Hypersurface support and prime ideal spectra for stable categories

TL;DR

This paper classifies thick ideals and prime spectra in stable categories of various finite-dimensional Hopf algebra representations using hypersurface support, revealing new spectral identifications.

Contribution

It introduces a hypersurface support framework to classify thick ideals and explicitly identifies prime spectra for multiple classes of finite-dimensional Hopf algebra stable categories.

Findings

Thick ideals are classified via hypersurface support.

Prime spectra are explicitly identified for several Hopf algebra categories.

In the case of finite group schemes, the spectrum relates to cohomology quotients.

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 $\pi_0(G)$ in $G$.