y}} f^{−1}Λ^{d_{L_{Y}}} (L_{Y}).

Proposition 3.4.1 Let Φ and Ψ be Lie algebroids morphisms from (L_{X}, ω_{X}) to (L_{Y} , ω_{Y}) and from (L_{Y} , ω_{Y}) to (L_{Z}, ω_{Z}) respectively . Then

\underline{Φ}^{−1} ∘ \underline{Ψ}^{−1} = (\underline{Ψ ∘ Φ})^{−1}.

The proof of proposition 3.4.1 is analogous to the D-modules case (see [Bo] p. 251).

Proposition 3.4.2 Let Φ be a Lie algebroid morphism from (L_{X}, ω_{X}) to

(L_{Y} , ω_{Y}). There is an isomorphism between \underline{Φ^{−1}} (M^{•} ⊗_{O_{Y}}^{L} N^{•}) and

\underline{Φ^{−1}} (M^{•}) ⊗_{O_{X}}^{L}\underline{Φ^{−1}} (N^{•}) for any M^{•}, N^{•} in D^{b} (D(L_{Y}))

The proof of proposition 3.4.2 is analogous to the D-modules case. We refer the reader to [Bj].

Proposition 3.4.3 Let Φ = (f, F) be a Lie algebroid morphism from (X, L_{X}) to (Y, L_{Y}) and let Φ^{ab} = (f, F^{ab}) be the Lie algebroid morphism it induces between the abelian Lie algebroids (X, L_{X}^{ab}) and (Y, L_{Y}^{ab}). Let R be a filtered FD(L_{Y})-module and let

L^{•} : ⋯ → ⊕_{i=1}^{n_{1}}FD(L_{Y})(r_{i,1}) → ⊕_{i=1}^{n_{0}}FD(L_{Y})(r_{i,0}) → R → 0

be a filtered resolution of R. Then \underline{Φ}^{−1}(R) is isomorphic to O_{f^{−1}V} ⊗_{f^{−1}O_{V}} f^{−1}L^{•}

and we filter it by the image of the complex O_{f^{−1}V} ⊗_{f^{−1}O_{V}} f^{−1}F_{k}L^{•}. One has

an isomorphism

Gr\underline{Φ^{−1}}(R) ≃ \underline{Φ^{ab}^{−1}}(GrR).

Proof of proposition 3.4.3 :

The image of the complex O_{f^{−1}V} ⊗_{f^{−1}O_{V}} f^{−1}F_{k}L^{•} in O_{f^{−1}V} ⊗_{f^{−1}O_{V}} f^{−1}FL^{•} is

isomorphic to the complex O_{f^{−1}V} ⊗_{f^{−1}O_{V}} f^{−1}F_{k}L^{•} itself because the compo-

nents of L^{•} are of the type FD(L_{Y})(r). As GrFL^{i} is a free O_{V} -module,

the complexes Gr (O_{f^{−1}V} ⊗_{f^{−1}O_{V}} f^{−1}FL^{•}) and (O_{f^{−1}V} ⊗_{f^{−1}O_{V}} f^{−1}GrFL^{•}) are

isomorphic. This finishes the proof of the proposition.

The following question arises naturally: Let R^{•} be an element of D_{coh}^{b} (D(L_{Y})). Give a sufficient condition so that \underline{Φ}^{−1}(R^{•}) is in D_{coh}^{b}(D(L_{X})).

To do so, we introduce, as in the D-modules case, the notion of non characteristicity.

Let L_{X} (respectively L_{Y}) be the vector bundle associated to L_{X} (respectively L_{Y}). We have the following diagram

L_{X}^{∗} ⟵^{^{t}F} X ×_{Y} L_{Y}^{∗} ⟶^{F_{π}} L_{Y}^{∗}

where, for x ∈ X and λ ∈ L_{f(x)}^{∗}, one has

\begin{matrix}^{t}F (x, f(x), λ) = (x,^{t} F (λ)) \\ F_{π} (x, f(x), λ) = (f(x), λ) . \end{matrix}

Denote by NS(X ×_{Y} L_{Y}^{∗}) the null section of X ×_{Y} L_{Y}^{∗}. Let R^{•} be in D_{coh}^{b} (D(L_{Y})). We will say that R^{•} is non characteristic with respect to Φ if the following inclusion holds.

F_{π}^{−1} (char(R^{•})) ⋂{(x, f(x), λ) ∈ X ×_{Y} L_{Y}^{∗} | λ ∘ F_{x} = 0} ⊂ NS(X ×_{Y} L_{Y}^{∗}).

