Arkiv för Matematik

Volume 60 (2022)

Number 2

On the existence of Auslander–Reiten $n$-exangles in $n$-exangulated categories

Pages: 365 – 385

DOI: https://dx.doi.org/10.4310/ARKIV.2022.v60.n2.a8

Authors

Jian He (Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu, China)

Jiangsheng Hu (School of Mathematics and Physics, Jiangsu University of Technology, Changzhou, Jiangsu, China)

Dongdong Zhang (Department of Mathematics, Zhejiang Normal University, Jinhua, Zhejiang, China)

Panyue Zhou (School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha, Hunan, China)

Abstract

Let $\mathscr{C}$ be an $n$-exangulated category. In this note, we show that if $\mathscr{C}$ is locally finite, then $\mathscr{C}$ has Auslander–Reiten $n$-exangles. This unifies and extends results of Xiao–Zhu, Zhu–Zhuang, Zhou, and Xie–Lu–Wang for triangulated, extriangulated, $(n+2)$-angulated and $n$-abelian categories, respectively.

Keywords

$n$-exangulated categories, Auslander–Reiten $n$-exangles, locally finite, extriangulated categories, triangulated categories, $n$-abelian categories

2010 Mathematics Subject Classification

18E10, 18G50

Received 27 October 2021

Received revised 10 January 2022

Accepted 19 January 2022

Published 26 October 2022