On Bass' ``Strong conjecture'' about projective modules
type: Item
instance of: scholarly article
title: On Bass' ``Strong conjecture'' about projective modules
zbMATH Open document ID: 0688.16013
DOI: 10.1016/0022-4049(89)90150-3
author: Andrzej Strojnowski
published in: Journal of Pure and Applied Algebra
publication date: 1989-01-01T00:00:00Z
review text: Let \({\mathbb{Z}}[G]\) be the integral group ring of a group G. Let P be a finitely generated projective \({\mathbb{Z}}[G]\)-module. It has a rank \(r_ p\) which can be viewed as an integer valued function on conjugacy classes of G. The Bass' ``Strong Conjecture'' [\textit{H. Bass}, Invent. Math. 35, 155- 196 (1976; Zbl 0365.20008), p. 156] claims that \(r_ p(g)=0\) for \(g\neq 1\) in G. Bass proved this conjecture for a class of torsion free groups including those with faithful linear representations. In the present paper the author proves the conjecture for a large class of groups which satisfy the following condition called WD: Suppose H is a finitely generated subgroup of G, \(s\in H\), N a positive integer and s is conjugate in H to \(s^{p^ N}\) for all primes p; then \(s=1\). The author shows that the following groups G satisfy condition WD: (i) G is a torsion group. (ii) G is a nilpotent-by-Noetherian group. (iii) (cf. \textit{H. Bass} [loc. cit.]) G is a linear group. (iv) (cf. \textit{P. A. Linnell} [Proc. Lond. Math. Soc., III. Ser. 47, 83-127 (1983; Zbl 0531.20002)]) The group G does not contain any subgroup isomorphic to the additive group of rational numbers. (v) Every finitely generated subgroup of G satisfies condition WD. (vi) G is a finite extension of some WD-groups. (vii) G is a subdirect product of (maybe infinitely many) WD-groups. (viii) G is a free product of (maybe infinitely many) WD-groups.
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 16D40
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 20E06
zbMATH DE Number: 4127430
zbMATH Keywords: integral group ring
zbMATH Keywords: finitely generated projective \({\mathbb{Z}}[G]\)-module
zbMATH Keywords: rank
zbMATH Keywords: integer valued function on conjugacy classes
zbMATH Keywords: Bass' ``Strong Conjecture''
zbMATH Keywords: torsion free groups
zbMATH Keywords: condition WD
zbMATH Keywords: finitely generated subgroup
zbMATH Keywords: subdirect product
zbMATH Keywords: WD-groups
zbMATH Keywords: free product
Wikidata QID: Q122903447
reviewed by: T. Akasaki
MaRDI profile type: Publication
full work available at URL: 0022-4049(89)90150-3
OpenAlex ID: W2085316076
cites work: Euler characteristics and characters of discrete groups
cites work: Q3859975
cites work: Zero Divisors and Idempotents in Group Rings
cites work: Decomposition of Augmentation Ideals and Relation Modules
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description: scientific article; zbMATH DE number 4127430