Minimal resolutions of geometric D-modules

In this paper, we study minimal free resolutions for modules over rings of linear differential operators. The resolutions we are interested in are adapted to a given filtration, in particular to the so-called V-filtrations (see Oaku and Takayama (2001) [18], and Granger and Oaku (2004) [9]). We are interested in the module Dx,tfs associated with germs of functions f1,…,fp, which we call a geometric module, and it is endowed with V-filtration along t1=⋯=tp=0. The Betti numbers of the minimal resolution associated with this data lead to analytical invariants for the germ of space defined by f1,…,fp. For p=1, we show that, under some natural conditions on f, the computation of the Betti numbers is reduced to a commutative algebra problem. This includes the case of an isolated quasi-homogeneous singularity, for which we give the Betti numbers explicitly. Moreover, for an isolated singularity, we characterize the quasi-homogeneity in terms of the minimal resolution.