arXiv subjects

Zhanping Wang

Publications and source records attributed to Zhanping Wang.

7 recordsLinked to original sources

arXiv2018

Gorenstein flat modules with respect to duality pairs

Let $\mathcal{X}$ be a class of left $R$-modules, $\mathcal{Y}$ be a class of right $R$-modules. In this paper, we introduce and study Gorenstein $(\mathcal{X}, \mathcal{Y})$-flat modules as a common generalization of some known modules such as Gorenstein flat modules \cite{EJT93}, Gorenstein $n$-flat modules \cite{SUU14}, Gorenstein $\mathcal{B}$-flat modules \cite{EIP17}, Gorenstein AC-flat modules \cite{BEI17}, $Ω$-Gorenstein flat modules \cite{EJ00} and so on. We show that the class of all Gorenstein $(\mathcal{X}, \mathcal{Y})$-flat modules have a strong stability. In particular, when $(\mathcal{X}, \mathcal{Y})$ is a perfect (symmetric) duality pair, we give some functorial descriptions of Gorenstein $(\mathcal{X}, \mathcal{Y})$-flat dimension, and construct a hereditary abelian model structure on $R$-Mod whose cofibrant objects are exactly the Gorenstein $(\mathcal{X}, \mathcal{Y})$-flat modules. These results unify the corresponding results of the aforementioned modules.

math.RT↗

arXiv2014

Gorenstein homological dimensions of modules over triangular matrix rings

Let $A$ and $B$ be rings, $U$ a $(B, A)$-bimodule and $T=\left(\begin{smallmatrix} A & 0 \\ U & B \\\end{smallmatrix}\right)$ be the triangular matrix ring. In this paper, we characterize the Gorenstein homological dimensions of modules over $T$, and discuss when a left $T$-module is strongly Gorenstein projective or strongly Gorenstein injective module.

math.RA↗

arXiv2014

Stability of Gorenstein objects in triangulated categories

Let $\mathcal{C}$ be a triangulated category with a proper class $ξ$ of triangles. Asadollahi and Salarian introduced and studied $ξ$-Gorenstein projective and $ξ$-Gorenstein injective objects, and developed Gorenstein homological algebra in $\mathcal{C}$. In this paper, we further study Gorenstein homological properties for a triangulated category. First, we discuss the stability of $ξ$-Gorenstein projective objects, and show that the subcategory $\mathcal{GP}(ξ)$ of all $ξ$-Gorenstein projective objects has a strong stability. That is, an iteration of the procedure used to define the $ξ$-Gorenstein projective objects yields exactly the $ξ$-Gorenstein projective objects. Second, we give some equivalent characterizations for $ξ$-Gorenstein projective dimension of object in $\mathcal{C}$.

math.CT↗

arXiv2014

Stability of strongly Gorenstein flat modules

A left $R$-module $M$ is called two-degree Ding projective if there exists an exact sequence $...\longrightarrow D_{1}\longrightarrow D_{0}\longrightarrow D_{-1}\longrightarrow D_{-2}\longrightarrow...$ of Ding projective left $R$-modules such that $M\cong\ker (D_{0}\longrightarrow D_{-1})$ and $\Hom_{R} (-, F)$ leaves the sequence exact for any flat (or Gorenstein flat) left $R$-module $F$. In this paper, we show that the two-degree Ding projective modules are nothing more than the Ding projective modules.

math.KT↗

arXiv2013

Injective Objects of Monomorphism Categories

For an acyclic quiver $Q$ and a finite-dimensional algebra $A$, we give a unified form of the indecomposable injective objects in the monomorphism category ${\rm Mon}(Q,A)$ and prove that ${\rm Mon}(Q, A)$ has enough injective objects. As applications, we show that for a given self-injective algebra $A$, a tilting object in the stable category $\underline{A}$-mod induces a natural tilting object in the stable monomorphism category $\underline{\rm Mon}(Q,A)$. We also realize the singularity category of the algebra $kQ\otimes_k A$ as the stable monomorphism category of the module category of $A$.

math.RT↗

arXiv2013

Special precovers and preenvelopes of complexes

The notion of an $\mathcal{L}$ complex (for a given class of $R$-modules $\mathcal{L}$) was introduced by Gillespie: a complex $C$ is called $\mathcal{L}$ complex if $C$ is exact and $\Z_{i}(C)$ is in $\mathcal{L}$ for all $i\in \mathbb{Z}$. Let $\widetilde{\mathcal{L}}$ stand for the class of all $\mathcal{L}$ complexes. In this paper, we give sufficient condition on a class of $R$-modules such that every complex has a special $\widetilde{\mathcal{L}}$-precover (resp., $\widetilde{\mathcal{L}}$-preenvelope). As applications, we obtain that every complex has a special projective precover and a special injective preenvelope, over a coherent ring every complex has a special FP-injective preenvelope, and over a noetherian ring every complex has a special $\widetilde{\mathcal{GI}}$-preenvelope, where $\mathcal{GI}$ denotes the class of Gorenstein injective modules.

math.KT↗

arXiv2013

Ding projective dimension of complexes

In this paper, we define and study a notion of Ding projective dimension for complexes of left modules over associative rings. In particular, we consider the class of homologically bounded below complexes of left R-modules, and show that Ding projective dimension has a nice functorial description.

math.AC↗