Minimal resolutions of geometric D-modules

Rémi Arcadias (1)

1 LAREMA - Laboratoire Angevin de Recherche en Mathématiques

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.

Domaines: Géométrie algébrique [math.AG]; Anneaux et algèbres [math.RA]

Rémi Arcadias. Minimal resolutions of geometric D-modules. Journal of Pure and Applied Algebra, 2010, 214, pp.1477-1496. ⟨10.1016/j.jpaa.2009.12.001⟩. ⟨hal-00368826⟩