Higher Auslander-Reiten sequences revisited

Abstract

Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an $n$-exangulated category with enough projectives and enough injectives, and $\mathcal{X}$ be a cluster-tilting subcategory of $\mathcal{C}$. Liu and Zhou have shown that the quotient category $\mathcal{C}/\mathcal{X}$ is an $n$-abelian category. In this paper, we prove that if $\mathcal{C}$ has Auslander-Reiten $n$-exangles, then $\mathcal{C}/\mathcal{X}$ has Auslander-Reiten $n$-exact sequences. Moreover, we also show that if a Frobenius $n$-exangulated category $\mathcal{C}$ has Auslander-Reiten $n$-exangles, then the stable category $\overline{\mathcal{C}}$ of $\mathcal{C}$ has Auslander-Reiten $(n+2)$-angles.

19 pages. arXiv admin note: text overlap with arXiv:2006.02223, arXiv:2108.07985, arXiv:2110.02476