Abstract
Xiao and Zhu have shown that if C is a locally finite triangulated category, then the Auslander-Reiten triangles generate the relations for the Grothendieck group of C. The notion of (d+2)-angulated categories is a “higher dimensional” analogue of triangulated categories. In this article, we show that a (d+2)-angulated category C with d odd is locally finite if and only if the Auslander-Reiten (d+2)-angles generate the relations for the Grothendieck group of 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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