Kashiwara has shown that, in the D-modules case, the non characteristicity condition ensures that \underline{Φ}^{−1}(R^{•}) is in D_{coh}^{b}(D(L_{X})) (see [S2]). We will now generalize this result to Lie algebroids.

Theorem 3.4.4 Let (X, L_{X}) and (Y, L_{Y}) be Lie algebroids over the complex manifolds X and Y respectively. Let Φ = (f, F) be a Lie algebroid morphism from (X, L_{X}) to (Y, L_{Y}). Let R^{•} be an element of D_{coh}^{b}(D(L_{Y})) which is supposed to be non characteristic with respect to Φ. Then \underline{Φ}^{−1}(R^{•}) is in D_{coh}^{b}(D(L_{X})) and

Char(\underline{Φ}^{−1}(R^{•})) ⊂^{t} F F_{π}(CharR^{•}).

Proof of theorem 3.4.4

Let R^{•} be an element of D_{coh}^{b}(D(L_{Y})) which is supposed to be non characteristic for Φ. By a standard induction argument on the number of non zero cohomology groups of R^{•}, one sees that it is enough to prove the theorem for a module which we will denote by R. The question being local, we may assume

\begin{matrix}L_{X} = O_{X}e_{1} ⊕ … ⊕ O_{X}e_{r} \\ L_{Y} = O_{Y} e_{r+1} ⊕ … ⊕ O_{Y} e_{m}. \end{matrix}

One may shrink Y so that R is endowed with a good filtration. The module GrR is then a graded coherent S(L_{Y})-module non characteristic for the morphism Φ^{ab} (the morphism induced by Φ on the abelian Lie algebroids L_{X}^{ab} and L_{Y}^{ab}).

Hence propositions 2.1.7 and 3.4.3 show that it is enough to prove the theorem in the case of Φ^{ab} ( that is to say the abelian case) for a graded coherent S(L_{Y})-module non characteristic for Φ^{ab}. Introduce the following maps :

\begin{matrix}t : X & → & X × Y \\ x & ↦ & (x, f(x)) \end{matrix}

\begin{matrix}q : X × Y & → & Y \\ (x, y) & ↦ & f(x) \end{matrix}

\begin{matrix}T : L_{X} & → & L_{X} ⊕ (O_{X} ⊗_{f^{−1}O_{Y}} f^{−1}L_{Y} ) \\ D & ↦ & D + F (D) \end{matrix}

and the following abelian Lie algebroids :

\begin{matrix}(X × Y, P_{X×Y}^{ab} = O_{X×Y} ⊗_{p_{1}^{−1}O_{X}}p_{1}^{−1}T (L_{X}^{ab})) \\ (X × Y, Q_{X×Y}^{ab} = O_{X×Y} ⊗_{q^{−1}O_{Y}} q^{−1}L_{Y}^{ab}) . \end{matrix}

(id, T) is a Lie algebroid isomorphism from (X, L_{X}^{ab}) to (X, T (L_{X}^{ab})).

The pair (t, id) defines a Lie algebroid morphism from (X, T (L_{X})^{ab}) to (X × Y, P_{X×Y}^{ab} ). Let ψ be the map

\begin{matrix}P_{X×Y}^{ab} & → & Q_{X×Y}^{ab} \\ b ⊗ (D + ∑_{i}a_{i} ⊗ Δ_{i}) & ↦ & b (∑_{i}a_{i} ⊗ Δ_{i}) . \end{matrix}

The pair Ψ^{ab} = (id, ψ) is a Lie algebroid morphism from P_{X×Y}^{ab} to Q_{X×Y}^{ab}. Lastly the pair (q, id) is a Lie algebroid morphism from (X × Y, Q_{X×Y}^{ab} ) to

(Y, L_{Y}^{ab}). Then we have (f, F^{ab}) = (q, id) ∘ (id, ψ) ∘ (t, id) ∘ (id, T). This factorization shows that it is enough to prove our proposition in the cases where (f, F^{ab}) = (id, F^{ab}) and (f, F^{ab}) = (f, id).

a) Case where Φ^{ab} = (f, id).

Necessarily L_{X}^{ab} = O_{X} ⊗_{f^{−1}O_{Y}} f^{−1}L_{Y}^{ab}. In this case any module is non char-

acteristic with respect to Φ^{ab} and \underline{Φ^{ab}}^{−1} (S_{O_{Y}} (L_{Y}^{ab})) = S_{O_{X}}(L_{X}^{ab}). Thus the coherence is established.

b) Case where Φ^{ab} = (id, F^{ab}).

