Higher-dimensional Auslander–Reiten theory on (d+2)-angulated categories
Abstract Let $\mathscr{C}$ be a $(d+2)$ -angulated category with d-suspension functor $\Sigma^d$ . Our main results show that every Serre functor on $\mathscr{C}$ is a $(d+2)$ -angulated functor. We also show that $\mathscr{C}$ has a Serre functor $\mathbb{S}$ if and only if $\mathscr{C}$ has Auslander–Reiten $(d+2)$ -angles. Moreover, $\tau_d=\mathbb{S}\Sigma^{-d}$ where $\tau_d$ is d-Auslander–Reiten translation. These results generalize work by Bondal–Kapranov and Reiten–Van den Bergh. As an application, we prove that for a strongly functorially finite subcategory $\mathscr{X}$ of $\mathscr{C}$ , the quotient category $\mathscr{C}/\mathscr{X}$ is a $(d+2)$ -angulated category if and only if $(\mathscr{C},\mathscr{C})$ is an $\mathscr{X}$ -mutation pair, and if and only if $\tau_d\mathscr{X} =\mathscr{X}$ .