Rank functions on triangulated categories

Abstract

We introduce the notion of a rank function on a triangulated category $\mathcal{C}$ which generalizes the Sylvester rank function in the case when $\mathcal{C}=\operatorname{Perf}(A)$ is the perfect derived category of a ring $A$. We show that rank functions are closely related to functors into simple triangulated categories and classify Verdier quotients into simple triangulated categories in terms of particular rank functions called localizing. If $\mathcal{C}=\operatorname{Perf}(A)$ as above, localizing rank functions also classify finite homological epimorphisms from $A$ into differential graded skew-fields or, more generally, differential graded Artining rings. To establish these results, we develop the theory of derived localization of differential graded algebras at thick subcategories of their perfect derived categories. This is a far-reaching generalization of Cohn's matrix localization of rings and has independent interest.