The Swan Mapping

Abstract

In his fundamental paper on group cohomology Swan (Ann. Math. 72:267–291, 1960) defined, for any finite group G, a homomorphism $(\mathbf{Z}/\vert G \vert)^{*} \rightarrow\widetilde{K_{0}}(\mathbf{Z}[G])$ which, in this restricted context, has since been used extensively both in the classification of projective modules (Swan in J. Reine Angew. Math. 342:66–172, 1983) and the algebraic homotopy theory of finite complexes (Johnson in Stable modules and D(2)-problem. LMS lecture notes in mathematics, vol. 301. Cambridge University Press, Cambridge, 2003). We extend the definition so that, for suitable modules J over reasonably general rings Λ, it takes the form $S_{J}: \mathrm{Aut}_{\mathcal{D}\mathrm{er}}(J)\rightarrow\widetilde{K_{0}}(\varLambda)$.