Higher Auslander-Reiten sequences revisited

TL;DR

This paper explores the existence and properties of Auslander-Reiten sequences and angles in quotient and stable categories derived from $n$-exangulated categories, extending classical results to higher homological algebra.

Contribution

It proves that Auslander-Reiten $n$-exangles induce Auslander-Reiten $n$-exact sequences in quotient categories and that stable categories of Frobenius $n$-exangulated categories have Auslander-Reiten $(n+2)$-angles.

Findings

Existence of Auslander-Reiten $n$-exact sequences in quotient categories.

Stable categories of Frobenius $n$-exangulated categories have Auslander-Reiten $(n+2)$-angles.

Extension of classical Auslander-Reiten theory to higher homological algebra.

Abstract

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