arXiv subjects

Zhongkui Liu

Publications and source records attributed to Zhongkui Liu.

12 recordsLinked to original sources

arXiv2026

Compatibility of HRS tilt and completion of $t$-structures in triangulated categories

Let $K$ be a field, and $\mathcal{T}$ a $K$-linear essentially small triangulated category equipped with an extendable $t$-structure $(\mathcal{T}^{\leq0}, \mathcal{T}^{\geq0})$ with respect to a good metric $\mathfrak{B}$. Given a torsion class $\mathcal{X}$ in the heart of $(\mathcal{T}^{\leq0}, \mathcal{T}^{\geq0})$, we prove that lifting the HRS-tilt of $(\mathcal{T}^{\leq0}, \mathcal{T}^{\geq0})$ at $\mathcal{X}$ along the completion of $\mathcal{T}$ coincides with the HRS-tilt of the lifted $t$-structure at the completion of $\mathcal{X}$. As an application, we provide the compatibility result between left silting mutation in $\mathcal{T}$ and HRS tilting in the ambient completion.

math.CT↗

arXiv2026

Three results on extension dimensions of syzygy module categories

This paper establishes three main results on the extension dimension of syzygy module categories:(1)we prove that excellent ring extensions preserve extension dimensions of syzygy module categories,including syzygy categories of modules of finite projective dimensions;(2) for cleft extensions, we investigate the behavior of extension dimensions under natural nilpotency and projective conditions;(3)for commutative Artin rings, we establish local-global characterisations for the extension dimension.

math.CT↗

arXiv2026

The extension dimension of group graded rings

In this paper, we introduce the concept of graded extension dimension for a group graded ring R, denoted by gr.ext.dim(R). We prove that when R is strongly graded, its graded extension dimension coincides with the non-graded extension dimension of both R itself and its degree-zero subring Re. Furthermore, we demonstrate that graded equivalence and graded separable equivalence preserve the extension dimension under appropriate conditions.

math.CT↗

arXiv2025

Improved Bounds for the s-multiplicity

Let (RmR), (SmS) and (TmT) be Noetherian local rings sharing the same residue eld k and prime characteristic p > 0. We establish some formulas relating the h-function and s-multiplicity of the ber product R T S in terms of the h-functions and s-multiplicities of R, T and S. Furthermore, we derive formulas that connect the h-function and s-multiplicity of the idealization ring R M to the corresponding invariants of R and M, where M is a nitely generated R-module. As applications of these results, we derive new estimates for the Taylor-Miller question and the Watanabe-Yoshida conjecture concerning s-multiplicity.

math.AC↗

arXiv2018

On triangle equivalences of stable categories

We apply the Auslander-Buchweitz approximation theory to show that the Iyama and Yoshino's subfactor triangulated category can be realized as a triangulated quotient. Applications of this realization go in three directions. Firstly, we recover both a result of Iyama and Yang and a result of the third author. Secondly, we extend the classical Buchweitz's triangle equivalence from Iwanaga-Gorenstein rings to Noetherian rings. Finally, we obtain the converse of Buchweitz's triangle equivalence and a result of Beligiannis, and give characterizations for Iwanaga-Gorenstein rings and Gorenstein algebras

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

Cartan-Eilenberg complexes and Auslander categories

Let $R$ be a commutative noetherian ring with a semi-dualizing module $C$. The Auslander categories with respect to $C$ are related through Foxby equivalence: $\xymatrix@C=50pt{\mathcal {A}_C(R) \ar@<0.4ex>[r]^{C\otimes^{\mathbf{L}}_{R} -} & \mathcal {B}_C(R) \ar@<0.4ex>[l]^{\mathbf{R}\mathrm{Hom}_{R}(C, -)}}$. We firstly intend to extend the Foxby equivalence to Cartan-Eilenberg complexes. To this end, C-E Auslander categories, C-E $\mathcal{W}$ complexes and C-E $\mathcal{W}$-Gorenstein complexes are introduced, where $\mathcal{W}$ denotes a self-orthogonal class of $R$-modules. Moreover, criteria for finiteness of C-E Gorenstein dimensions of complexes in terms of resolution-free characterizations are considered.

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

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↗

arXiv2012

Stability of Gorenstein flat categories with respect to a semidualizing module

In this paper, we first introduce $\mathcal {W}_F$-Gorenstein modules to establish the following Foxby equivalence: $\xymatrix@C=80pt{\mathcal {G}(\mathcal {F})\cap \mathcal {A}_C(R) \ar@<0.5ex>[r]^{C\otimes_R-} & \mathcal {G}(\mathcal {W}_F) \ar@<0.5ex>[l]^{\textrm{Hom}_R(C,-)}} $ where $\mathcal {G}(\mathcal {F})$, $\mathcal {A}_C(R) $ and $\mathcal {G}(\mathcal {W}_F)$ denote the class of Gorenstein flat modules, the Auslander class and the class of $\mathcal {W}_F$-Gorenstein modules respectively. Then, we investigate two-degree $\mathcal {W}_F$-Gorenstein modules. An $R$-module $M$ is said to be two-degree $\mathcal {W}_F$-Gorenstein if there exists an exact sequence $\mathbb{G}_\bullet=\indent ...\longrightarrow G_1\longrightarrow G_0\longrightarrow G^0\longrightarrow G^1\longrightarrow...$ in $\mathcal {G}(\mathcal {W}_F)$ such that $M \cong$ $\im(G_0\rightarrow G^0) $ and that $\mathbb{G}_\bullet$ is Hom$_R(\mathcal {G}(\mathcal {W}_F),-)$ and $\mathcal {G}(\mathcal {W}_F)^+\otimes_R-$ exact. We show that two notions of the two-degree $\mathcal {W}_F$-Gorenstein and the $\mathcal {W}_F$-Gorenstein modules coincide when R is a commutative GF-closed ring.

math.RA↗

arXiv2007

Rota-Baxter operators on generalized power series rings

An important instance of Rota-Baxter algebras from their quantum field theory application is the ring of Laurent series with a suitable projection. We view the ring of Laurent series as a special case of generalized power series rings with exponents in an ordered monoid. We study when a generalized power series ring has a Rota-Baxter operator and how this is related to the ordered monoid.

math.RA↗