arXiv subjects

Zhenggang He

Publications and source records attributed to Zhenggang He.

4 recordsLinked to original sources

arXiv2025

Quasi-resolving subcategories and dimensions in extriangulated categories

Let $\mathcal{C}=(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category with a proper class $ξ$ of $\mathbb{E}$-triangles. In this paper, we introduce and study quasi-resolving subcategories in $\mathcal{C}$. More precisely, we first introduce the notion of $\mathcal{X}$-resolution dimensions for a quasi-resolving subcategory $\mathcal{X}$ of $\mathcal{C}$ and then give some equivalent characterizations of objects which have finite $\mathcal{X}$-resolution dimensions. As an application, we introduce Gorenstein quasi-resolving subcategories, denoted by $\mathcal{GQP}_{\mathcal{X}}(ξ)$, in term of a quasi-resolving subcategory $\mathcal{X}$, and prove that $\mathcal{GQP}_{\mathcal{X}}(ξ)$ is also a quasi-resolving subcategory of $\mathcal{C}$. Moreover, some classical known results are generalized in $\mathcal{GQP}_{\mathcal{X}}(ξ)$.

math.CT↗

arXiv2025

Relative quasi-Gorensteinness in extriangulated categories

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.

math.RT↗

arXiv2021

Gorenstein Derived Functors for Extriangulated Categories

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 Gorenstein derived functors for extriangulated categories. More precisely, we first introduce the notion of the proper $ξ$-Gorenstein projective resolution for any object in $\mathcal{C}$ and define the functors $ξ\text{xt}_{\mathcal{GP}(ξ)}$ and $ξ\text{xt}_{\mathcal{GI}(ξ)}$. Under some assumptions, we give some equivalent characterizations for any object with finite $ξ$-Gorenstein projective dimension. Next we get some nice results by using derived functors. As an application, our main results generalize their work by Ren-Liu. Moreover, our proof is not far from the usual module categories or triangulated categories.

math.CT↗

arXiv2021

Gorenstein Objects in Extriangulated Categories

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

math.CT↗