Auslander-Reiten-Serre duality for n-exangulated categories

TL;DR

This paper establishes a duality condition in n-exangulated categories, linking Auslander-Reiten-Serre duality with the existence of Auslander-Reiten n-exangles, and explores conditions for Serre duality.

Contribution

It provides a characterization of Auslander-Reiten-Serre duality in n-exangulated categories and relates it to the existence of Auslander-Reiten n-exangles, offering new insights into their structure.

Findings

Equivalence between Auslander-Reiten-Serre duality and Auslander-Reiten n-exangles.

An equivalent condition for the existence of Serre duality.

A bijection triangle involving Auslander bijection and duality.

Abstract

Let $(\mathscr{C},\mathbb{E},\mathfrak{s})$ be an Ext-finite, Krull-Schmidt and $k$-linear $n$-exangulated category with $k$ a commutative artinian ring. In this note, we prove that $\mathscr{C}$ has Auslander-Reiten-Serre duality if and only if $\mathscr{C}$ 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 $\mathscr{C}$ has Auslander-Reiten-Serre duality. We exploit a bijection triangle, which involves the restricted Auslander bijection and the Auslander-Reiten-Serre duality.