Generic Module Theory Original Research Article

The process of restricting modules to cyclic shifted subgroups is a fundamental technique in the modular representation theory of elementary abelianp-groups. IfEis elementary abelian ofp-rankrandkis an algebraically closed field of characteristicp, then each point inkr−{0} determines a cyclic shifted subgroup. Because the restriction of akE-module to this shifted subgroup depends only upon the corresponding point in projective space, it is often convenient to work with2kr−1instead ofkr−{0}. Roughly speaking, this paper shows that ifVis an irreducible subvariety of3kr−1andMis akE-module, then for almost all points inVthe direct sum decomposition ofMis the same upon restriction; moreover, this decomposition is completely determined by the behavior ofMupon restriction to the cyclic shifted subgroup corresponding to the generic point ofV. A similar idea provides a stratification of the rank variety ofMinto 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 orderp.