Rank functions on triangulated categories

TL;DR

This paper introduces rank functions on triangulated categories, generalizing Sylvester rank, and uses them to classify Verdier quotients and homological epimorphisms, developing a broad localization theory for differential graded algebras.

Contribution

It defines and studies rank functions on triangulated categories, linking them to simple categories and classifying quotients and epimorphisms, extending localization theory.

Findings

Rank functions generalize Sylvester rank to triangulated categories.

Classifies Verdier quotients via localizing rank functions.

Develops derived localization theory for differential graded algebras.

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.