On some cohomological invariants for large families of infinite groups
type: Item
instance of: scholarly article
title: On some cohomological invariants for large families of infinite groups
zbMATH Open document ID: 1472.18006
author: Rudradip Biswas
published in: The New York Journal of Mathematics
publication date: 2021-08-16T00:00:00Z
full work available at URL: 27-31.html
review text: In this article, the author investigates some cohomological invariants of the class of infinite groups denoted \(LH\mathcal F_{\phi,A}\). This class is defined using the groups of type \(\Phi\), introduced by Talelli over \(\mathbb Z\) in [\textit{O. Talelli}, Arch. Math. 89, No. 1, 24--32 (2007; Zbl 1158.20026)], and groups derived from a hierarchy of groups introduced by \textit{P. H. Kropholler} [J. Pure Appl. Algebra 90, No. 1, 55--67 (1993; Zbl 0816.20042)]. In the article under review, the author extends some results that were known over the ring of integers to arbitrary commutative rings of finite global dimension. In particular, one of the main result states that if a group \(\Gamma\) is in \(LH\mathcal F_{\phi,A}\) with \(A\) of finite global dimension, say \(t\), then the projective dimension of the module of bounded functions \(B(\Gamma,A)\), of \(\Gamma\) on \(A\) over \(A\Gamma\), is equal to the Gorenstein cohomological dimension of \(\Gamma\) over \(A\). Moreover, this quantity, say \(\theta\), is bounded above by the finistic dimension of \(A\Gamma\), which is itself bounded above by \(\theta+t\).
reviewed by: Nadia Mazza
Mathematics Subject Classification ID: 18G05
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 20K40
zbMATH DE Number: 7382052
zbMATH Keywords: Kropholler's hierarchy
zbMATH Keywords: Gorenstein cohomological dimension
zbMATH Keywords: complete resolutions
zbMATH Keywords: Gorenstein projectives
zbMATH Keywords: finitistic dimension
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 7382052