Coefficients for the Farrell-Jones conjecture
type: Item
instance of: scholarly article
title: Coefficients for the Farrell-Jones conjecture
zbMATH Open document ID: 1161.19003
author: Holger Reich
author: Arthur Bartels
published in: Advances in Mathematics
publication date: 2007-02-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0510602
review text: The Farrell-Jones conjecture asserts that the \(K\)- or \(L\)-theory of a group ring \(RG\) can be obtained from that of the virtually cyclic subgroups of \(G\); more precisely, \(K_*(RG)\) is isomorphic, via the assembly map, to the direct limit \({\mathrm{colim}}\;H^G_*(E_{H \in \mathcal{F}}G;\mathbf{K}_R)\) where \(\mathcal{F}\) is the family of virtually cyclic subgroups of \(G\), \(H^G_*(-;\mathbf{K}_R)\) is \(G\)-equivariant homology and \(E_{\mathcal{F}}G\) is the classifying space for \(\mathcal{F}\), and likewise for \(L\)-groups. This paper presents a generalization of the Farrell-Jones conjecture which allows coefficients in additive categories with \(G\)-action. There are many advantages obtained from this generalization. The conjecture passes to subgroups and more generally pulls-back under arbitrary group homomorphisms. The injectivity and surjectivity parts of the conjecture have these inheritence properties separately. Further, the correct generalizations of the conjecture are obtained for twisted group rings and crossed product rings. The construction of the corresponding additive categories is matter of interest in its own right and can be expected to have wider applications in representation theory.
Mathematics Subject Classification ID: 19D50
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 20C07
Mathematics Subject Classification ID: 18E05
zbMATH DE Number: 5123975
zbMATH Keywords: algebraic \(K\)-theory, group ring
zbMATH Keywords: Farrell-Jones conjecture, controlled algebra, additive category with group action
Wikidata QID: Q123358989
MaRDI profile type: Publication
OpenAlex ID: W1978742629
arXiv ID: math/0510602
cites work: Squeezing and higher algebraic \(K\)-theory
cites work: On the Farrell-Jones conjecture for higher algebraic 𝐾-theory
cites work: On the isomorphism conjecture in algebraic \(K\)-theory.
cites work: Controlled algebra and the Novikov conjectures for \(K\)- and \(L\)-theory
cites work: Spaces over a category and assembly maps in isomorphism conjectures in \(K\)- and \(L\)-theory
cites work: Isomorphism Conjectures in Algebraic K-Theory
cites work: Counterexamples to the Baum-Connes conjecture
cites work: Identifying assembly maps in \(K\)- and \(L\)-theory
cites work: Q2770573
cites work: Q3436930
cites work: Q5718772
cites work: Q5639839
cites work: Induction theorems for infinite groups
cites work: Q3995022
cites work: Q3719492
cites work: Q4003859
cites work: Splitting with continuous control in algebraic \(K\)-theory
cites work: On the algebraic \(K\)- and \(L\)-theory of word hyperbolic groups.
cites work: Q3701607
DOI: 10.1016/J.AIM.2006.05.005
description: scientific article