2023/01/05 by Tobias Barthel, Barthel, Tobias, Natàlia Castellana +7 · 1 citation
Mathematics · #Advanced Topics in Algebra #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2301.02212
openalex publication_date 2023/01/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
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 \underlineEn, 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 \underlineEn, thereby establishing a finite height analogue of the work of Benson, Iyengar, and Krause in modular representation theory.