Coassembly and the $K$-theory of finite groups

TL;DR

This paper investigates the $K$-theory and Swan theory of group rings for finite groups, revealing new relationships between assembly, coassembly, and equivariant norms, with implications for localization and torsion computations.

Contribution

It introduces the concept of coassembly in the context of finite groups and establishes its relation to the equivariant norm, providing new splitting results and computational tools.

Findings

Splitting of assembly and coassembly after $K(n)$-localization.

A new map connecting Whitehead torsion and Tate cohomology.

Partial computation of $K$-theory of representations in spectra.

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.