On the continuous (co)homology of locally profinite groups and the Künneth theorem
type: Item
instance of: scholarly article
title: On the continuous (co)homology of locally profinite groups and the Künneth theorem
zbMATH Open document ID: 0866.22002
author: Amara Hitta
published in: Journal of Algebra
publication date: 1997-07-07T00:00:00Z
review text: Let \(G\) be a locally compact and totally disconnected group, which we assume to be separated, metrizable, and countable at infinity. The neighbourhood basis of the neutral element \(1_G\) of \(G\) is formed by compact open subgroups, i.e., \(G\) is a locally profinite group. The purpose of this paper is to develop some aspects of a (co)homology (smooth and continuous) of the group \(G\) in a topological locally convex quasi-complete separated complex vector space \(V\). We study two categories: the category \({\mathcal C}_G\) of continuous \(G\)-modules and the category \(\text{Alg}(G)\) of smooth \(G\)-modules, i.e., each vector of the representation space is invariant by an open compact subgroup. The relation between smooth and continuous (co)homologies of the group \(G\) and those of its closed normal subgroup \(H\) is coded in the notion of spectral sequence, which was discovered by G. Hochschild and J.-P. Serre for discrete groups and is extended in this paper to locally compact totally disconnected groups. One of the main difficulties is the fact that it is no longer evident (as for discrete or profinite groups) that the Hecke algebra \(C^\infty_c(G)\) of locally constant functions with compact supports is injective as an \(H\)-module. One might hope that any injective \(G\)-module in \(\text{Alg}(G)\) is \(H\)-acyclic, which guarantees the convergence of the analogue of the Hochschild-Serre spectral sequence. For \(H\) a \(p\)-adic subgroup of \(G\), this is done in [\textit{A. Borel} and \textit{N. Wallach}, Continuous cohomology, discrete subgroups, and representations of reductive groups, Ann. Math. Stud., No. 94 (1980; Zbl 0443.22010), p. 315]. Our result is based on a new method using the fact that the space \(\text{Hom}_{\mathbb{C}}(C^\infty_c(G);V)_\infty\), endowed with the action \(\sigma\) of \(G\) defined by \((\sigma(x)\varphi)(g)=\pi(x\mid\varphi(x^{-1}g))\), is injective as a smooth \(G\)-module, where \((\pi,V)\in\text{Alg}(G)\). This researc…
Mathematics Subject Classification ID: 22D05
Mathematics Subject Classification ID: 22E41
Mathematics Subject Classification ID: 57T10
zbMATH DE Number: 554217
zbMATH Keywords: locally profinite group
zbMATH Keywords: complex vector space \(V\)
zbMATH Keywords: Hecke algebra
zbMATH Keywords: Künneth theorem
MaRDI profile type: Publication
full work available at URL: jabr.1994.1028
OpenAlex ID: W2074269714
DOI: 10.1006/JABR.1994.1028
description: scientific article; zbMATH DE number 554217