An algebraic model for rational torus-equivariant spectra

TL;DR

This paper establishes an algebraic model for rational G-spectra when G is a torus, simplifying the understanding of rational G-equivariant cohomology theories through explicit algebraic structures.

Contribution

It introduces a universal algebraic model for rational G-spectra for tori, utilizing isotropy separation, fixed points, and differential graded algebras, advancing the algebraic understanding of equivariant spectra.

Findings

Rational G-spectra are Quillen equivalent to an algebraic model.

The model simplifies computations in rational G-equivariant cohomology.

Systematic cellularization is key to the model's construction.

Abstract

We show that the category of rational G-spectra for a torus G is Quillen equivalent to an explicit small and practical algebraic model, thereby providing a universal de Rham model for rational G-equivariant cohomology theories. The result builds on the first author's Adams spectral sequence, the second author's functors making rational spectra algebraic. There are several steps, some perhaps of wider interest (1) isotropy separation (replacing the category of G-spectra by modules over a diagram of isotropically simple ring G-spectra) (2) passage to fixed points on ring and module categories (replacing diagrams of ring G-spectra by diagrams of ring spectra) (3) replacing diagrams of ring spectra by diagrams of differential graded algebras (4) rigidity (replacing diagrams of DGAs by diagrams of graded rings). Systematic use of cellularization of model categories is central.