arXiv subjects

Zhaoyong Huang

Publications and source records attributed to Zhaoyong Huang.

At least 19 recordsLinked to original sources

arXiv2026

Generalized Gorenstein Categories

Let $\mathscr{A}$ be an abelian category and let $\mathscr{C}$ and $\mathscr{D}$ be additive subcategories of $\mathscr{A}$. As a generalization of Gorenstein categories, we introduce one-sided $n$-$(\C,\D)$-Gorenstein categories with $n\geq 0$. Under certain conditions, we give some equivalent characterizations of one-sided $n$-$(\C,\D)$-Gorenstein categories in term of the finiteness of projective and injective dimensions relative to one-sided Gorenstein subcategories, which induce some new equivalent characterizations of Gorenstein categories. Then we apply these results to categories of interest. In particular, a necessary condition is obtained for the validity of the Wakamatsu tilting conjecture.

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arXiv2025

Normed modules and the categorification of integrations, series expansions, and differentiations

We explore the assignment of norms to $\mathitΛ$-modules over a finite-dimensional algebra $\mathitΛ$, resulting in the establishment of normed $\mathitΛ$-modules. Our primary contribution lies in constructing two new categories $\mathscr{N}\!\!or^p$ and $\mathscr{A}^p$, where each object in $\mathscr{N}\!\!or^p$ is a normed $\mathitΛ$-module $N$ limited by a special element $v_N\in N$ and a special $\mathitΛ$-homomorphism $δ_N: N^{\oplus 2^{\dim\mathitΛ}} \to N$, the morphism in $\mathscr{N}\!\!or^p$ is a $\mathitΛ$-homomorphism $θ: N\to M$ such that $θ(v_N) = v_M$ and $θδ_N = δ_Mθ^{\oplus 2^{\dim\mathitΛ}}$, and $\mathscr{A}^p$ is a full subcategory of $\mathscr{N}\!\!or^p$ generated by all Banach modules. By examining the objects and morphisms in these categories. We establish a framework for understanding the categorification of integration, series expansions, and derivatives. Furthermore, we obtain the Stone--Weierstrass approximation theorem in the sense of $\mathscr{A}^p$.

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arXiv2025

Homological dimensions over almost gentle algebras

We provide a method for computing the global dimension and self-injective dimension of almost gentle algebras,and prove that an almost gentle algebra is Gorenstein if it satisfies the Auslander condition.

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arXiv2024

Auslander-type conditions and weakly Gorenstein algebras

Let $R$ be an Artin algebra. Under certain Auslander-type conditions, we give some equivalent characterizations of (weakly) Gorenstein algebras in terms of the properties of Gorenstein projective modules and modules satisfying Auslander-type conditions. As applications, we provide some support for several homological conjectures. In particular, we prove that if $R$ is left quasi Auslander, then $R$ is Gorenstein if and only if it is (left and) right weakly Gorenstein; and that if $R$ satisfies the Auslander condition, then $R$ is Gorenstein if and only if it is left or right weakly Gorenstein. This is a reduction of an Auslander--Reiten's conjecture, which states that $R$ is Gorenstein if $R$ satisfies the Auslander condition.

math.RA↗

arXiv2023

Homological Dimensions of Gentle Algebras via Geometric Models

Let $A=kQ/I$ be a finite dimensional basic algebra over an algebraically closed field $k$ which is a gentle algebra with the marked ribbon surface $(\mathcal{S}_A,\mathcal{M}_A,Γ_A)$. It is known that $\mathcal{S}_A$ can be divided into some elementary polygons $\{Δ_i\mid 1\le i\le d\}$ by $Γ_A$ which has exactly one side in the boundary of $\mathcal{S}_A$. Let $\mathfrak{C}(Δ_i)$ be the number of sides of $Δ_i$ belonging to $Γ_A$ if the unmarked boundary component of $\mathcal{S}_A$ is not a side of $Δ_i$; otherwise, $\mathfrak{C}(Δ_i)=\infty$, and let $\mathsf{f}\text{-}Δ$ be the set of all non-$\infty$-elementary polygons and $\mathcal{F}_A$ (respectively, ${\mathsf{f}\text{-}\mathcal{F}}_A$) the set of all forbidden threads (respectively, of finite length). Then we have \begin{enumerate} \item[{\rm (1)}] The global dimension of $A=\max\limits_{1\leq i\leq d}{\mathfrak{C}(Δ_i)}-1 =\max\limits_{\mathitΠ\in\mathcal{F}_A} l(\mathitΠ)$, where $l(\mathitΠ)$ is the length of $\mathitΠ$. \item[{\rm (2)}] The left and right self-injective dimensions of $A=$ \begin{center} $\begin{cases} 0,\ \mbox{\text{if {\it Q} is either a point or an oriented cycle with full relations};}\\ \max\limits_{Δ_i\in{\mathsf{f}\text{-}Δ}}\big\{1, {\mathfrak{C}(Δ_i)}-1 \big\}= \max\limits_{\mathitΠ\in{\mathsf{f}\text{-}\mathcal{F}}_A} l(\mathitΠ),\ \mbox{\text{otherwise}.} \end{cases}$ \end{center} \end{enumerate} As a consequence, we get that the finiteness of the global dimension of gentle algebras is invariant under AG-equivalence. In addition, we get that the number of indecomposable non-projective Gorenstein projective modules over gentle algebras is also invariant under AG-equivalence.

