Infinitely generated pseudocompact modules for finite groups and Weiss' Theorem

Abstract

One of the most beautiful results in the integral representation theory of finite groups is a theorem of A. Weiss that detects a permutation R-lattice for the finite p-group G in terms of the restriction to a normal subgroup N and the N-fixed points of the lattice, where R is a finite extension of the p-adic integers. Using techniques from relative homological algebra, we generalize Weiss' Theorem to the class of infinitely generated pseudocompact lattices for a finite p-group, allowing R to be any complete discrete valuation ring in mixed characteristic. A related theorem of Cliff and Weiss is also generalized to this class of modules. The existence of the permutation cover of a pseudocompact module is proved as a special case of a more general result. The permutation cover is explicitly described.