Generic Module Theory

Mathematics Research Report MRR95-078

Generic Module Theory

Wayne W. Wheeler

Abstract: The process of restricting modules to cyclic shifted subgroups is a fundamental technique in the modular representation theory of elementary abelian p-groups. If E is elementary abelian of p-rank r and k is an algebraically closed field of characteristic p, then each point in k^r-{0} determines a cyclic shifted subgroup. Because the restriction of a kE-module to this shifted subgroup depends only upon the corresponding point in projective space, it is often convenient to work with P_k^r-1 instead of k^r-{0}. Roughly speaking, this paper shows that if V is an irreducible subvariety of P_k^r-1 and M is a kE-module, then for almost all points in V the direct sum decomposition of M is the same upon restriction; moreover, this decomposition is completely determined by the behavior of M upon restriction to the cyclic shifted subgroup corresponding to the generic point of V. A similar idea provides a stratification of the rank variety of M into a disjoint union of locally closed subspaces. The closures of these subspaces are then described in terms of deformations of modules over a group of order p.

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