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Hypersurface support and prime ideal spectra for stable categories

Cris Negron and Julia Pevtsova

Vol. 8 (2023), No. 1, 25–79
Abstract

We use hypersurface support to classify thick (two-sided) ideals in the stable categories of representations for several families of finite-dimensional integrable Hopf algebras: bosonized quantum complete intersections, quantum Borels in type A, Drinfeld doubles of height 1 Borels in finite characteristic, and rings of functions on finite group schemes over a perfect field. We then identify the prime ideal (Balmer) spectra for these stable categories. In the curious case of functions on a finite group scheme 𝒢, the spectrum of the category is identified not with the spectrum of cohomology, but with the quotient of the spectrum of cohomology by the adjoint action of the subgroup of connected components π0(𝒢) in 𝒢.

Keywords
tensor triangular geometry, Balmer spectrum, quantum groups, quantum complete intersections
Mathematical Subject Classification
Primary: 16E30, 16T05, 18G80, 18M05
Milestones
Received: 26 August 2021
Revised: 31 August 2022
Accepted: 6 November 2022
Published: 1 May 2023
Authors
Cris Negron
Department of Mathematics
University of Southern California
Los Angeles, CA
United States
Julia Pevtsova
Department of Mathematics
University of Washington
Seattle, WA
United States