On a decomposition theorem in equivariant generalized homology theories for finite group actions

TL;DR

This paper extends a decomposition theorem for rational equivariant algebraic K-theory to broader algebraic (co)homology theories with the Mackey property, revealing new indexing by conjugacy classes of subgroups.

Contribution

It generalizes Vistoli's decomposition theorem to algebraic (co)homology theories with the Mackey property, incorporating non-abelian subgroup indexing.

Findings

Decomposition theorem extended to broader theories.

Indexing by conjugacy classes of subgroups, including non-abelian.

Application to Borne's modular K-theory showing new subgroup indexing.

Abstract

A. Vistoli proved a decomposition theorem for the rational equivariant algebraic K-theory of a variety under the action of a finite group $G$. We generalize his result to more general algebraic (co)homology theories having the Mackey property and admitting localization long exact sequences. In general, the pieces are indexed by conjugacy classes of subgroups of $G$. Our construction is based on some result about a decomposition of the rational Burnside ring of a finite group, which stands behind the classical splitting theorems for equivariant spectra in stable equivariant homotopy theory. Applying this result to the case of Borne's modular K-theory we exhibit a case where the splitting is indexed by not necessarily abelian subgroups.