Quillen stratification in equivariant homotopy theory

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 Borelequivariant Lubin–Tate E-theory En, 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 ⊗-ideals of the category of equivariant modules over En, thereby establishing a finite height analogue of the work of Benson, Iyengar, and Krause in modular representation theory.