We introduce the abelian Lie algebroid over Z = X×Y , (Z, 𝔞_{Z} = P_{Z}^{ab} ⊕ Q_{Z}^{ab}) and we factorize (id, F^{ab}) = (id, Π^{ab}) ∘ (id, I^{ab}) where

\begin{matrix}I^{ab} : P_{Z}^{ab} & → & 𝔞_{Z} \\ D & ↦ & D + F^{ab}(D) \end{matrix}

\begin{matrix}Π^{ab} : 𝔞_{Z} & → & Q_{Z}^{ab} \\ D + Δ & ↦ & Δ \end{matrix}

The case of (id, Π^{ab}) is easy. Let us now treat the case of (id, I^{ab}). Put P_{Z}^{′ab} = I^{ab}(P_{Z}^{ab}). Introduce the following abelian Lie algebroids with zero anchor (over Z):

V_{j} = P_{Z}^{′ab} ⊕ O_{Z}e_{r+1} … ⊕ O_{Z}e_{r+j}.

We have the following sequence of Lie algebroids embeddings :

V_{0} = P_{Z}^{′ab} ↪ V_{1} ↪ … ↪ V_{m−r} = 𝔞_{Z}.

So we may assume that m − r = 1 and we are in the following situation

I^{ab} : P_{Z}^{′ab} ↪ P_{Z}^{′ab} ⊕ O_{Z}e_{r+1}.

Let η be the element of 𝔞_{Z}^{∗} such that η_{|P′_{Z}} = 0 and η(e_{r+1}) = 1.

Restricting Z, we can take (σ_{1}, …, σ_{q}) to be a generating system of the S(𝔞_{Z})-module R. For each j, denote by I_{j} the annihilating (graded) ideal of

σ_{j} in S(𝔞_{Z}). The module \frac{S(𝔞_{Z})}{I_{j}} is non characteristic for Φ^{ab}.

Let z be in Z. If t ≠ 0, the pair (z, te_{r+1}^{∗}) is not in char(\frac{S(𝔞_{Z})}{I_{j}}). So there

exists an homogeneous element s_{j} in I_{j} such that s_{j}(z, te_{r+1}) ≠ 0. Put

s_{j} = ∑_{l=0}^{p_{j}}α_{l,j}e_{r+1}^{l}

with α_{l,j} ∈ S^{p_{j}−l}(P′_{Z}). Shrinking Z, we may assume that α_{p_{j},j} is an invertible element of O_{Z}. We have a natural epimorphism

⊕_{j=1}^{q} \frac{S_{O_{Z}}(𝔞_{Z})}{< s_{j} >} → GrR → 0.

So it is enough to prove that \frac{S_{O_{Z}}(𝔞_{Z})}{< s_{j} >} is a coherent S_{O_{Z}}(P_{Z}^{′ab})-module. And

this is obvious because Euclidean division in S_{O_{Z}}(𝔞_{Z}) provides an isomor-

phism of S_{O_{Z}}(P′_{Z})-modules χ : \frac{S_{O_{Z}}(𝔞_{Z})}{< s_{j} >} → S_{O_{U}} (P′_{Z})^{p_{j}}. This finishes the

proof of the proposition.

4 A duality theorem

This section is devoted to the generalization of a duality theorem due to Kashiwara, Kawai and Sato in the D-modules case (see [SKK]).

4.1 Statement

We will construct a functorial arrow \underline{D}_{L_{X}} \underline{Φ}^{−1}(R^{•}) → \underline{Φ}^{−1}\underline{D}_{L_{Y}} (R^{•}) for any element R^{•} in D^{b}(D(L_{Y})) such that \underline{Φ}^{−1}(R^{•}) is in D_{coh}^{b} (D(L_{X})). Our construction will be in particular valid if R^{•} is non characteristic for Φ.

