## Un théorème de dualité relative pour les modules différentiels. (A relative duality theorem for differential modules).(French)Zbl 0605.14016

The purpose of this paper is to prove a duality theorem in the theory of differential modules. Let $$f: X\to Y$$ be a morphism between complex analytic manifolds and let $${\mathcal M}$$ be a coherent right $${\mathcal D}_ X$$-module where $${\mathcal D}_ X$$ is the sheaf of differential operators on X. Let us denote by \b{f}$${}_*$$ (resp. \b{f}$${}_ !)$$ the ordinary (resp. extraordinary) direct image functor for differential modules and let $${\mathbb{D}}_ X$$ (resp. $${\mathbb{D}}_ Y)$$ denote the differential dualizing functor on X (resp. Y). In this paper, the author constructs a canonical morphism $$\underline f_ !: {\mathbb{D}}_ X({\mathcal M})\to {\mathbb{D}}_ Y(\underline f_*{\mathcal M})$$ and establishes that it is an isomorphism if $${\mathcal M}$$ is f-coherent. The assumption of f-coherency is an adaptation to arbitrary analytic maps of the assumption formulated by C. Houzel and P. Schapira in the direct image theorem.
Reviewer: M.Muro

### MSC:

 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 32C38 Sheaves of differential operators and their modules, $$D$$-modules