# A Flatness Property for Filtered <em>D</em>-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 *DX* of linear differential operators on an analytic manifold *X* and let _Z_1 , . . . , *Zk* 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 *Mx* 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 <em>D</em>-modules. Publ. Res. Inst. Math. Sci. 43 (2007), no. 1, pp. 121–141

DOI 10.2977/PRIMS/1199403810