Auslander-Reiten-Serre duality revisited

Abstract

Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category with a right Auslander-Reiten-Serre (ARS for short) duality $(\tau, \eta)$ in the sense of Iyama, Nakaoka and Palu. We show $(\tau, \eta)$ induces right ARS dualities on the relative theories of $(\mathcal{C},\mathbb{E},\mathfrak{s})$. Under relative structures, we show that the functor $\tau$ is indeed an exact functor between extriangulated categories and preserves almost split exangles. Finally, we give some applications and examples on these results. For instance, we generalize a recent result by A. Hubery to a categorical framework.