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 \({\mathcal M}\).

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) |