The spectrum of excisive functors

• 成果类型: Article
• 署名作者: Arone, Gregory; Barthel, Tobias; Heard, Drew; Sanders, Beren
• 署名单位: Stockholm University; Max Planck Society; Norwegian University of Science & Technology (NTNU); University of California System; University of California Santa Cruz
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-025-01338-9
• 发表日期: 2025
• 页码: 363-464
• 关键词: TENSOR-TRIANGULAR GEOMETRY; BALMER SPECTRUM; NILPOTENCY; HOMOLOGY; CLASSIFICATION; localization; EXTENSIONS; SUBGROUPS; calculus; THEOREMS

摘要

We prove a thick subcategory theorem for the category of d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$d$\end{document}-excisive functors from finite spectra to spectra. This generalizes the Hopkins-Smith thick subcategory theorem (the d=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$d=1$\end{document} case) and the C2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{2}$\end{document}-equivariant thick subcategory theorem (the d=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$d=2$\end{document} case). We obtain our classification theorem by completely computing the Balmer spectrum of compact d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$d$\end{document}-excisive functors. A key ingredient is a non-abelian blueshift theorem for the generalized Tate construction associated to the family of non-transitive subgroups of products of symmetric groups. Also important are the techniques of tensor triangular geometry and striking analogies between functor calculus and equivariant homotopy theory. In particular, we introduce a functor calculus analogue of the Burnside ring and describe its Zariski spectrum & agrave; la Dress. The analogy with equivariant homotopy theory is strengthened further through two applications: We explain the effect of changing coefficients from spectra to HZ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\operatorname{H}\hspace {-0.2em}\mathbb{Z}}$\end{document}-modules and we establish a functor calculus analogue of transchromatic Smith-Floyd theory as developed by Kuhn-Lloyd. Our work offers a new perspective on functor calculus which builds upon the previous approaches of Arone-Ching and Glasman.