A Flatness Property for Filtered $\mathcal{D}$-modules

Abstract

Let $\mathcal{M}$ be a coherent module over the ring ${\mathcal{D}}_X$ of linear differential operators on an analytic manifold $X$ and let $Z_1,...,Z_k$ be $k$ germs of transverse hypersurfaces at a point $x\in X$. The Malgrange–Kashiwara $\mathrm{V}$-filtrations along these hypersurfaces, associated with a given presentation of the germ of $\mathcal{M}$ at $x$, give rise to a multifiltration $U_{•}(\mathcal{M})$ of ${\mathcal{M}}_x$ 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.