Varieties of modules and representations of Frobenius kernels of reductive groups

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An algebra A is a type of mathematical object with a structure satisfying certain properties; a representation for A is a space on which A acts in a way which is compatible with its structure. Varieties of modules arise when we consider the set of *all* possible representations (of a given dimension) for A. Beginning with elementary examples, one obtains surprisingly rich and complex geometric structures parametrizing the n-dimensional modules.One natural question to ask is the following: given an algebra A, can we identify the irreducible components of the variety of n-dimensional A-modules? In general, this turns out to be a hard question. The existing methods for tackling it mostly depend on fairly restrictive properties of the algebra A. In the research proposed here, we will investigate varieties of modules for a particular class of algebras: group algebras of elementary abelian p-groups of rank 2. The problem of describing these varieties of modules has an alternative interpretation in relation to cohomology of the second Frobenius kernel of the group of invertible n x n matrices, due to work of Suslin, Friedlander and Bendel. In order to tackle this specific problem, we will have to develop some new methods for studying varieties of modules, adapting earlier results of Crawley-Boevey and Schroer.