A universal multicoefficient theorem for the Kasparov groups
type: Item
instance of: scholarly article
title: A universal multicoefficient theorem for the Kasparov groups
zbMATH Open document ID: 0881.46048
DOI: 10.1215/S0012-7094-96-08412-4
author: Terry A. Loring
author: Marius Dadarlat
published in: Duke Mathematical Journal
publication date: 1998-02-19T00:00:00Z
review text: A universal coefficient theorem of Rosenberg and Schochet for the Kasparov groups \(KK(A,B)\) is extended to the Bockstein category. Namely, let \(\underline K(A)\) denote the direct sum of all \(K\)-theory groups of a \(C^*\)-algebra \(A\) in all degrees and with all cyclic coefficient groups. Let \(\Hom_\Lambda(\underline K(A),\underline K(B))\) denote the set of all homomorphisms that respect the direct sum decomposition and the action of certain natural transformations; some of these transformations have degree zero and are induced by coefficient morphisms \(\mathbb{Z}/n\to\mathbb{Z}/m\), and the others have degree one and are known as the Bockstein operations. The aim of this paper is to establish a universal multicoefficient exact sequence \[ 0\to\text{Pext}(K_*(A), K_*(B))\to KK(A,B)\to \Hom_\Lambda(\underline K(A),\underline K(B))\to 0 \] that holds in the same generality as the universal coefficient theorem of Rosenberg and Schochet. (Recall that for groups \(G\) and \(H\) a group \(\text{Pext}(G,H)\) is the subgroup of pure extensions in the group \(\text{Ext}(G,H)\).).
Mathematics Subject Classification ID: 46L80
Mathematics Subject Classification ID: 19K35
Mathematics Subject Classification ID: 46M20
zbMATH DE Number: 926951
zbMATH Keywords: Kasparov \(KK\)-theory
zbMATH Keywords: universal coefficient theorem
zbMATH Keywords: Kasparov groups
zbMATH Keywords: Bockstein category
zbMATH Keywords: cyclic coefficient groups
zbMATH Keywords: direct sum decomposition
zbMATH Keywords: universal multicoefficient exact sequence
reviewed by: V. M. Deundjak
MaRDI profile type: Publication
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full work available at URL: s0012-7094-96-08412-4
OpenAlex ID: W1507672198
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description: scientific article; zbMATH DE number 926951