Cambridge Journal of Mathematics

Volume 11 (2023)

Number 4

Stratification in tensor triangular geometry with applications to spectral Mackey functors

Pages: 829 – 915

DOI: https://dx.doi.org/10.4310/CJM.2023.v11.n4.a2

Authors

Tobias Barthel (Max Planck Institute for Mathematics, Bonn, Germany)

Drew Heard (Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway)

Beren Sanders (Department of Mathematics, University of California, Santa Cruz, Calif., U.S.A.)

Abstract

We systematically develop a theory of stratification in the context of tensor triangular geometry and apply it to classify the localizing tensor-ideals of certain categories of spectral $G$-Mackey functors for all finite groups $G$. Our theory of stratification is based on the approach of Stevenson which uses the Balmer–Favi notion of big support for tensor-triangulated categories whose Balmer spectrum is weakly noetherian. We clarify the role of the local-to-global principle and establish that the Balmer–Favi notion of support provides the universal approach to weakly noetherian stratification. This provides a uniform new perspective on existing classifications in the literature and clarifies the relation with the theory of Benson–Iyengar–Krause. Our systematic development of this approach to stratification, involving a reduction to local categories and the ability to pass through finite étale extensions, may be of independent interest. Moreover, we strengthen the relationship between stratification and the telescope conjecture. The starting point for our equivariant applications is the recent computation by Patchkoria–Sanders–Wimmer of the Balmer spectrum of the category of derived Mackey functors, which was found to capture precisely the height $0$ and height $\infty$ chromatic layers of the spectrum of the equivariant stable homotopy category. We similarly study the Balmer spectrum of the category of $E(n)$-local spectral Mackey functors noting that it bijects onto the height $\leq n$ chromatic layers of the spectrum of the equivariant stable homotopy category; conjecturally the topologies coincide. Despite our incomplete knowledge of the topology of the Balmer spectrum, we are able to completely classify the localizing tensor-ideals of these categories of spectral Mackey functors.

Keywords

stratification, support, localizing ideals, tensor triangular geometry, spectral Mackey functors

2010 Mathematics Subject Classification

14Fxx, 55P42, 55P91, 55U35, 18F99

Received 24 February 2022

Published 29 September 2023