Barthel, Tobias and Castellana, Natalia and Heard, Drew and Naumann, Niko and Pol, Luca (2025) Quillen stratification in equivariant homotopy theory. INVENTIONES MATHEMATICAE, 239 (1). pp. 219-285. ISSN 0020-9910, 1432-1297
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We prove a version of Quillen's stratification theorem in equivariant homotopy theory for a finite group G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$G$\end{document}, 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$E$\end{document}-theory En_\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\underline{E_{n}}$\end{document}, for any finite height n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n$\end{document} and any finite group G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$G$\end{document}, 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 circle times-ideals of the category of equivariant modules over En_\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\underline{E_{n}}$\end{document}, thereby establishing a finite height analogue of the work of Benson, Iyengar, and Krause in modular representation theory.
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | TENSOR-TRIANGULAR GEOMETRY; STABLE-HOMOTOPY; BALMER SPECTRUM; K-THEORY; COHOMOLOGY; SUBCATEGORIES; RESTRICTION; NILPOTENCY; EXTENSIONS; CATEGORIES; |
| Subjects: | 500 Science > 510 Mathematics |
| Divisions: | Mathematics > Prof. Dr. Niko Naumann |
| Depositing User: | Dr. Gernot Deinzer |
| Date Deposited: | 24 Mar 2026 09:11 |
| Last Modified: | 24 Mar 2026 09:11 |
| URI: | https://pred.uni-regensburg.de/id/eprint/64808 |
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