Noncommutative tensor triangular geometry
Daniel K. Nakano, Kent B. Vashaw, Milen Yakimov
Abstract
We develop a general noncommutative version of Balmer's tensor triangular geometry that is applicable to arbitrary monoidal triangulated categories (MCs). Insight from noncommutative ring theory is used to obtain a framework for prime, semiprime, and completely prime (thick) ideals of an MC, K, and then to associate to K a topological space-the Balmer spectrum Spc K. We develop a general framework for (noncommutative) support data, coming in three different flavors, and show that Spc 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 MC. The third type (quasi support) is used in another theorem that provides a method for the explicit classification of the thick right ideals of K, which in turn can be applied to classify the thick two-sided ideals and Spc K.