An algebraic model for free rational G-spectra

TL;DR

This paper provides an algebraic classification of free rational G-spectra for compact Lie groups, translating the problem into torsion modules over a twisted group ring, thus enabling easier computation of natural transformations.

Contribution

It introduces an algebraic model equating free rational G-spectra with torsion modules over a twisted group ring, simplifying classification and calculations.

Findings

Equivalence between free rational G-spectra and torsion modules over H^*(BN)[W]

Provides a practical method for computing natural transformations

Simplifies the algebraic understanding of rational G-equivariant cohomology theories

Abstract

We show that for any compact Lie group $G$ with identity component $N$ and component group $W=G/N$, the category of free rational $G$-spectra is equivalent to the category of torsion modules over the twisted group ring $H^*(BN)[W]$. This gives an algebraic classification of rational $G$-equivariant cohomology theories on free $G$-spaces and a practical method for calculating the groups of natural transformations between them. This uses the methods of arXiv:1101.2511, and some readers may find the simpler context of the present paper highlights the main thread of the argument.