An algebraic model for free rational G-spectra for connected compact Lie groups G

TL;DR

This paper establishes an algebraic model for free rational G-spectra of connected compact Lie groups G, linking them to torsion differential graded modules over the polynomial cohomology ring of BG, using advanced algebraic techniques.

Contribution

It introduces a Quillen equivalence between free rational G-spectra and torsion DG modules over H*(BG), providing a new algebraic framework for these spectra.

Findings

Category of free rational G-spectra is Quillen equivalent to torsion DG modules

Uses Morita equivalences, Koszul duality, and subcategory arguments

Simplifies the understanding of rational G-spectra for connected compact Lie groups

Abstract

We show that the category of free rational G-spectra for a connected compact Lie group G is Quillen equivalent to the category of torsion differential graded modules over the polynomial cohomology ring on the classifying space, H*(BG). The ingredients are enriched Morita equivalences, functors making rational spectra algebraic, and Koszul duality and thick subcategory arguments based on the simplicity of the derived category of a polynomial ring.