Abstract | 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 D x , t f s associated with germs of functions f 1 , … , f p , which we call a geometric module, and it is endowed with V -filtration along t 1 = ⋯ = t p = 0 . The Betti numbers of the minimal resolution associated with this data lead to analytical invariants for the germ of space defined by f 1 , … , f p . 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. |