Noncommutative Tensor Triangular Geometry and Its Applications to Representation Theory

Abstract

One of the cornerstones of the representation theory of Hopf algebras and finite tensor categories is the theory of support varieties. Balmer introduced tensor triangular geometry for symmetric monoidal triangulated categories, which united various support variety theories coming from disparate areas such as homotopy theory, algebraic geometry, and representation theory. In this thesis a noncommutative version will be introduced and developed. We show that this noncommutative analogue of Balmer's theory can be determined in many concrete situations via the theory of abstract support data, and can be used to classify thick tensor ideals. We prove an analogue of prime ideal contraction, connecting the Balmer spectrum of a stable category of a finite tensor category with the stable category of its Drinfeld center. We classify the Balmer spectra for various examples arising in representation theory, such as Drinfeld doubles of cosemisimple Hopf algebras, the smash coproducts studied by Benson and Witherspoon, and the small quantum Borels. Lastly, we leverage the theory to prove the tensor product property for cohomological support varieties in a family of small quantum Borels, a conjecture of Negron and Pevtsova.