A Flatness Property for Filtered <em>D</em>-modules

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.