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arXiv · 1912.11397

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

Panyue Zhou
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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.

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Panyue Zhou. 2019-12-22. Grothendieck groups and Auslander-Reiten (d+2)-angles. https://arxiv.org/abs/1912.11397

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