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Algebra Seminar

Monday, April 14, 2008 - 4:00pm

Jakub Byszewski

Univ. of Leiden

Location

University of Pennsylvania

DRL 4N30

This is joint work with Gunther Cornelissen. We determine exactly which formal deformation functors of representations of a finite group as weakly ramified automorphisms of a power series ring over a perfect field of positive characteristic are prorepresentable. Examples of such representations are provided by a group action on an ordinary curve: the action of a ramification group on the completed local ring of any point on such a curve is weakly ramified. The optimal situation in deformation theory occurs when a universal object exists --- when a deformation functor is (pro-)representable. For example, this happens in the formal deformation theory of a group action on a projective curve of genus g >= 2, or for absolutely irreducible Galois representations. The latter example played a decisive role in the proof of Fermat's Last Theorem. Equally often, one doesn't expect or cannot establish (pro-)representability, and the remedy is the construction of a so called 'versal hull' for the deformation functor. This is the classical approach to the local version of the first example: the action of a finite group on the completed local ring of a point of a curve. In this talk, I will prove that some of these versal hulls are actually universal, though not by a standard method.