manuscripta mathematica

, Volume 50, Issue 1, pp 49-71

First online:

Sur les Images Directes deD-Modules

  • Bernard MalgrangeAffiliated withInstitut Fourier, Université de Grenoble I

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Let\(Y\xrightarrow{f}X\) be a morphism of compact analytic manifolds, and M a right coherentD Y-module admitting a good filtration; if V⊂T*Y denotes the characteristic variety of M, one can define [M]V as the class of gr M in some suitable Grothendieck group of sheaves with support in V. Let\(T*Y\xleftarrow{F}Y\mathop { \times T*X\xrightarrow{{\bar f}}T*X}\limits_X \) be the morphisms naturally defined by f. A result of Kashiwara says that, for all i, the characteristic variety of ∫ f i M is contained in\(W = \bar fF^{ - 1} V\). Here we prove the following K-theoretic version of this result:\(\sum {( - 1)^i [\int_f^i {M]} _W } = \bar f_* F*[M]_V \).