The Hochschild-Serre property for some p-adic analytic group actions

TL;DR

This paper investigates the Hochschild-Serre property for certain p-adic Lie group actions, extending known results to cases where the subgroup is not subnormal, under the assumption of analytic actions relevant to p-adic Hodge theory.

Contribution

It extends the Hochschild-Serre spectral sequence results to non-subnormal subgroups in p-adic Lie groups with analytic actions, broadening the applicability in p-adic Hodge theory.

Findings

Vanishing of H-cohomology implies vanishing of G-cohomology under new conditions.

Extension of Hochschild-Serre property to non-subnormal subgroups.

Application to p-adic Hodge theory contexts.

Abstract

Let $H \subseteq G$ be an inclusion of $p$-adic Lie groups. When $H$ is normal or even subnormal in $G$, the Hochschild-Serre spectral sequence implies that any continuous $G$-module whose $H$-cohomology vanishes in all degrees also has vanishing $G$-cohomology. With an eye towards applications in $p$-adic Hodge theory, we extend this to some cases where $H$ is not subnormal, assuming that the $G$-action is analytic in the sense of Lazard.