arXiv subjects

Zhenggang He

Publications and source records attributed to Zhenggang He.

5 recordsLinked to original sources

arXiv2026

Relative quasi-Gorenstein homological dimensions in extriangulated categories

Let $(\mathcal{C},\mathbb{E},\mathfrak s)$ be an extriangulated category equipped with a proper class $ξ$ of $\mathbb E$-triangles.We establish stability under iteration for the quasi-$ξ$-Gorenstein projective and injective subcategories and extend the relative Ext-vanishing criteria for their finite homological dimensions to larger coefficient subcategories. An example in a bounded homotopy category shows that $\mathcal{P}(ξ)\subsetneq\mathcal{QGP}(ξ)\subsetneq\mathcal{GP}(ξ)$. For module categories, every quasi-Gorenstein projective module over a left perfect ring or a commutative Noetherian ring of finite Krull dimension is projective. When a commutative ring admits a finite chain of trace and nilpotent quotients ending in an Artinian ring and the endomorphism rings at the trace steps have finite right global dimension, we construct finite families of quasi-projective test modules that detect prescribed upper bounds on projective dimension for modules of bounded cardinality through Ext vanishing in finitely many consecutive degrees.

math.CT↗

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↗