Minimal resolutions of geometric D-modules

TL;DR

This paper investigates minimal free resolutions of geometric D-modules with V-filtration, linking Betti numbers to analytical invariants of singularities, and provides explicit calculations for certain cases like quasi-homogeneous singularities.

Contribution

It introduces a method to compute Betti numbers of minimal resolutions for geometric D-modules, connecting algebraic invariants to singularity properties, including explicit results for quasi-homogeneous cases.

Findings

Betti numbers relate to analytical invariants of singularities

Explicit Betti numbers for quasi-homogeneous singularities

Characterization of quasi-homogeneity via minimal resolutions

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. 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.