# A Flatness Property for Filtered $D$-modules

### Francisco J. Castro-Jiménez

Universidad de Sevilla, Spain### Michel Granger

Université d'Angers, France

## Abstract

Let $M$ be a coherent module over the ring $D_{X}$ of linear differential operators on an analytic manifold $X$ and let $Z_{1},...,Z_{k}$ be $k$ germs of transverse hypersurfaces at a point $x∈X$. The Malgrange–Kashiwara $V$-ﬁltrations along these hypersurfaces, associated with a given presentation of the germ of $M$ at $x$, give rise to a multiﬁltration $U_{•}(M)$ of $M_{x}$ as in Sabbah’s paper [9] and to an analytic standard fan in a way similar to [3]. We prove here that this standard fan is adapted to the multiﬁltration, in the sense of C. Sabbah. This result completes the proof of the existence of an adapted fan in [9], for which the use of [8] is not possible.

## Cite this article

Francisco J. Castro-Jiménez, Michel Granger, A Flatness Property for Filtered $D$-modules. Publ. Res. Inst. Math. Sci. 43 (2007), no. 1, pp. 121–141

DOI 10.2977/PRIMS/1199403810