Cohomological aspects of equivariant deformation theory

Abstract

Deformation theory in its modern form arose from the work of Kunihiko Kodaira and Donald C. Spencer on complex analytic manifolds. The methods applied by Kodaira and Spencer have been later on translated by Alexander Grothendieck into the language of algebraic geometry. The motivation for this thesis comes from algebraic geometry. Given a curve with a finite group of its automorphisms, one may try to deform the curve together with the group action, that is in such a way that the deformation preserves symmetries of the curve. A point on a curve with a non-trivial stabilizer leads to a local action of a group in a formal neighbourhood of the point and deformations of the curve lead to deformations of the local actions. It follows from the local-global principle that all the obstructions to deforming a curve arise from obstructions to deforming these local actions. In order to classify all the deformations, one constructs a so-called local (infinitesimal) deformation functor. To this functor, one associates several invariants, in particular its versal deformation ring. In the case when the functor is pro-representable, the ring carries all of the information about the functor. We answer the question when precisely are the local deformation functors pro-representable in the case of weak ramification. It turns out that the non-pro-representable examples occur only in characteristic 2 when the group is elementary abelian of order 2 or 4. This came as a surprise, since local deformation functors have plenty of local automorphisms, which is often a hindrance to pro-representability. The main problem studied in this thesis is the problem of d鶩ssage for local deformation functors. Given a local action of a group G and a normal subgroup N of G, we can consider three deformation functors relative to the groups G, N and G/N. The cohomological invariants related to these functors are given by group cohomology and are related by several group cohomological morphisms, in particular by the Hochschild-Serre spectral sequence. We show that many of these operations lift to the level of deformation functors. On this basis, we formulate a conjectural relation between the three functors. In order to motivate this conjecture, we study deformations on a higher level, deforming not the actions, but a morphism between deformation functors. Our main result states that the morphism between the deformation functor of G and the product of the deformation functors of N and G/N has the expected obstruction space in the sense of Fantechi and Manetti. Tangential to the main interests of the thesis, we study an example of a deformation functor of a linear representation due to Bleher and Chinburg, whose versal deformation ring is not a complete intersection ring. We give an elementary proof of this result using deformation theory, but not modular representation theory, which plays a role in the original proof.