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

TL;DR

This paper extends higher-dimensional Auslander-Reiten theory to $(d+2)$-angulated categories, establishing conditions for Serre functors, Auslander--Reiten $(d+2)$-angles, and mutation pairs, generalizing classical results.

Contribution

It demonstrates that Serre functors are $(d+2)$-angulated functors and characterizes the existence of Serre functors via Auslander--Reiten $(d+2)$-angles, generalizing prior work.

Findings

Serre functors on $ ext{(d+2)}$-angulated categories are $(d+2)$-angulated functors.

Existence of a Serre functor is equivalent to having Auslander--Reiten $(d+2)$-angles.

Quotient categories are $(d+2)$-angulated if and only if certain mutation conditions hold.

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 $\tau_d\mathscr{X}=\mathscr{X}$.