  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. 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 the
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 explicitely the Betti numbers. Moreover, for an isolated
singularity, we characterize the quasi-homogeneity in terms of the minimal
resolution.

Algebraic Geometry;Rings and Algebras