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arXiv2022

The derived and extension dimensions of abelian categories

For an abelian category $\mathcal{A}$, we establish the relation between its derived and extension dimensions. Then for an artin algebra $Λ$, we give the upper bounds of the extension dimension of $Λ$ in terms of the radical layer length of $Λ$ and certain relative projective (or injective) dimension of some simple $Λ$-modules, from which some new upper bounds of the derived dimension of $Λ$ are induced.

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arXiv2021

Homological Dimensions Relative to Preresolving Subcategories II

Let $\mathscr{A}$ be an abelian category having enough projective and injective objects, and let $\mathscr{T}$ be an additive subcategory of $\mathscr{A}$ closed under direct summands. A known assertion is that in a short exact sequence in $\mathscr{A}$, the $\mathscr{T}$-projective (respectively, $\mathscr{T}$-injective) dimensions of any two terms can sometimes induce an upper bound of that of the third term by using the same comparison expressions. We show that if $\mathscr{T}$ contains all projective (respectively, injective) objects of $\mathscr{A}$, then the above assertion holds true if and only if $\mathscr{T}$ is resolving (respectively, coresolving). As applications, we get that a left and right Noetherian ring $R$ is $n$-Gorenstein if and only if the Gorenstein projective (respectively, injective, flat) dimension of any left $R$-module is at most $n$. In addition, in several cases, for a subcategory $\mathscr{C}$ of $\mathscr{T}$, we show that the finitistic $\mathscr{C}$-projective and $\mathscr{T}$-projective dimensions of $\mathscr{A}$ are identical.

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arXiv2020

Duality Pairs Induced by One-Sided Gorenstein Subcategories

For a ring $R$ and an additive subcategory $\C$ of the category $\Mod R$ of left $R$-modules, under some conditions we prove that the right Gorenstein subcategory of $\Mod R$ and the left Gorenstein subcategory of $\Mod R^{op}$ relative to $\C$ form a coproduct-closed duality pair. Let $R,S$ be rings and $C$ a semidualizing ($R,S$)-bimodule. As applications of the above result, we get that if $S$ is right coherent and $C$ is faithfully semidualizing, then $(\mathcal{GF}_C(R),\mathcal{GI}_C(R^{op}))$ is a coproduct-closed duality pair and $\mathcal{GF}_C(R)$ is covering in $\Mod R$, where $\mathcal{G}\mathcal{F}_C(R)$ is the subcategory of $\Mod R$ consisting of $C$-Gorenstein flat modules and $\mathcal{G}\mathcal{I}_C(R^{op})$ is the subcategory of $\Mod R^{op}$ consisting of $C$-Gorenstein injective modules; we also get that if $S$ is right coherent, then $(\mathcal{A}_C(R^{op}),l\mathcal{G}(\mathcal{F}_C(R)))$ is a coproduct-closed and product-closed duality pair and $\mathcal{A}_C(R^{op})$ is covering and preenveloping in $\Mod R^{op}$, where $\mathcal{A}_C(R^{op})$ is the Auslander class in $\Mod R^{op}$ and $l\mathcal{G}(\mathcal{F}_C(R))$ is the left Gorenstein subcategory of $\Mod R$ relative to $C$-flat modules.

math.CT↗

arXiv2020

One-Sided Gorenstein Subcategories

We introduce the right (left) Gorenstein subcategory relative to an additive subcategory $\C$ of an abelian category $\A$, and prove that the right Gorenstein subcategory $r\mathcal{G}(\mathscr{C})$ is closed under extensions, kernels of epimorphisms, direct summands and finite direct sums. When $\C$ is self-orthogonal, we give a characterization for objects in $r\mathcal{G}(\mathscr{C})$, and prove that any object in $\A$ with finite $r\mathcal{G}(\C)$-projective dimension is isomorphic to a kernel (resp. a cokernel) of a morphism from an object in $\A$ with finite $\C$-projective dimension to an object in $r\mathcal{G}(\C)$. As an application, we obtain a weak Auslander-Buchweitz context related to the kernel of a hereditary cotorsion pair in $\A$ having enough injectives.

