An example concerning the Bass conjecture
type: Item
instance of: scholarly article
title: An example concerning the Bass conjecture
zbMATH Open document ID: 0806.20004
published in: Michigan Mathematical Journal
publication date: 1995-02-13T00:00:00Z
review text: Let \(R\) be a commutative ring with a 1 and let \(G\) be a group. If \(P\) is a finitely generated projective \(RG\)-module then there is a map \(r_ P: G \to R\), called the rank of \(P\), which is constant on the conjugacy classes of \(G\). This map can be considered as an extension of the usual trace map from square matrices over \(RG\) to \(RG\). The precise definition of \(r_ P\) is given by \textit{H. Bass} [see Lond. Math. Soc. Lect. Note Ser. 36, 1-26 (1979; Zbl 0425.20006)]. If \(F\) is a finitely generated free \(RG\)-module then \(r_ F(x) = 0\) for all \(x \in G \setminus 1\), and if \(Q\) is a finitely generated projective \(RG\)-module then \(r_{P \oplus Q} = r_ P + r_ Q\). One form of the strong Bass conjecture is the following: Let \(P\) be a finitely generated projective \(RG\)-module, and let \(1 \neq x \in G\). If \(R\) is a subring of \(\mathbb{C}\) such that \(R \cap\mathbb{Q}=\mathbb{Z}\), then \(r_ P = 0\). The following question was posed by Schafer: If \(G\) is a finitely generated group containing a subgroup \(A \cong\mathbb{Q}^ +\) such that \(A\) is contained in only finitely many conjugacy classes of \(G\), must \(G\) be of infinite cohomological dimension? [see also \textit{J. A. Schafer}, Mich. Math. J. 38, 103-109 (1991; Zbl 0722.20002)]. If the answer to this question were `yes' then the strong Bass conjecture would be true for all groups \(G\) of finite cohomological dimension. In the paper under review the author shows that the answer to the question is `no'. He gives an example of a 2-generator subgroup \(G\) of cohomological dimension 2 which contains a subgroup \(A \cong\mathbb{Q}^ +\) such that \(A \setminus 1\) is contained in a single conjugacy class of \(G\).
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 16D40
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 20J05
zbMATH DE Number: 488365
zbMATH Keywords: finitely generated projective \(RG\)-module
zbMATH Keywords: rank
zbMATH Keywords: conjugacy classes
zbMATH Keywords: trace map
zbMATH Keywords: strong Bass conjecture
zbMATH Keywords: finite cohomological dimension
author: Peter A. Linnell
reviewed by: Nikolai I. Osetinski
Wikidata QID: Q122963172
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029004683
OpenAlex ID: W2053113850
DOI: 10.1307/MMJ/1029004683
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description: scientific article; zbMATH DE number 488365