Résolutions minimales de d-modules géométriques

Abstract : Let D be the ring of germs at the origin of linear dierential operators with analytic coefficients. We study minimal free resolutions of D-modules, introduced by M. Granger, T. Oaku and N. Takayama. More precisely we consider modules endowed with a V -filtration along a smooth hypersurface, and the resolutions are adapted to this filtration. We focus on the ranks of such a resolution, which we call Betti numbers, they are invariant for the module considered. First, we give some general results : we reduce the computation of the Betti numbers to a commutative algebra problem, and we dene generic minimal resolutions. Next, we consider a complex hypersurface singularity f = 0 and the module N = D x , t Fs introduced by B. Malgrange, whose restriction along t = 0 gives the algebraic local cohomology of the sheaf of analytic functions with support in f = 0. The module N is naturally endowed with the V -filtration along t = 0, we study the Betti numbers associated to this data. Those numbers are analytical invariants for the hypersurface f = 0. We compute them in the quasi homogeneous isolated singularity case and in the monomial case. In the isolated singularity case, we characterize the quasi-homogeneity in terms of the Betti numbers.
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  • HAL Id : tel-00451962, version 1

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Rémi Arcadias. Résolutions minimales de d-modules géométriques. Mathématiques [math]. Université d'Angers, 2009. Français. ⟨tel-00451962⟩

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