Using corollary 3.2.4 , the element 1 of Hom_{D^{b}(D(L_{Y} ))}(R^{•}, R^{•}) provides an arrow

O_{Y} → \underline{D}_{L_{Y}} (R^{•}) ⊗_{O_{Y}}^{L} R^{•}

in D^{b} (D(L_{Y})). Hence an arrow

\begin{aligned}\underline{Φ}^{−1}(O_{Y}) = O_{X} & → & \underline{Φ}^{−1} (\underline{D}_{L_{Y}} (R^{•}) ⊗_{O_{Y}}^{L} R^{•}) \\ & & ≃ \underline{Φ}^{−1} (\underline{D}_{L_{Y}} (R^{•})) ⊗_{O_{X}}^{L}\underline{Φ}^{−1} (R^{•}) \end{aligned}

Using the fact that \underline{D}_{L_{X}}^{2}(\underline{Φ}^{−1}(R^{•})) ≃ \underline{Φ}^{−1}(R^{•}) and corollary 3.2.4 again, we get an arrow

\underline{D}_{L_{X}} \underline{Φ}^{−1}(R^{•}) → \underline{Φ}^{−1}\underline{D}_{L_{Y}} (R^{•}).

Theorem 4.1.1 Let Φ be a Lie algebroid morphism from (X, L_{X}) to (Y, L_{Y}). Let R^{•} be in D_{coh}^{b} (D(L_{Y})) which is supposed to be non characteristic with respect to Φ. The functorial morphism from \underline{D}_{L_{X}} \underline{Φ}^{−1}(R^{•}) to \underline{Φ}^{−1}\underline{D}_{L_{Y}} (R^{•}) constructed above is an isomorphism.

4.2 Proof of theorem 4.1.1

As the proof of theorem 3.4.4, the proof of theorem 4.1.1 is into two steps. First, we will reduce to the abelian case, then we will prove the theorem in the abelian case.

a) Reduction to the abelian case.

Preliminary remark : Let Q^{•} be a bounded complex of locally free D(L_{X})-modules and N^{•} be a bounded complex of D(L_{X})-modules. Let C_{X} ⟶^{α} S^{•} be a resolution of C_{X} by c-soft sheaves. One has an isomorphism (see [K-S 3] exercise 4.7)

Hom_{D^{b}(D(L_{X}))}(Q^{•}, N^{•}) ≃ H^{0} (X, Hom_{D(L_{X})}^{•}(Q^{•}, N^{•}) ⊗_{C_{X}}S^{•}) .

There is a morphism

\begin{matrix}H^{0} (Hom_{D(L_{X})}^{•} (Q^{•}, N^{•})) & → & Hom_{D^{b}(D(L_{X}))}(Q^{•}, N^{•}) \\ [ϕ] & ↦ & [ϕ ⊗ α(1)]. \end{matrix}

If [ϕ] is an isomorphism, so is [ϕ ⊗ α(1)]. We will say that the morphism [ϕ ⊗ α(1)] is induced by ϕ.

To show that the morphism we have constructed is an isomorphism, we may assume that R^{•} is a module (denoted R) which is non characteristic with respect to the morphism Φ and endowed with a good filtration. Then GrR is a coherent S(L_{Y})-module non characteristic for Φ^{ab}. As we may reason locally, we may take P^{•} to be a resolution of R by finite free D(L_{Y})- modules. With the notation of corollary 3.2.3, we see that the arrow O_{Y} → \underline{D}_{L_{Y}} (R) ⊗_{O_{Y}}^{L} R in D^{b}(D(L_{Y})) is induced by the arrow Γ (X, χ_{L_{Y}} (P^{•}, P^{•})) (id)

from K_{L_{Y}}^{•} to Hom_{D(L_{Y} )} (P^{•}, H_{L_{Y}}) ⊗_{O_{Y}} P^{•}[d_{L_{Y}}].

The arrow O_{X} → \underline{Φ}^{−1} (\underline{D}_{L_{Y}} (R)) ⊗_{O_{X}}^{L}\underline{Φ}^{−1} (R) in D^{b}(D(L_{X})) is induced

by the morphism id_{O_{X}} ⊗ Γ (X, χ_{L_{Y}} (P^{•}, P^{•})) (id) from O_{X} ⊗_{f^{−1}O_{Y}} f^{−1}K_{L_{Y}}^{•} to

O_{X} ⊗_{f^{−1}O_{Y}} f^{−1} (Hom_{D(L_{Y} )} (P^{•}, H_{L_{Y}}) ⊗_{O_{Y}} P^{•}) [d_{L_{Y}}]. One sees that the arrow

\underline{D}_{L_{X}} ∘ \underline{Φ}^{−1}(R) → \underline{Φ}^{−1} ∘ \underline{D}_{L_{Y}} (R) is also induced by a morphism and we want to see that this morphism is an isomorphism.

