Abstract:We introduce the notion of a rank function on a $(d+2)$-angulated category $\mathcal{C}$ which generalises the notion of a rank function on a triangulated category. Inspired by work of Chuang and Lazarev, for $d$ an odd positive integer, we prove that there is a bijective correspondence between rank functions defined on objects in $\mathcal{C}$ and rank functions defined on morphisms in $\mathcal{C}$. Inspired by work of Conde, Gorsky, Marks and Zvonareva, for $d$ an odd positive integer, we show there is a bijective correspondence between rank functions on $\operatorname{\mathsf{Proj}}A$ and additive functions on $\operatorname{\mathsf{mod}}(\operatorname{\mathsf{Proj}}A)$, where $\operatorname{\mathsf{Proj}}A$ is endowed with the Amiot-Lin $(d+2)$-angulated category structure. This allows us to show that every integral rank function on $\operatorname{\mathsf{Proj}}A$ can be decomposed into irreducible rank functions.
From: David Nkansah [view email]
[v1]
Wed, 29 May 2024 12:33:14 UTC (33 KB)
[v2]
Tue, 5 Aug 2025 18:05:41 UTC (37 KB)
[v3]
Fri, 12 Jun 2026 12:34:37 UTC (34 KB)
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