Sur les Images Directes deD-Modules

Abstract

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 ∫ ⁱ_f M is contained in$W = \bar fF^{ - 1} V$. Here we prove the following K-theoretic version of this result:$\sum {( - 1)^i [\int_f^i {M]} _W } = \bar f_* F*[M]_V$.