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Sur la catégorie dérivée des $${\mathcal D}$$-modules filtrés. (French) Zbl 0551.14006
Algebraic geometry, Proc. Jap.-Fr. Conf., Tokyo and Kyoto 1982, Lect. Notes Math. 1016, 151-237 (1983).
[For the entire collection see Zbl 0511.00009.]
This paper deals with the purely algebraic proof of a result obtained by M. Kashiwara on the coherency of the direct image of coherent modules over the ring of differential operators on complex algebraic varieties. Let $$f: X\to Y$$ be a projective morphism between two complex quasi- projective varieties. Let $${\mathcal D}_ X$$ (resp. $${\mathcal D}_ Y)$$ be the sheaf of algebraic differential operators on X (resp. Y) and let D($${\mathcal D}_ X)$$ (resp. D($${\mathcal D}_ Y))$$ be the category of the complexes of $${\mathcal D}_ X$$-modules (resp. $${\mathcal D}_ Y$$-modules). M. Kashiwara defined a functor $$\int_{f}:D({\mathcal D}_ X)\to D({\mathcal D}_ Y),$$ which is called the direct image. Coherent $${\mathcal D}_ X$$-(resp. $${\mathcal D}_ Y$$-)modules $${\mathcal M}$$ are considered as generalization of a system of linear differential equations on X (resp. Y) in a natural way. The characteristic varieties and the characteristic cycles of $${\mathcal M}$$ are defined as subvarieties or cycles on them in the cotangent bundle $$T^*X$$ (resp. $$T^*Y)$$. The functor $$\int_{f}$$ corresponds to the integral along the fiber of f. M. Kashiwara has shown that for a coherent $${\mathcal D}_ X$$-module $${\mathcal M}$$, all the cohomological sheaves of $$\int_{f}{\mathcal M}$$ are coherent $${\mathcal D}_ Y$$-modules and has estimated the characteristic variety of $$\int_{f}{\mathcal M}$$. This is one of the fundamental results on $${\mathcal D}$$-modules. - The proof given by M. Kashiwara was analytic one and it is carried out by microlocalizing $${\mathcal M}$$. However, since all the objects are defined in the algebraic nature, there may be a proof by using only algebraic tools. The author of this paper gives such an algebraic proof and obtains a formula for the calculations of the characteristic cycles of $$\int_{f}{\mathcal M}$$ as a function of that of $${\mathcal M}$$.
Reviewer: M.Muro

##### MSC:
 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX) 32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results 32K15 Differentiable functions on analytic spaces, differentiable spaces 47E05 General theory of ordinary differential operators (should also be assigned at least one other classification number in Section 47-XX)