Balmer spectra and Drinfeld centers

Vashaw, Kent B.

doi:10.2140/ant.2024.18.1081

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Balmer spectra and Drinfeld centers

Kent B. Vashaw

Vol. 18 (2024), No. 6, 1081–1116

DOI: 10.2140/ant.2024.18.1081

Abstract

The Balmer spectrum of a monoidal triangulated category is an important geometric construction which is closely related to the problem of classifying thick tensor ideals. We prove that the forgetful functor from the Drinfeld center of a finite tensor category $C$ to $C$ extends to a monoidal triangulated functor between their corresponding stable categories, and induces a continuous map between their Balmer spectra. We give conditions under which it is injective, surjective, or a homeomorphism. We apply this general theory to prove that Balmer spectra associated to finite-dimensional cosemisimple quasitriangular Hopf algebras (in particular, group algebras in characteristic dividing the order of the group) coincide with the Balmer spectra associated to their Drinfeld doubles, and that the thick ideals of both categories are in bijection. An analogous theorem is proven for certain Benson–Witherspoon smash coproduct Hopf algebras, which are not quasitriangular in general.

Keywords

Hopf algebra, stable module category, tensor triangulated category, thick ideal

Mathematical Subject Classification

Primary: 16T05, 18G65, 18G80, 18M05, 18M15

Milestones

Received: 1 December 2021

Revised: 5 June 2023

Accepted: 3 September 2023

Published: 30 April 2024

Authors

Kent B. Vashaw
Department of Mathematics Massachusetts Institute of Technology Cambridge, MA United States

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Vol. 18, No. 6, 2024