Relative quasi-Gorensteinness in extriangulated categories

TL;DR

This paper explores the concept of quasi-Gorensteinness in extriangulated categories, introducing new notions of projective objects and characterizing their properties and dimensions, generalizing existing module category results.

Contribution

It introduces quasi-$\xi$-projective and quasi-$\xi$-Gorenstein projective objects in extriangulated categories, expanding the understanding of their properties and dimensions.

Findings

Defined quasi-$\xi$-projective and quasi-$\xi$-Gorenstein projective objects.

Provided characterizations of objects with finite quasi-$\xi$-Gorenstein projective dimension.

Generalized results from module categories to extriangulated categories.

Abstract

Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category with a proper class $\xi$ of $\mathbb{E}$-triangles. In this paper, we study the quasi-Gorensteinness of extriangulated categories. More precisely, we introduce the notion of quasi-$\xi$-projective and quasi-$\xi$-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-$\xi$-Gorenstein projective dimension. As an application, our main results generalize Mashhad and Mohammadi's work in module categories.