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arXiv2020

Almost Split Triangles and Morphisms Determined by Objects in Extriangulated Categories

Let $(\mathfrak{C},\mathbb{E},\mathfrak{s})$ be an Ext-finite, Krull-Schmidt and $k$-linear extriangulated category with $k$ a commutative artinian ring. We define an additive subcategory $\mathfrak{C}_r$ (respectively, $\mathfrak{C}_l$) of $\mathfrak{C}$ in terms of the representable functors from the stable category of $\mathfrak{C}$ modulo $\mathfrak{s}$-injectives (respectively, $\mathfrak{s}$-projectives) to $k$-modules, which consists of all $\mathfrak{s}$-projective (respectively, $\mathfrak{s}$-injective) objects and objects isomorphic to direct summands of finite direct sums of all third (respectively, first) terms of almost split $\mathfrak{s}$-triangles. We investigate the subcategories $\mathfrak{C}_r$ and $\mathfrak{C}_l$ in terms of morphisms determined by objects, and then give equivalent characterizations on the existence of almost split $\mathfrak{s}$-triangles.

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arXiv2020

Silting Modules over Triangular Matrix Rings

Let $Λ,Γ$ be rings and $R=\left(\begin{array}{cc}Λ& 0 \\ M & Γ\end{array}\right)$ the triangular matrix ring with $M$ a $(Γ,Λ)$-bimodule. Let $X$ be a right $Λ$-module and $Y$ a right $Γ$-module. We prove that $(X, 0)$$\oplus$$(Y\otimes_ΓM, Y)$ is a silting right $R$-module if and only if both $X_Λ$ and $Y_Γ$ are silting modules and $Y\otimes_ΓM$ is generated by $X$. Furthermore, we prove that if $Λ$ and $Γ$ are finite dimensional algebras over an algebraically closed field and $X_Λ$ and $Y_Γ$ are finitely generated, then $(X, 0)$$\oplus$$(Y\otimes_ΓM, Y)$ is a support $τ$-tilting $R$-module if and only if both $X_Λ$ and $Y_Γ$ are support $τ$-tilting modules, $\Hom_Λ(Y\otimes_ΓM,τX)=0$ and $\Hom_Λ(eΛ, Y\otimes_ΓM)=0$ with $e$ the maximal idempotent such that $\Hom_Λ(eΛ, X)=0$.

math.RT↗

arXiv2020

An Upper Bound for the Dimension of Bounded Derived Categories

Let $Λ$ be an artin algebra. We give an upper bound for the dimension of the bounded derived category of the category $\mod Λ$ of finitely generated right $Λ$-modules in terms of the projective and injective dimensions of certain class of simple right $Λ$-modules as well as the radical layer length of $Λ$. In addition, we give an upper bound for the dimension of the singularity category of $\mod Λ$ in terms of the radical layer length of $Λ$.

math.RA↗

arXiv2020

Support $τ$-Tilting Modules under Split-by-Nilpotent Extensions

Let $Γ$ be a split extension of a finite-dimensional algebra $Λ$ by a nilpotent bimodule $_ΛE_Λ$, and let $(T,P)$ be a pair in $\modΛ$ with $P$ projective. We prove that $(T\otimes_ΛΓ_Γ, P\otimes_ΛΓ_Γ)$ is a support $τ$-tilting pair in $\mod Γ$ if and only if $(T,P)$ is a support $τ$-tilting pair in $\mod Λ$ and $\Hom_Λ(T\otimes_ΛE,τT_Λ)=0=\Hom_Λ(P,T\otimes_ΛE)$. As applications, we obtain a necessary and sufficient condition such that $(T\otimes_ΛΓ_Γ, P\otimes_ΛΓ_Γ)$ is support $τ$-tilting pair for a cluster-tilted algebra $Γ$ corresponding to a tilted algebra $Λ$; and we also get that if $T_1,T_2\in\modΛ$ such that $T_1\otimes_ΛΓ$ and $T_2\otimes_ΛΓ$ are support $τ$-tilting $Γ$-modules, then $T_1\otimes_ΛΓ$ is a left mutation of $T_2\otimes_ΛΓ$ if and only if $T_1$ is a left mutation of $T_2$.

