A flatness property for filtered D-modules

Abstract

Let M be a coherent module over the ring DX of linear differential operators on an analytic manifold X and let Z1, · · · , 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] C. Sabbah, Proximité evanescente I. La structure polaire d’un D–module Compositio Math. 62 (1987) 283-319 and to an analytic standard fan in a way similar to [3] A. Assi., F. Castro-Jiménez and M. Granger, The analytic standard fan of a D-module, J. Pure Appl. Algebra 164 (2001) 3-21. 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] C. Sabbah, Proximité evanescente I. La structure polaire d’un D–module Compositio Math. 62 (1987) 283-319, for which the use of [8] C. Sabbah and F. Castro, Appendice à “proximité evanescente” I. La structure polaire d’un D–module, Compositio Math. 62 (1987) 320-328. is not possible.