Auslander-Reiten-Serre duality for n-exangulated categories

Abstract

Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an Ext-finite, Krull-Schmidt and $k$-linear $n$-exangulated category with $k$ a commutative artinian ring. In this note, we prove that $\mathcal{C}$ has Auslander-Reiten-Serre duality if and only if $\mathcal{C}$ has Auslander-Reiten $n$-exangles. Moreover, we also give an equivalent condition for the existence of Serre duality (which is a special type of Auslander-Reiten-Serre duality). Finally, assume further that $\mathcal{C}$ has Auslander-Reiten-Serre duality. We exploit a bijection triangle, which involves the restricted Auslander bijection and the Auslander-Reiten-Serre duality.

arXiv comment24 pages. arXiv admin note: substantial text overlap with arXiv:2111.06522; text overlap with arXiv:2110.02476

References in corpus (6)

• $n$-angulated categories
• An introduction to higher Auslander-Reiten theory
• Auslander-Reiten theory in quasi-abelian and Krull-Schmidt categories
• On the existence of Auslander-Reiten (d+2)-angles in (d+2)-angulated categories
• $n$-exact categories arising from $n$-exangulated categories
• Higher Auslander-Reiten sequences via morphisms determined by objects