Permutation modules over cyclic $p$-groups

TL;DR

This paper generalizes the characterization of permutation modules over cyclic p-groups to cases where p is ramified in the ring, using elementary cohomological methods instead of advanced categorical tools.

Contribution

It extends previous results on permutation modules to ramified p, employing simpler cohomological techniques rather than complex categorical frameworks.

Findings

Extended characterization of permutation modules to ramified p cases.

Proofs rely solely on basic group cohomology and Weiss' Theorem.

Avoided complex categorical methods in proofs.

Abstract

Let $G$ be a cyclic $p$-group for some prime number $p>0$ and let $R$ be a complete discrete valuation ring in mixed characteristic. In this paper, we present a generalization of two results that characterize $RG$-permutation modules, extending previous work by B. Torrecillas and Th. Weigel. Their original results were established under the assumption that $ p$ is unramified in $R$, whereas we extend their characterization to the case where $p$ may be ramified. Unlike prior approaches, our proofs rely solely on fundamental facts from group cohomology and a version of Weiss' Theorem, avoiding deeper categorical techniques.