We have

GrΓ (X, χ_{L_{Y}} (P^{•}, P^{•})) (id) = Γ (X, χ_{L_{Y}^{ab}}(GrP^{•}, GrP^{•})) (id)

\begin{matrix}Gr (id_{O_{X}} ⊗ f^{−1}Γ (X, χ_{L_{Y}} (P^{•}, P^{•})) (id)) & = \\ id_{O_{X}}⊗ & f^{−1}Γ (X, χ_{L_{Y}^{ab}}(GrP^{•}, GrP^{•})) (id). \end{matrix}

Using the last remark of section 3.2 (equality #), one sees that it is enough to show the theorem 4.1.1 for the case of Φ^{ab}.

b) Proof of theorem 4.1.1 in the abelian case.

From now on, we assume that Φ = Φ^{ab} and that R is a graded S(L_{Y})- module non characteristic for Φ^{ab}. As in the proof of theorem 3.4.4, we introduce the maps t, q, T as well as the abelian Lie algebroids P_{X×Y}^{ab} and Q_{X×Y}^{ab} and we factorize (f, F^{ab}) into

(f, F^{ab}) = (q, id) ∘ (id, ψ) ∘ (t, id) ∘ (id, T).

This factorization shows that it is enough to prove the theorem in the case where (f, F^{ab}) = (id, F^{ab}) and (f, F^{ab}) = (f, id).

Let us first treat the case where Φ^{ab} = (f, id). Necessarily L_{X} = O_{X} ⊗_{f^{−1}O_{Y}} f^{−1}L_{Y}. In this case any module is non characteristic with re-

spect to Φ^{ab} so that we can assume that R = S(L_{Y}). Then \underline{Φ^{ab}}^{−1} (S_{O_{Y}} (L_{Y}^{ab})) = S_{O_{X}}(L_{X}^{ab}) and the morphism of functor \underline{D}_{L_{X}} ∘ (\underline{Φ}^{ab})^{−1} → (\underline{Φ}^{ab})^{−1} ∘ \underline{D}_{L_{Y}} is the identity as easily checked for R = S(L_{Y}). So the theorem is proved in the case where Φ^{ab} = (f, Id).

Let us now treat the case where Φ^{ab} = (id, F). As in the proof of theorem 3.4.4, we introduce the abelian Lie algebroid over Z = X × Y, (Z, 𝔞_{Z} = P_{Z}^{ab} ⊕ Q_{Z}^{ab}) and we factorize (id, F^{ab}) into

(id, F^{ab}) = (id, Π^{ab}) ∘ (id, I^{ab}).

We can treat the cases where (id, F^{ab}) = (id, Π^{ab}) and (id, F^{ab}) = (id, I^{ab}) independently.

Assume that (id, F^{ab}) = (id, Π^{ab}). Recall that \underline{D}_{𝔞_{Z}} (\frac{S(𝔞_{Z})}{S(𝔞_{Z})P_{Z}^{ab}} ) is iso-

morphic to S(Q_{Z}^{ab}) ⊗_{O_{Z}}Λ^{d_{Q{ab \atop Z}}}(Q_{Z}^{ab})[d_{Q_{Z}^{ab}}] in D^{b} (S(Q_{Z}^{ab}) ⊗ S(Q_{Z}^{ab})^{op}) ([C2]).

These remarks allow us to write the following sequence of isomorphisms in D^{b}(S (Q_{Z}^{ab})) :

\underline{D}_{𝔞_{Z}} (R^{•}) ≃ \underline{D}_{𝔞_{Z}} (\frac{S(𝔞_{Z})}{S(𝔞_{Z})P_{Z}^{ab}} ⊗_{S(Q_{Z}^{ab})}R^{•}))
≃ RHom_{S(Q_{Z})} (R^{•}, \underline{D}_{𝔞_{Z}} (\frac{S(𝔞_{Z})}{S(𝔞_{Z})P_{Z}^{ab}} ))
≃ \underline{D}_{Q_{Z}} (R^{•}) .

Let us now treat the case where Φ^{ab} = (id, I^{ab}). Put P′_{Z} = I^{ab}(P_{Z}^{ab}). Reasoning as in the proof of theorem 3.4.4 and keeping the notation , we may assume that Q_{Y} is a one dimensional free O_{Y} -module and put Q_{Y} = O_{Y} e_{r+1}. We have then the following Lie algebroids embedding

P′_{Z} ↪ 𝔞_{Z} = P′_{Z} ⊕ O_{Z}e_{r+1}.

R is a graded coherent S(𝔞_{Z})-module non characteristic for the morphism Φ^{ab} = (id, I^{ab}).

