Classifying Rational G-Spectra for Finite G

TL;DR

This paper provides a new proof establishing a symmetric monoidal Quillen equivalence between rational G-equivariant spectra for finite groups G and a product of algebraic categories, facilitating algebraic understanding of these spectra.

Contribution

It introduces a novel proof of the algebraic model for rational G-spectra that emphasizes symmetric monoidal Quillen equivalences, enhancing the algebraic understanding of module categories.

Findings

Rational G-equivariant spectra are Quillen equivalent to algebraic models.

The equivalences are symmetric monoidal, preserving algebraic structures.

The approach simplifies understanding modules over ring spectra in algebraic terms.

Abstract

We give a new proof that for a finite group G, the category of rational G-equivariant spectra is Quillen equivalent to the product of the model categories of chain complexes of modules over the rational group ring of the Weyl group of H in G, as H runs over the conjugacy classes of subgroups of G. Furthermore the Quillen equivalences of our proof are all symmetric monoidal. Thus we can understand categories of algebras or modules over a ring spectrum in terms of the algebraic model.