Grothendieck groups and Auslander-Reiten (d+2)-angles

TL;DR

This paper extends Xiao and Zhu's result by showing that in locally finite $(d+2)$-angulated categories, Auslander-Reiten $(d+2)$-angles generate the Grothendieck group relations, establishing a bidirectional equivalence.

Contribution

It generalizes the relation between Auslander-Reiten angles and Grothendieck groups from triangulated to higher $(d+2)$-angulated categories, including the converse implication.

Findings

Auslander-Reiten $(d+2)$-angles generate Grothendieck group relations in locally finite categories

The equivalence between local finiteness and the generation of relations by Auslander-Reiten $(d+2)$-angles

Extension of Xiao and Zhu's result to higher-dimensional categories

Abstract

Xiao and Zhu has shown that if $\cal C$ is a locally finite triangulated category, then the Auslander-Reiten triangles generate the relations for the Grothendieck group of $\cal C$. The notion of $(d +2)$-angulated categories is a "higher dimensional" analogue of triangulated categories. In this article, we show that if A $(d+2)$-angulated category $\cal C$ is locally finite if and only if the Auslander-Reiten $(d+2)$-angles generate the relations for the Grothendieck group of $\cal C$. This extends the result of Xiao and Zhu, and gives the converse of Xiao and Zhu's result is also true.