We know from the proof of theorem 3.4.4 that there exists an epimorphism

⊕_{j=1}^{q} \frac{S_{O_{Z}}(𝔞_{Z})}{< u_{j} >} → R → 0

with u_{j} of the form

u_{j} = ∑_{l=0}^{p_{j}}α_{l,j}e_{r+1}^{l}

with α_{l,j} ∈ S^{p_{j}−l}(P′_{Z}) and α_{p_{j},j} invertible in O_{Z}.

Consequently, it is enough to show the theorem for R = \frac{S_{O_{Z}}(𝔞_{Z})}{< u_{j} >} and in

this case it is easy to make the arrow

\underline{D}_{P′_{Z}} ∘ (\underline{Φ}^{ab})^{−1}(R) → (\underline{Φ}^{ab})^{−1} ∘ \underline{D}_{𝔞_{Z}} (R)

explicit, and to see that it is an isomorphism. This finishes the proof of theorem 4.1.1.

Corollary 4.2.1 Let Ψ be a Lie algebroid morphism from (X, L_{X}) to (Y, L_{Y}). Let R be a left D(L_{Y})-module which is coherent as an O_{Y} -module. For short, put N^{∗} = RHom_{O_{Y}} (N , O_{Y}) with its D(L_{Y})-module structure defined in section 3.2. There is a functorial isomorphism of

right D(L_{X})-modules from Ext_{D(L_{X})}^{d_{L_{X}} +i} (D_{L_{X}→L_{Y}} ⊗_{f^{−1}D(L_{Y} )}^{L}f^{−1}N , D(L_{X})) to

T or_{i}^{f^{−1}D(L_{Y} )} (f^{−1} (N^{∗} ⊗ Λ^{d_{L_{Y}}} (L_{Y}^{∗})) , D_{L_{Y} ←L_{X}} ) .

5 Adjunction formulas and applications

5.1 Adjunction formulas

Generalizing the results of [K-S 2] ( chapter 7), we obtain the following adjunction formulas.

Theorem 5.1.1 Let (X, L_{X}) and (Y, L_{Y}) be two Lie algebroids over X and Y respectively. Set d_{L_{X}} = rank(L_{X}) and d_{L_{Y}} = rank(L_{Y}). Let Φ = (f, F) be a Lie algebroid morphism from (X, L_{X}) to (Y, L_{Y}).

a) Let M ∈ D^{b}(D(L_{Y})) and N ∈ D^{b}(D(L_{X})^{op}). Then \underline{Φ}_{!}(N) ⊗_{D(L_{Y} )}^{L}M and

Rf_{!} (N ⊗_{D(L_{X})}^{L}\underline{Φ}^{−1}(M)) are isomorphic in D^{b}(C_{Y}).

b) Assume that M is in D_{coh}^{b}(D(L_{Y})) and that Φ is non characteristic for M. The objects Rf_{!}RHom_{D(L_{X})} (\underline{Φ}^{−1}(M), N ) [d_{L_{X}} − d_{L_{Y}}] and RHom_{D(L_{Y} )} (M, \underline{Φ}_{!}(N)) are isomorphic in D^{b}(C_{Y})

c) Assume that N is in D_{good}^{b} (D(L_{X})) and that f is proper on SuppN. Then Rf_{∗}RHom_{D(L_{X})} (N , \underline{Φ}^{−1}M) [dimX − dimY]

and RHom_{D(L_{Y} )} (\underline{Φ}_{!}(Ω_{X} ⊗_{O_{X}}N), Ω_{Y} ⊗_{O_{Y}} M) are isomorphic in D^{b}(C_{Y})

Proof of theorem 5.1.1 : a) is straightforward , b) follows from theorem 4.1.1 and c) follows from theorem 3.3.4.

5.2 Applications

Application 1

Note that the proof of theorem 4.1.1 holds in the case of Lie algebras over any field k. As a corollary of Theorem 5.1.1, we get the result below where we adopt the following notation : Let V be an n dimensional vector space, and let det(V) = Λ^{n}(V).

Corollary 5.2.1 Let 𝔥 and 𝔤 be two finite dimensional Lie algebras over a field and let ϕ be a Lie algebra morphism from 𝔥 to 𝔤. Let M be a finitely generated U(𝔤)-module. Assume that

char(M) ∩ {λ ∈ 𝔤^{∗} | λ_{|ϕ(𝔥)} = 0} = {0}.