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arXiv2019

Higher differential objects in additive categories

Given an additive category $\mathcal{C}$ and an integer $n\geqslant 2$. We form a new additive category $\mathcal{C}[ε]^n$ consisting of objects $X$ in $\mathcal{C}$ equipped with an endomorphism $ε_X$ satisfying ${ε^n_X}=0$. First, using the descriptions of projective and injective objects in $\mathcal{C}[ε]^n$, we not only establish a connection between Gorenstein flat modules over a ring $R$ and $R[t]/(t^n)$, but also prove that an Artinian algebra $R$ satisfies some homological conjectures if and only if so does $R[t]/(t^n)$. Then we show that the corresponding homotopy category $\K(\mathcal{C}[ε]^n)$ is a triangulated category when $\mathcal{C}$ is an idempotent complete exact category. Moreover, under some conditions for an abelian category $\mathcal{A}$, the natural quotient functor $Q$ from $\K(\mathcal{A}[ε]^n)$ to the derived category $\D(\mathcal{A}[ε]^n)$ produces a recollement of triangulated categories. Finally, we prove that if $\mathcal{A}$ is an Ab4-category with a compact projective generator, then $\D(\mathcal{A}[ε]^n)$ is a compactly generated triangulated category.

math.RT↗

arXiv2019

Gorenstein Projective Objects in Comma Categories

Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories and $\mathbf{F}:\mathcal{A}\to \mathcal{B}$ an additive and right exact functor which is perfect, and let $(\mathbf{F},\mathcal{B})$ be the left comma category. We give an equivalent characterization of Gorenstein projective objects in $(\mathbf{F},\mathcal{B})$ in terms of Gorenstein projective objects in $\mathcal{B}$ and $\mathcal{A}$. We prove that there exists a left recollement of the stable category of the subcategory of $(\mathbf{F},\mathcal{B})$ consisting of Gorenstein projective objects modulo projectives relative to the same kind of stable categories in $\mathcal{B}$ and $\mathcal{A}$. Moreover, this left recollement can be filled into a recollement when $\mathcal{B}$ is Gorenstein and $\mathbf{F}$ preserves projectives.

math.RA↗

arXiv2019

Cograde conditions and cotorsion pairs

Let $R$ and $S$ be rings and $_Rω_S$ a semidualizing bimodule. We study when the double functor $\Tor^S_i(ω, \Ext^i_{R}(ω,-))$ preserves epimorphisms and the double functor $\Ext_{R}^i(ω, \Tor_i^{S}(ω,-))$ preserves monomorphisms in terms of the (strong) cograde conditions of modules. Under certain cograde condition of modules, we construct two complete cotorsion pairs. In addition, we establish the relation between some relative finitistic dimensions of rings and the right and left projective dimensions of $ω$.

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arXiv2019

The Extension Dimension of Abelian Categories

Let $\A$ be an abelian category having enough projective objects and enough injective objects. We prove that if $\A$ admits an additive generating object, then the extension dimension and the weak resolution dimension of $\A$ are identical, and they are at most the representation dimension of $\A$ minus two. By using it, for a right Morita ring $\La$, we establish the relation between the extension dimension of the category $\mod \La$ of finitely generated right $Λ$-modules and the representation dimension as well as the right global dimension of $Λ$. In particular, we give an upper bound for the extension dimension of $\mod Λ$ in terms of the projective dimension of certain class of simple right $Λ$-modules and the radical layer length of $Λ$. In addition, we investigate the behavior of the extension dimension under some ring extensions and recollements.

math.RT↗

arXiv2018

Duality Pairs Induced by Auslander and Bass Classes

Let $R$ and $S$ be any rings and $_RC_S$ a semidualizing bimodule, and let $\mathcal{A}_C(R^{op})$ and $\mathcal{B}_C(R)$ be the Auslander and Bass classes respectively. Then both the pairs $$(\mathcal{A}_C(R^{op}),\mathcal{B}_C(R))\ {\rm and}\ (\mathcal{B}_C(R),\mathcal{A}_C(R^{op}))$$ are coproduct-closed and product-closed duality pairs and both $\mathcal{A}_C(R^{op})$ and $\mathcal{B}_C(R)$ are covering and preenveloping; in particular, the former duality pair is perfect. Moreover, if $\mathcal{B}_C(R)$ is enveloping in $\Mod R$, then $\mathcal{A}_C(S)$ is enveloping in $\Mod S$. Then some applications to the Auslander projective dimension of modules are given.

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