Permutation modules over cyclic $p$-groups

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.