Let N be a 𝔥-module. Then, for all i in ZZ, we have an isomorphism

Ext_{U(𝔥)}^{i+dim𝔥} (M_{|𝔥}, N) ≃ Ext_{U(𝔤)}^{i+dim𝔤} (M ⊗ det(𝔤^{∗}), (N ⊗ det(𝔥^{∗})) ⊗_{U(𝔥)}^{L}U(𝔤)) .

Application 2

Let X be an analytic Poisson manifold. We consider the Lie algebroid (X, Ω_{X}^{1}) (see section 2.4). Let R^{•} be an element of D^{b} (D(Ω_{X}^{1})). The cohomology of the complex RHom_{D(Ω_{X}^{1})} (O_{X}, R^{•}) is called the canonical cohomology of X with values in R^{•} ([Li],[Hu]). We set

C_{can}^{•} (R^{•}) = RHom_{D(Ω_{X}^{1})} (O_{X}, R^{•}) .

Let (Y, Ω_{Y}^{1} ) be another holomorphic manifold endowed with a complex structure and let f : X → Y be a Poisson map. Consider the correspondence

\xymatrix{ & O_{X} ⊗_{f^{−1}O_{Y}} f^{−1}Ω_{Y}^{1} \ar@{-->}[dl]_{Φ=(id,F)} \ar@{-->}[dr]^{Ψ=(f,id)} \\ (X, Ω_{X}^{1}) &  & (Y, Ω_{Y}^{1} ),}

where F is defined as follows : for all α in O_{X} and all β in O_{Y},

\begin{matrix}O_{X} ⊗_{f^{−1}O_{Y}} f^{−1}Ω_{Y}^{1} & → & Ω_{X}^{1} \\ α ⊗ βdq & ↦ & α(β ∘ f)d(q ∘ f). \end{matrix}

Let N be a left D (O_{X} ⊗_{f^{−1}O_{Y}} f^{−1}Ω_{Y}^{1} )-module. Taking M = O_{Y}, the

adjunction formula tells us that we have the following isomorphisms

\begin{aligned}RHom_{D(O_{X} ⊗_{f^{−1}O_{Y}} f^{−1}Ω_{Y}^{1} )} (O_{X}, N) [dimY − dimX] & ≃ & RHom_{D(Ω_{X}^{1})} (O_{X}, \underline{Φ}_{!}N) \\ Rf_{!}RHom_{D(O_{X} ⊗_{f^{−1}O_{Y}} f^{−1}Ω_{Y}^{1} )} (O_{X}, N) & ≃ & RHom_{D(Ω_{Y}^{1} )} (O_{Y} , \underline{Ψ}_{!}N) \end{aligned}

Noticing that \underline{Ψ}_{!}(N) is nothing but N considered as a left D (Ω_{Y}^{1} )-module, we deduce the isomorphism

C_{can,Y}^{•} (N) ≃ Rf_{!}C_{can,X}^{•}(\underline{Φ}_{!}N)[dimX − dimY].

Application 3

Let (X, L_{X}) be a Lie algebroid over X. Let x_{0} be a point of X. Define the sheaf M_{x_{0}} of ideals of O_{X} by :

M_{x_{0}}(U) = {f ∈ O_{X}(U) | f(x_{0}) = 0}

for any open subset U of X. Define the (not necessarily locally free) Lie algebroid L(M_{x_{0}}) by

L(M_{x_{0}}) = {D ∈ L_{X} | D(M_{x_{0}}) ⊂ M_{x_{0}}}.

Let L(x_{0}) be the Lie algebra

\frac{O_{X}}{M_{x_{0}}} ⊗_{O_{X}}L_{X}(M_{x_{0}}) = L_{X}(x_{0}).

Let us consider the Lie algebroid morphism I = (ϵ, I) : ({pt}, L_{X}(x_{0})) → (X, L_{X}) where ϵ({pt}) = x_{0} and I is defined by tensoring the canonical embedding L_{X}(M_{x_{0}}) ↪ L_{X}. For any D(L_{X})-module R supposed to be non characteristic with respect to I, the adjunction formula gives

Ext_{U(L_{X}(x_{0}))}^{i+dimL_{X}(x_{0})} (\frac{R}{M_{x_{0}}R}, W ) ≃ Ext_{D(L_{X})}^{i+rkL_{X}} (R, \underline{I}_{!}(W)) .

In particular,

H^{i+dimL_{X}(x_{0})} (L_{X}(x_{0}), W) ≃ Ext_{D(L_{X})}^{i+rkL_{X}} (O_{X}, \underline{I}_{!}(W)) .

