Auslander-Reiten-Serre duality for n-exangulated categories

Abstract

Let (C , E, 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 C has Auslander-Reiten-Serre duality if and only if 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 C has Auslander-Reiten-Serre duality. We exploit a bijection triangle, which involves the restricted Auslander bijection and the Auslander-Reiten-Serre duality.