Gorenstein Objects in Extriangulated Categories

TL;DR

This paper explores Gorenstein objects within extriangulated categories, defining new types of projective resolutions and characterizing their properties to advance understanding of relative Gorenstein theory.

Contribution

It introduces the notion of $\xi$-$ ext{G}$projective resolutions and characterizes $\xi$-$n$-strongly $ ext{G}$projective objects, expanding the framework of Gorenstein objects in extriangulated categories.

Findings

Established equivalences between different types of $\xi$-projective resolutions.

Defined and characterized $\xi$-$n$-strongly $ ext{G}$projective objects.

Provided properties and relations of these objects within the category.

Abstract

This paper mainly studies the relative Gorenstein objects in the extriangulated category $\mathcal{C}=(\mathcal{C},\mathbb{E},\mathfrak{s})$ with a proper class $\xi$ and the related properties of these objects. In the first part, we define the notion of the $\xi$-$\mathcal{G}$projective resolution, and study the relation between $\xi$-projective resolution and $\xi$-$\mathcal{G}$projective resolution for any object $A$ in $\mathcal{C}$, i.e. $A$ has a $\mathcal{C}(-,\mathcal{P}(\xi))$-exact $\xi$-projective resolution if and only if $A$ has a $\mathcal{C}(-,\mathcal{P}(\xi))$-exact $\xi$-$\mathcal{G}$projective resolution. In the second part, we define a particular $\xi$-Gorenstein projective object in $\mathcal{C}$ which called $\xi$-$n$-strongly $\mathcal{G}$projective object. On this basis, we study the relation between $\xi$-$m$-strongly $\mathcal{G}$projective object and $\xi$-$n$-strongly $\mathcal{G}$projective object whenever $m\neq n$, and give some equivalent characterizations of $\xi$-$n$-strongly $\mathcal{G}$projective objects. What is more, we give some nice propsitions of $\xi$-$n$-strongly $\mathcal{G}$projective objects.