Quillen stratification in equivariant homotopy theory

• 成果类型: Article
• 署名作者: Barthel, Tobias; Castellana, Natalia; Heard, Drew; Naumann, Niko; Pol, Luca
• 署名单位: Max Planck Society; Autonomous University of Barcelona; Centre de Recerca Matematica (CRM); Norwegian University of Science & Technology (NTNU); University of Regensburg
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-024-01301-0
• 发表日期: 2025
• 页码: 219-285
• 关键词: TENSOR-TRIANGULAR GEOMETRY; STABLE-HOMOTOPY; BALMER SPECTRUM; K-THEORY; COHOMOLOGY; SUBCATEGORIES; RESTRICTION; NILPOTENCY; EXTENSIONS; CATEGORIES

摘要

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.