A flatness property for filtered D-modules

Abstract

Let M be a coherent module over the ring D of linear differential operators on an analytic manifold X and let us consider k germs of transverse hypersurfaces at a point x in X. The Malgrange-Kashiwara V-filtrations along these hypersurfaces, associated with a given presentation of the germ of M at the point x, give rise to a multifiltration U(M) of M_x introduced by Sabbah and to an analytic standard fan as developed by Assi-Castro-Granger. 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 (see References).