Deformations and derived categories

Bleher, Frauke M.; Chinburg, Ted

Annales de l'Institut Fourier, Tome 55 (2005) no. 7, pp. 2285-2359.

Dans cet article, nous généralisons la théorie des déformations de représentations d’un groupe profini dévélopée par Schlessinger et Mazur aux déformations d’objets d’une catégorie dérivée de complexes limités de modules pseudocompacts. Nous prouvons que de tels objets ont des déformations verselles selon certaines conditions naturelles, et nous déterminons une condition suffisante pour que ces déformations soient universelles. De plus, nous considérons des applications en des déformations de classes de cohomologie Galoisienne et de la hypercohomologie étale de ${\mu }_p$ sur certaines CM courbes elliptiques affines.

In this paper we generalize the deformation theory of representations of a profinite group developed by Schlessinger and Mazur to deformations of objects of the derived category of bounded complexes of pseudocompact modules for such a group. We show that such objects have versal deformations under certain natural conditions, and we find a sufficient condition for these versal deformations to be universal. Moreover, we consider applications to deforming Galois cohomology classes and the étale hypercohomology of ${\mu }_p$ on certain affine CM ellitpic curves.

Classification : 20CXX, 18E30, 18G40, 11F80
Keywords: Versal and universal deformations, derived categories, hypercohomology, CM elliptic curves, Versal and universal deformations, derived categories, hypercohomology, CM elliptic curves
Mot clés : déformations verselles et universelles, catégories derivées, cohomologie Galoisienne, hypercohomologie, CM courbes elliptiques

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