Noncommutative tensor triangular geometry

Abstract

We develop a general noncommutative version of Balmer's tensor triangular geometry that is applicable to arbitrary monoidal triangulated categories (M$\Delta$Cs). Insight from noncommutative ring theory is used to obtain a framework for prime, semiprime, and completely prime (thick) ideals of an M$\Delta$C, ${\bf K}$, and then to associate to ${\bf K}$ a topological space--the Balmer spectrum $\operatorname{Spc} {\bf K}$. We develop a general framework for (noncommutative) support data, coming in three different flavors, and show that $\operatorname{Spc} {\bf K}$ is a universal terminal object for the first two notions (support and weak support). The first two types of support data are then used in a theorem that gives a method for the explicit classification of the thick (two-sided) ideals and the Balmer spectrum of an M$\Delta$C. The third type (quasi support) is used in another theorem that provides a method for the explicit classification of the thick right ideals of ${\bf K}$, which in turn can be applied to classify the thick two-sided ideals and $\operatorname{Spc} {\bf K}$. As a special case, our approach can be applied to the stable module categories of arbitrary finite dimensional Hopf algebras that are not necessarily cocommutative (or quasitriangular). We illustrate the general theorems with classifications of the Balmer spectra and thick two-sided/right ideals for the stable module categories of all small quantum groups for Borel subalgebras, and classifications of the Balmer spectra and thick two-sided ideals of Hopf algebras studied by Benson and Witherspoon.