Permutation modules over cyclic $p$-groups

TL;DR

The paper addresses the problem of characterizing RG-permutation modules for cyclic $p$-groups over a complete discrete valuation ring $R$ in mixed characteristic, extending Torrecillas–Weigel to ramified $p$. It adopts a cohomological approach, leveraging $H^1$-vanishing and a Weiss-type splitting theorem to show that suitable $N$-local permutation structure enforces a global permutation structure for $U$. The work establishes equivalences among permutation-ness, $G$-coflasque-ness, and the behavior of quotients $U_N$, and proves two main results (Theorems referred to as 'myresult' and 'corollo'). These findings broaden applicability to profinite groups and Picard groups of blocks, and accommodate ramified $p$ in $R$ (including the case $p=2$).

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.

Theorems & Definitions (18)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Lemma 2.1
• proof
• Lemma 2.2
• Lemma 2.3
• proof