Example :

Assume that X = V is a finite dimensional vector space on which a connected algebraic group G acts locally freely (that is max_{v∈V} dimG · v = dimG). Put LieG = 𝔤. If L_{V} = O_{V} ⊗ 𝔤 (as in example 5 ) and x_{0} = 0_{V}, then L(0_{V}) = 𝔤. Consider the Lie algebroid M_{V} = (V, ω_{V} (O_{V} ⊗ 𝔤)). We have the isomorphism

H^{i+dim𝔤} (𝔤, W) ≃ Ext_{D(M_{V} )}^{i+rkM_{V}} (O_{V} , \underline{ω_{V} ∘ I}_{!}(W)) .

But dim𝔤 = rkM_{V} ([P] p. 186). Assume that 𝔤 is semi-simple. Then W can be written as a direct sum of irreducible representations. Let p be the multiplicity of k in W. Then

Ext_{D(M_{V} )}^{i} (O_{V} , \underline{ω_{V} ∘ I}_{!}(W)) = (Λ^{i}(𝔤)^{𝔤})^{p} .

6 Bibliography

[A-K] Almeida R. - Kumpera A. : Structure produit dans la catégorie des algébroides de Lie, Ann. Acad. Brasil Cienc. 53, 247-250 (1981).

[Be] Bernstein J. : Unpublished manuscript.

[Bj] Bjork J-E : Analytic D-modules and applications . Kluwer Academic Publishers (1993).

[Bo] Borel A. : Algebraic D-modules. Academic press (1987).

[C1] Chemla S. : Poincaré duality for k − A-Lie superalgebras, Bull. Sté. Math. France. 122, 371-397 (1994).

[C2] Chemla S. : Operations for modules on Lie-Rinehart superalgebras. Manuscripta Mathematica 87, 199 - 223 (1995).

[C3] Chemla S. : A duality property for complex Lie algebroids. Mathematische Zeitschrift 367-388, (1999).

[Ho] Hotta R. : Introduction to D-modules. Series of lectures given at the Institute of Mathematical Sciences, Madras, India.

[Hu] Huebschmann J. : Poisson cohomology and quantization, J. Reine. Angew. Math., 408, 1990, p 57-113.

[Ka] Kashiwara M. : b-functions and holonomic systems, Inventiones Mathematicae 38 (1976), 33-53. [K-S 1] Kashiwara M. - Schapira P. : Sheaves on manifolds, Springer Verlag (1990). [K-S 2] Kashiwara M. - Schapira P. : Moderate and formal cohomology associated with constructible sheaves, Mémoires de la Société Mathématique de France 64 (1996). [K-S 3] Kashiwara M. - Schapira P. : Ind - Sheaves, Astérisque (2001).

[Li] Lichnerowicz A. : Les variétés de Poisson et leurs algèbres associées, Journal of differential geometry 12 (1977), 253-300.

[M] Mackenzie K. - Lie groupoids and Lie algebroids in differential geometry - London mathematical Society Lecture Notes Series 124, Cambridge University Press.

[Me1] Mebkhout Z. : Théoreme de dualité pour les D_{X}-modules cohérents, C.R.Acad. Sci. Paris, 287, (1977), 785-787.

[Me2] Mebkhout Z. : Théorème de dualité pour les D_{X}-modules cohérents, Math. Scand. 50 (1982), 25-43.

[P] Panyushev D. - The Jacobian modules of a representation of a Lie algebra and geometry of commuting varieties, Compositio Mathematica 94 (1994), p. 181 - 199.

[R] Rinehart G.S - Differential forms on general commutative algebras - Trans. Amer. Math. Soc. 108, 195-222 (1963). [S-K-K] Sato M.- Kawai T.- Kashiwara M. : Hyperfunctions and pseudodifferential equations. In : Hyperfunctions and Pseudo-Differential Equations, H. Komatsu, editor. Lecture Notes in Mathematics 287 p. 265-529, Springer-Verlag (1973).

[S1] Schneiders J.P. : Thèse, Université de Liège, Octobre 1986.

[S2] Schneiders J.P. : An introduction to D-modules, Bulletin de la société royale des sciences de Liège, 63, 3-4, 223-295 (1994). [S-S] Schapira P. - Schneiders J.P. : Elliptic pairs I. Relative finiteness and duality, Astérisque 224 (1994).

Sophie Chemla, Université Pierre et Marie Curie, Institut de mathématiques, Analyse algébrique, 175 rue du Chevaleret, 75013 Paris. Sophie.Chemla@math.jussieu.fr .