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


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