Rank functions on triangulated categories

Joseph Chuang,Andrey Lazarev

doi:10.1515/crelle-2021-0052

Abstract

Abstract We introduce the notion of a rank function on a triangulated category 𝒞 {\mathcal{C}} which generalizes the Sylvester rank function in the case when 𝒞 = 𝖯𝖾𝗋𝖿 ⁢ ( A ) {\mathcal{C}=\mathsf{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 𝒞 = 𝖯𝖾𝗋𝖿 ⁢ ( A ) {\mathcal{C}=\mathsf{Perf}(A)} as above, localizing rank functions also classify finite homological epimorphisms from A into differential graded skew-fields or, more generally, differential graded Artinian 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.

Highlights

The dimension of a vector space V over afield K is a basic characteristic of V and it is an elementary fact that it is an invariant, i.e. does not depend on the choice of a basis in V
One possibility to obviate this difficulty is to start with a homomorphism f W A ! K where K is a skew-field and define the rank of an A-module as the rank of the corresponding K-module
Different homomorphisms f give rise to possibly different ranks. This suggests that ranks are closely related to homomorphisms into skew-fields

Summary

Introduction

The dimension of a vector space V over a (possibly skew-)field K is a basic characteristic of V and it is an elementary fact that it is an invariant, i.e. does not depend on the choice of a basis in V. In the present paper we build on this work to construct derived localization of a dg ring A with respect to an arbitrary thick subcategory of Perf.A/ This extends the notion of matrix localization since the latter is the nonderived version of the localization with respect to the thick subcategory generated by a collection of free complexes of length 2. / is simple, i.e. equivalent to the perfect derived category of a graded skew-field These are the so-called localizing rank functions.

Rank functions on perfect derived categories of ordinary rings

Further properties of rank functions

Derived localization of differential graded algebras

Localizing rank functions and fraction fields