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- Title
A functorial approach to rank functions on triangulated categories.
- Authors
Conde, Teresa; Gorsky, Mikhail; Marks, Frederik; Zvonareva, Alexandra
- Abstract
We study rank functions on a triangulated category 풞 via its abelianisation mod C . We prove that every rank function on 풞 can be interpreted as an additive function on mod C . As a consequence, every integral rank function has a unique decomposition into irreducible ones. Furthermore, we relate integral rank functions to a number of important concepts in the functor category Mod C . We study the connection between rank functions and functors from 풞 to locally finite triangulated categories, generalising results by Chuang and Lazarev. In the special case C = T c for a compactly generated triangulated category 풯, this connection becomes particularly nice, providing a link between rank functions on 풞 and smashing localisations of 풯. In this context, any integral rank function can be described using the composition length with respect to certain endofinite objects in 풯. Finally, if C = per (A) for a differential graded algebra 퐴, we classify homological epimorphisms A → B with per (B) locally finite via special rank functions which we call idempotent.
- Subjects
TRIANGULATED categories; ADDITIVE functions; DIFFERENTIAL algebra; INTEGRAL functions; NUMBER concept; C*-algebras; IDEMPOTENTS
- Publication
Journal für die Reine und Angewandte Mathematik, 2024, Vol 2024, Issue 811, p135
- ISSN
0075-4102
- Publication type
Article
- DOI
10.1515/crelle-2024-0009