Coassembly and the $K$-theory of finite groups

Cary Malkiewich

Published 2015-03-23Version 1

We study the $K$-theory and Swan theory of the group ring $R[G]$, when $G$ is a finite group and $R$ is any discrete ring or ring spectrum. In this setting, the well-known assembly map for $K(R[G])$ has a lesser-known companion called the coassembly map. We prove that their composite is the equivariant norm of $K(R)$. As a result, we get a splitting of both assembly and coassembly after $K(n)$-localization, and an apparently new map from the Whitehead torsion group of $G$ over $R$ to the Tate cohomology of $BG$ with coefficients in $K(R)$.