A universal multicoefficient theorem for the Kasparov groups (Q1922127)

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 \(\text{Ext}(G,H)\).).