?(ξ)-Gorenstein objects in extriangulated categories

摘要

We fix a proper class of ?-triangles ξ in an extriangulated category ?. Let ? be a class of objects in ? such that ξxtξi(W,W′)=0, ξxtξ0(−,W)≅?(−,W) and ξxtξ0(W,−)≅?(W,−) for all W,W′∈? and all i≥1. In this paper, we study ?(ξ)-Gorenstein objects, and use the relative homological methods to show that the stability of the Gorenstein category ??(ξ) in ?. Moreover, we give some equivalent characterizations of ?(ξ)-Gorenstein objects and finite ??(ξ)-(co)resolution dimensions for any object in ?. As an application, we consider the equivalent characterizations of ξ-?projective objects, ξ-?projective dimension and their duals.