Relative Auslander Bijection in Extriangulated Categories

Abstract

Extriangulated categories were introduced by Nakaoka and Palu, which unify exact categories and extension-closed subcategories of triangulated categories. In this paper, we develop the relative homology aspect of Auslander bijection in extriangulated categories. Namely, we introduce the notion of generalized ARS-duality relative to an additive subfunctor, and prove that there is a bijective triangle which involves the generalized ARS-duality and the restricted Auslander bijection relative to the subfunctor. We give the Auslander’s defect formula in terms of the generalized ARS-duality, show the interplay of morphisms being determined by objects and a kind of extriangles, and characterize a class of objects which are somehow controlled by this kind of extriangles. We also give a realization for a functor relating the Auslander bijection and the generalized ARS-duality.