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Rank functions on triangulated categories

Chuang, J. & Lazarev, A. (2021). Rank functions on triangulated categories. Journal für die reine und angewandte Mathematik, 2021(781), pp. 127-164. doi: 10.1515/crelle-2021-0052


We introduce the notion of a rank function on a triangulated category C which generalizes the Sylvester rank function in the case when C D 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 D 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.

Publication Type: Article
Additional Information: © 2021 Joseph Chuang and Andrey Lazarev, published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 International License.
Subjects: Q Science > QA Mathematics
Departments: School of Science & Technology > Mathematics
Text - Published Version
Available under License Creative Commons: Attribution International Public License 4.0.

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