Coassembly and the K-theory of finite groups

Abstract

We study the K-theory and Swan theory of the group ring R[G], when G is a finite group and R is any ring or ring spectrum. In this setting, the well-known assembly map for K(R[G]) has a companion called the coassembly map. We prove that their composite is the equivariant norm of K(R). This gives a splitting of both assembly and coassembly after K(n)-localization, a new map between Whitehead torsion and Tate cohomology, and a partial computation of K-theory of representations in the category of spectra.