# Grothendieck groups and Auslander-Reiten -angles^{†}^{†}footnotetext: Panyue Zhou was supported by the National Natural Science Foundation of China (Grant No. 11901190 and 11671221), and by the Hunan Provincial Natural Science Foundation of China (Grant No. 2018JJ3205), and by the Scientific Research Fund of Hunan Provincial Education Department (Grant No. 19B239).

###### Abstract

Xiao and Zhu has shown that if is a locally finite triangulated category, then
the Auslander-Reiten triangles generate the relations for the Grothendieck group of .
The notion of -angulated categories is a “higher dimensional” analogue
of triangulated categories. In this article, we show that if is a locally finite -angulated category where is odd, then
the Auslander-Reiten -angles generate the relations for the Grothendieck group of .
A partial converse result is given when a -angulate category has a cluster tilting subcategory.

Key words: -angulated categories; Auslander-Reiten -angles; locally finite.
cluster tilting subcategories; triangulated categories.

2010 Mathematics Subject Classification: 16G70; 18E30.

## 1 Introduction

Auslander-Reiten theory was introduced by Auslander and Reiten in [ar1, ar2]. Since its introduction, Auslander-Reiten theory has become a fundamental tool for studying the representation theory of Artin algebras. It is well-known that a module category of over an Artin algebra has almost split sequences. For an Artin algebra of finite type, Butler [bu] proved that the relations of its Grothendieck group are generated by all Auslander-Reiten sequences. Soon later Auslander showed that the converse is true in [au]. The notion of Auslander-Reiten triangles in a triangulated category was introduced by Happel in [ha]. In contrast to module categories over Artin algebras, not all triangulated categories have Auslander-Reiten triangles [ha]. It was proved in [ha] that the derived category of a finite dimensional algebra has Auslander-Reiten triangles if and only if the global dimension of the algebra is finite. Reiten and Van den Bergh [rv] proved that the existence of Auslander-Reiten triangles if and only if the existence of Serre functor in a triangulated category. Recently, Xiao and Zhu [xz] showed that if is a locally finite triangulated category, then has Auslander-Reiten triangles. Beligiannis [be] proved the converse of this result holds when is a compactly generated triangulated category. In more recent times, many authors have shown the reverse direction of Xiao and Zhu is true in some special cases [h1, pppp]. Extriangulated categories were introduced by Nakaoka and Palu [np] as a simultaneous generalization of exact categories and triangulated categories. Hence, many results hold on exact categories and triangulated categories can be unified in the same framework. Iyama, Nakaoka and Palu [inp] introduced the notion of almost split extensions and Auslander-Reiten-Serre duality for extriangulated categories, and gave explicit connections between these notions and also with the classical notion of dualizing -varieties. Zhu and Zhuang [zz] has shown that a locally finite extriangulated category has Auslander-Reiten -triangles and the relations of Grothendieck group are generated by the Auslander-Reiten -triangles. A partial converse result is given when restricting to a triangulated category with a cluster tilting subcategory.

In [gko], Geiss, Keller and Oppermann introduced -angulated categories. These are are a “higher dimensional” analogue of triangulated categories, in the sense that triangles are replaced by -angles, that is, morphism sequences of length . Thus a -angulated category is precisely a triangulated category. They appear for example as certain cluster tilting subcategories of triangulated categories. Iyama and Yoshino defined the notion of Auslander-Reiten -angles in special -angulated categories. This notion was generalized to arbitrary -angulated categories by Fedele [fe]. Fedele also proved that there are Auslander-Reiten -angles in certain subcategories of -angulated categories. The author [z1] showed that if a -angulated category has Auslander-Reiten -angles if and only if has a Serre functor. Moreover, the author [z2] also proved if is a locally -angulated category, then has Auslander-Reiten -angles.

Bergh and Thaule [bt2] defined the Grothendieck group of a -angulated category. As in the triangulated case, it is the free abelian group on the set of isomorphism classes of objects, modulo the Euler relations corresponding to the -angles. They showed that when is odd, the set of subgroups corresponds bijectively to the complete and dense -angulated subcategories. Recently, Herschend, Liu and Nakaoka [hln] defined -exangulated categories as a “higher dimensional” analogue of extriangulated categories. Many categories studied in representation theory turn out to be -exangulated. In particular, -exangulated categories simultaneously generalize -angulated and -exact categories [ja]. Haugland [h2] defined the Grothendieck group of a -exangulated category, and classified dense complete subcategories of -exangulated categories with a -(co)generator in terms of subgroups of the Grothendieck group.

The aim of this article to discuss the relation between Grothendieck groups in -angulated categories and Auslander-Reiten -angles in -angulated categories. We show that the relations of the Grothendieck group of a locally finite -angulated category where is odd are generated by all Auslander-Reiten -angles, see Theorem LABEL:main1. A partial converse result is given when a -angulated category admits a cluster tilting subcategory, see Theorem LABEL:main2.

This article is organised as follows: In Section 2, we review some elementary definitions that we need to use, including -angulated categories and Auslander-Reiten angles. In Section 3, we show our first and second main results.

## 2 Preliminaries

In this section, we first recall the definition and basic properties of -angulated categories from [gko]. Let be an additive category with an automorphism , and an integer greater than or equal to one.

A -- in is a sequence of objects and morphisms

Its left rotation is the --sequence

A *morphism* of --sequences is a sequence of morphisms such that the following diagram commutes