The index with respect to a rigid subcategory of a triangulated category

Peter Jørgensen, Amit Shah*

*Corresponding author for this work

Research output: Contribution to journal/Conference contribution in journal/Contribution to newspaperJournal articleResearchpeer-review

3 Citations (Scopus)

Abstract

Palu defined the index with respect to a cluster tilting object in a suitable triangulated category, in order to better understand the Caldero–Chapoton map that exhibits the connection between cluster algebras and representation theory. We push this further by proposing an index with respect to a contravariantly finite, rigid subcategory, and we show this index behaves similarly to the classical index. Let C be a skeletally small triangulated category with split idempotents, which is thus an extriangulated category (C, E, s). Suppose X is a contravariantly finite, rigid subcategory of C. We define the index ind X (C) of an object C ∈ C with respect to X as the K 0-class [C] X in Grothendieck group K 0(C, E X , s X ) of the relative extriangulated category (C, E X , s X ). By analogy to the classical case, we give an additivity formula with error term for ind X on triangles in C. In case X is contained in another suitable subcategory T of C, there is a surjection Q: K 0(C, E T , s T ) → K 0(C, E X , s X ). Thus, in order to describe K 0(C, E X , s X ), it suffices to determine K 0(C, E T , s T ) and Ker Q. We do this under certain assumptions.

Original languageEnglish
JournalInternational Mathematics Research Notices
Volume2024
Issue4
Pages (from-to)3278-3309
Number of pages32
ISSN1073-7928
DOIs
Publication statusPublished - Feb 2024

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