Infinitely generated pseudocompact modules for finite groups and Weiss' Theorem

TL;DR

This paper extends Weiss' theorem from finite to infinitely generated pseudocompact modules over finite p-groups, using homological algebra techniques, and provides explicit descriptions of permutation covers.

Contribution

It generalizes Weiss' theorem to infinitely generated pseudocompact modules over finite p-groups, broadening the scope of integral representation theory.

Findings

Generalization of Weiss' theorem to pseudocompact modules

Existence and explicit description of permutation covers

Extension of Cliff and Weiss's related 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.