Quillen stratification in equivariant homotopy theory

TL;DR

This paper generalizes Quillen's stratification theorem to equivariant homotopy theory using equivariant ring spectra and categorifies it for modules, providing new insights into the structure of equivariant categories and their spectra.

Contribution

It extends Quillen's stratification to equivariant settings with ring spectra and categorifies it for modules, linking equivariant tensor-triangular geometry with geometric fixed points.

Findings

Established a stratification theorem in equivariant tensor-triangular geometry.

Computed the Balmer spectrum for equivariant Lubin--Tate E-theory.

Provided a cohomological parametrization of localizing tensor ideals.

Abstract

We prove a version of Quillen's stratification theorem in equivariant homotopy theory for a finite group $G$, generalizing the classical theorem in two directions. Firstly, we work with arbitrary commutative equivariant ring spectra as coefficients, and secondly, we categorify it to a result about equivariant modules. Our general stratification theorem is formulated in the language of equivariant tensor-triangular geometry, which we show to be tightly controlled by the non-equivariant tensor-triangular geometry of the geometric fixed points. We then apply our methods to the case of Borel-equivariant Lubin--Tate $E$-theory $\underline{E_n}$, for any finite height $n$ and any finite group $G$, where we obtain a sharper theorem in the form of cohomological stratification. In particular, this provides a computation of the Balmer spectrum as well as a cohomological parametrization of all localizing $\otimes$-ideals of the category of equivariant modules over $\underline{E_n}$, thereby establishing a finite height analogue of the work of Benson, Iyengar, and Krause in modular representation theory.