An algebraic model for rational torus-equivariant spectra

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.

Minor corrections to v5. This version includes reference to arxiv:1801.09766 which permits one to use ordinary equivariant orthogonal spectra as the underlying category. To appear in Journal of Topology

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Cited by in corpus (7)

• G-symmetric monoidal categories of modules over equivariant commutative ring spectra
• An algebraic model for free rational G-spectra
• The Left Localization Principle, completions, and cofree $G$-spectra
• Torsion models for tensor-triangulated categories: the one-step case
• Flatness and Shipley's algebraicization theorem
• Algebraic models of change of groups functors in (co)free rational equivariant spectra
• An algebraic model for rational toral G-spectra