Cyclic homology of groups and the Bass conjecture
type: Item
instance of: scholarly article
title: Cyclic homology of groups and the Bass conjecture
zbMATH Open document ID: 0613.20034
DOI: 10.1007/BF02621911
author: Beno Eckmann
published in: Commentarii Mathematici Helvetici
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/140048
review text: Cyclic homology of the rational group algebra \({\mathbb{Q}}G\) of a group G has a direct sum decomposition \[ HC_ i({\mathbb{Q}}G)=\oplus '_{[x]}[H_*(C_ x/<x>;{\mathbb{Q}})\otimes H_*({\mathbb{C}}P^{\infty};{\mathbb{Q}})]_ i\oplus \oplus ''_{[x]}H_ i(C_ x/<x>;{\mathbb{Q}}), \] where the summands are indexed by the conjugacy classes [x] of G, \(x\in G\), \(C_ x\) is the centralizer of x in G, \(\oplus '\) (resp. \(\oplus '')\) is summation over all [x] with x of finite (resp. infinite) order [\textit{D. Burghelea}, Comment. Math. Helv. 60, 354- 365 (1985; Zbl 0595.16022)]. The author shows that if G is of finite homology dimension \(hd_{{\mathbb{Q}}}G=n\) then the \(\oplus ''\)-part vanishes for \(i\geq n\) provided (a) G is soluble, or (b) \(n=2\), or (c) G is linear over a field of characteristic 0. As a consequence he obtains that for those groups G the image of the Connes character \(Ch^{\ell}: K_ 0({\mathbb{Q}}G)\to HC_{2\ell}({\mathbb{Q}}G)\), which is compatible with the Connes-Gysin map \(HC_{2\ell}({\mathbb{Q}}G)\to H_{2\ell -2}({\mathbb{Q}}G)\), is concentrated on the \(\oplus '\) summands. This applies, in particular, to \(Ch^ 0: K_ 0({\mathbb{Q}}G)\to HC_ 0({\mathbb{Q}}G)\)- which is the same as the Hattori- Stallings rank of G - and hence is a contribution towards the strong Bass-Conjecture that the Hattori-Stallings rank of a finitely generated projective \({\mathbb{Q}}G\)-module is concentrated on the conjugacy classes of elements of finite order.
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 18F25
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 20F16
Mathematics Subject Classification ID: 55R40
Mathematics Subject Classification ID: 16E40
zbMATH DE Number: 3991781
zbMATH Keywords: Cyclic homology
zbMATH Keywords: rational group algebra
zbMATH Keywords: direct sum decomposition
zbMATH Keywords: homology dimension
zbMATH Keywords: Connes character
zbMATH Keywords: Connes-Gysin map
zbMATH Keywords: Hattori-Stallings rank
zbMATH Keywords: Bass-Conjecture
zbMATH Keywords: finitely generated projective \({bbfQ}G\)-module
zbMATH Keywords: conjugacy classes
reviewed by: Robert Bieri
MaRDI profile type: Publication
Wikidata QID: Q123213642
OpenAlex ID: W1972479746
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description: scientific article; zbMATH DE number 3991781