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.

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

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