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

Abstract

Xiao and Zhu has shown that if $\mathcal{C}$ is a locally finite triangulated category, then the Auslander-Reiten triangles generate the relations for the Grothendieck group of $\mathcal{C}$. 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 $\mathcal{C}$ is locally finite if and only if the Auslander-Reiten $(d+2)$-angles generate the relations for the Grothendieck group of $\mathcal{C}$. This extends the result of Xiao and Zhu, and gives the converse of Xiao and Zhu's result is also true.

Updated version. arXiv admin note: text overlap with arXiv:1912.07983

References in corpus (4)

• Associahedra for finite type cluster algebras and minimal relations between $\mathbf{g}$-vectors
• On the existence of Auslander-Reiten (d+2)-angles in (d+2)-angulated categories
• Grothendieck groups in extriangulated categories
• Idempotent completion of n-angulated categories