Quillen stratification in equivariant homotopy theory

Abstract We prove a version of Quillen’s stratification theorem in equivariant homotopy theory for a finite group $G$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> , 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 $E$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>E</mml:mi> </mml:math> -theory $\underline{E_{n}}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:munder> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>_</mml:mo> </mml:munder> </mml:math> , for any finite height $n$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> </mml:math> and any finite group $G$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> , 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 $\underline{E_{n}}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:munder> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>_</mml:mo> </mml:munder> </mml:math> , thereby establishing a finite height analogue of the work of Benson, Iyengar, and Krause in modular representation theory.