Noncommutative Tensor Triangular Geometry and Applications

In this talk, I will show how to develop a general noncommutative version of Balmer's tensor triangular geometry that is applicable to arbitrary monoidal triangulated categories (M$\Delta$C). Insights from noncommutative ring theory are used to obtain a framework for prime, semiprime, and completely prime (thick) ideals of an M$\Delta$C, $\mathbf K$, and then to associate to $\mathbf K$ a topological space--the Balmer spectrum $\Spc {\mathbf K}$. 

We develop a general framework for (noncommutative) support data, coming in three different flavors, and show that $\Spc \bK$ 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 $\mathbf K$, which in turn can be applied to classify the thick two-sided ideals and $\Spc {\mathbf K}$. 

Applications will be given for quantum groups and non-cocommutative finite-dimensional Hopf algebras studied by Benson and Witherspoon. 

This work represents results with Brian Boe and Jonathan Kujawa, and with Milen Yakimov and Kent Vashaw]. 

Meeting-ID: 868 9560 8815
Password: 348162