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

Zhou, Panyue 

Glasgow mathematical journal, Tome 64 (2022) no. 3, pp. 527-547

Résumé

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

Détail

DOI : 10.1017/S0017089521000343

Mots-clés : triangulated categories, (d+2)-angulated categories, Auslander-Reiten (d+2)-angles, Serre functor