Balmer spectra and Drinfeld centers

Abstract

The Balmer spectrum of a monoidal triangulated category is an important geometric construction which, in many cases, can be used to obtain a classification of thick tensor ideals. Many interesting examples of monoidal triangulated categories arise as stable categories of finite tensor categories. Given a finite tensor category, its Drinfeld center is a braided finite tensor category constructed via half-braidings; in the case that the original category is the category of representations of a Hopf algebra, then its Drinfeld center is the category of representations of its Drinfeld double. 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. We then prove that this functor induces a continuous map between the Balmer spectra of these two stable categories. In the finite-dimensional Hopf algebra setting, we give conditions under which the image of this continuous map can be realized as a certain intersection of open sets in the Balmer spectrum, and prove conditions under which it is injective, surjective, or a homeomorphism. We apply this general theory to the cases that the finite tensor category C is the category of modules for group algebras of finite groups G in characteristic p dividing the order of G, more generally to cosemisimple Hopf algebras, and for Benson-Witherspoon smash coproduct Hopf algebras HG,L. In the first case, we are able to prove that the Balmer spectrum for the stable module category of the Drinfeld double of any cosemisimple quasitriangular Hopf algebra H is homeomorphic to the Balmer spectrum for the stable module category of H itself, and that thick ideals of the two categories are in bijection.

Hypersurface support and prime ideal spectra for stable categories

“…One can also see [60] for an exploration of the relationship between spectra for stab(C ) and spectra for stab(Z(C )) in some extreme settings. 6.3.…”