Higher-dimensional Auslander-Reiten theory on $(d+2)$-angulated categories

Abstract

Let $\mathcal{C}$ be a $(d+2)$-angulated category with $d$-suspension functor ${\Sigma }^d$. Our main results show that every Serre functor on $\mathcal{C}$ is a $(d+2)$-angulated functor. We also show that $\mathcal{C}$ has a Serre functor $\mathbb{S}$ if and only if $\mathcal{C}$ 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 $\mathcal{X}$ of $\mathcal{C}$, the quotient category $\mathcal{C}/\mathcal{X}$ is a $(d+2)$-angulated category if and only if $(\mathcal{C},\mathcal{C})$ is an $\mathcal{X}$-mutation pair, and if and only if ${\tau }_d\mathcal{X}=\mathcal{X}$.

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