Relative quasi-Gorensteinness in extriangulated categories
Zhenggang He
TL;DR
This work advances relative homological algebra in extriangulated categories by introducing quasi-$\xi$-projective and quasi-$\xi$-Gorenstein projective notions within a framework $(\mathcal{C},\mathbb{E},\mathfrak{s})$ equipped with a proper class $\xi$ of $\mathbb{E}$-triangles. It develops foundational properties, including closure and resolving behavior, and defines corresponding dimension theories via $\xi$-cohomology functors $\xi\text{xt}$ and $\xi$-exact resolutions. The paper also provides equivalent characterizations for objects of finite quasi-$\xi$-Gorenstein projective dimension and extends Mashhad and Mohammadi's module-category results to the extriangulated setting. Overall, it lays out a robust relative-homological-algebra toolkit for studying quasi-Gorenstein phenomena beyond exact or triangulated categories, with potential applications to broad categorical contexts.
Abstract
Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category with a proper class $ξ$ of $\mathbb{E}$-triangles. In this paper, we study the quasi-Gorensteinness of extriangulated categories. More precisely, we introduce the notion of quasi-$ξ$-projective and quasi-$ξ$-Gorenstein projective objects, investigate some of their properties and their behavior with respect to $\mathbb{E}$-triangles. Moreover, we give some equivalent characterizations of objects with finite quasi-$ξ$-Gorenstein projective dimension. As an application, our main results generalize Mashhad and Mohammadi's work in module categories.