arXiv subjects

Zhaoyong Huang

Publications and source records attributed to Zhaoyong Huang.

At least 55 records · Page 3Linked to original sources

arXiv2011

Injective Envelopes and (Gorenstein) Flat Covers

We characterize left Noetherian rings in terms of the duality property of injective preenvelopes and flat precovers. For a left and right Noetherian ring $R$, we prove that the flat dimension of the injective envelope of any (Gorenstein) flat left $R$-module is at most the flat dimension of the injective envelope of $_RR$. Then we get that the injective envelope of $_RR$ is (Gorenstein) flat if and only if the injective envelope of every Gorenstein flat left $R$-module is (Gorenstein) flat, if and only if the injective envelope of every flat left $R$-module is (Gorenstein) flat, if and only if the (Gorenstein) flat cover of every injective left $R$-module is injective, and if and only if the opposite version of one of these conditions is satisfied.

math.RA↗

arXiv2011

Torsionfree Dimension of Modules and Self-Injective Dimension of Rings

Let $R$ be a left and right Noetherian ring. We introduce the notion of the torsionfree dimension of finitely generated $R$-modules. For any $n\geq 0$, we prove that $R$ is a Gorenstein ring with self-injective dimension at most $n$ if and only if every finitely generated left $R$-module and every finitely generated right $R$-module have torsionfree dimension at most $n$, if and only if every finitely generated left (or right) $R$-module has Gorenstein dimension at most $n$. For any $n \geq 1$, we study the properties of the finitely generated $R$-modules $M$ with $\Ext_R^i(M, R)=0$ for any $1\leq i \leq n$. Then we investigate the relation between these properties and the self-injective dimension of $R$.

math.RA↗

arXiv2010

Gorenstein Syzygy Modules

For any ring $R$ and any positive integer $n$, we prove that a left $R$-module is a Gorenstein $n$-syzygy if and only if it is an $n$-syzygy. Over a left and right Noetherian ring, we introduce the notion of the Gorenstein transpose of finitely generated modules. We prove that a module $M\in \mod R^{op}$ is a Gorenstein transpose of a module $A\in \mod R$ if and only if $M$ can be embedded into a transpose of $A$ with the cokernel Gorenstein projective. Some applications of this result are given.

math.RA↗

arXiv2010

Homological behavior of Auslander's $k$-Gorenstein rings

In this paper we mainly study the homological properties of dual modules over $k$-Gorenstein rings. For a right quasi $k$-Gorenstein ring $Λ$, we show that the right self-injective dimension of $Λ$ is at most $k$ if and only if each $M \in$mod $Λ$ satisfying the condition that Ext$_Λ^i(M, Λ)=0$ for any $1\leq i \leq k$ is reflexive. For an $\infty$-Gorenstein ring, we show that the big and small finitistic dimensions and the self-injective dimension of $Λ$ are identical. In addition, we show that if $Λ$ is a left quasi $\infty$-Gorenstein ring and $M\in$mod $Λ$ with grade$M$ finite, then Ext$_Λ^i($Ext$_{Λ^{op}}^i($Ext$_Λ^{{\rm grade}M}(M, Λ), Λ), Λ)=0$ if and only if $i\neq$grade$M$. For a 2-Gorenstein ring $Λ$, we show that a non-zero proper left ideal $I$ of $Λ$ is reflexive if and only if $Λ/I$ has no non-zero pseudo-null submodule.

math.RA↗

arXiv2010

Canonical Filtrations of Gorenstein Injective Modules

The principle "Every result in classical homological algebra should have a counterpart in Gorenstein homological algebra" is given in [3]. There is a remarkable body of evidence supporting this claim (cf. [2] and [3]). Perhaps one of the most glaring exceptions is provided by the fact that tensor products of Gorenstein projective modules need not be Gorenstein projective, even over Gorenstein rings. So perhaps it is surprising that tensor products of Gorenstein injective modules over Gorenstein rings of finite Krull dimension are Gorenstein injective. Our main result is in support of the principle. Over commutative, noetherian rings injective modules have direct sum decompositions into indecomposable modules. We will show that Gorenstein injective modules over Gorenstein rings of finite Krull dimension have filtrations analogous to those provided by these decompositions. This result will then provide us with the tools to prove that all tensor products of Gorenstein injective modules over these rings are Gorenstein injective.

math.RA↗

arXiv2010

$n$-Strongly Gorenstein Projective, Injective and Flat modules

In this paper, we study the relation between $m$-strongly Gorenstein projective (resp. injective) modules and $n$-strongly Gorenstein projective (resp. injective) modules whenever $m \neq n$, and the homological behavior of $n$-strongly Gorenstein projective (resp. injective) modules. We introduce the notion of $n$-strongly Gorenstein flat modules. Then we study the homological behavior of $n$-strongly Gorenstein flat modules, and the relation between these modules and $n$-strongly Gorenstein projective (resp. injective) modules.

math.RA↗

arXiv2009

Trivial Maximal 1-Orthogonal Subcategories For Auslander's 1-Gorenstein Algebras

Let $Λ$ be an Auslander's 1-Gorenstein Artinian algebra with global dimension two. If $Λ$ admits a trivial maximal 1-orthogonal subcategory of $\modΛ$, then for any indecomposable module $M \in \mod Λ$, we have that the projective dimension of $M$ is equal to one if and only if so is its injective dimension and that $M$ is injective if the projective dimension of $M$ is equal to two. In this case, we further get that $Λ$ is a tilted algebra.

math.RT↗

arXiv2009

The Auslander-Type Condition of Triangular Matrix Rings

Let $R$ be a left and right Noetherian ring and $n,k$ any non-negative integers. $R$ is said to satisfy the Auslander-type condition $G_n(k)$ if the right flat dimension of the $(i+1)$-st term in a minimal injective resolution of $R_R$ is at most $i+k$ for any $0 \leq i \leq n-1$. In this paper, we prove that $R$ is $G_n(k)$ if and only if so is a lower triangular matrix ring of any degree $t$ over $R$.

math.RA↗

arXiv2009

From Auslander Algebras to Tilted Algebras

This paper has been withdrawn by the authors because it has been combined with "Higher Auslander Algebras Admitting Trivial Maximal Orthogonal Subcategories" (arXiv:0903.0761) together. Please see the new version of the latter paper for the details.

math.RT↗

arXiv2009

Higher Auslander Algebras Admitting Trivial Maximal Orthogonal Subcategories

For an Artinian $(n-1)$-Auslander algebra $Λ$ with global dimension $n(\geq 2)$, we show that if $Λ$ admits a trivial maximal $(n-1)$-orthogonal subcategory of $\modΛ$, then $Λ$ is a Nakayama algebra and the projective or injective dimension of any indecomposable module in $\modΛ$ is at most $n-1$. As a result, for an Artinian Auslander algebra with global dimension 2, if $Λ$ admits a trivial maximal 1-orthogonal subcategory of $\modΛ$, then $Λ$ is a tilted algebra of finite representation type. Further, for a finite-dimensional algebra $Λ$ over an algebraically closed field $K$, we show that $Λ$ is a basic and connected $(n-1)$-Auslander algebra $Λ$ with global dimension $n(\geq 2)$ admitting a trivial maximal $(n-1)$-orthogonal subcategory of $\modΛ$ if and only if $Λ$ is given by the quiver: $$\xymatrix{1 & \ar[l]_{β_{1}} 2 & \ar[l]_{β_{2}} 3 & \ar[l]_{β_{3}} ... & \ar[l]_{β_{n}} n+1} $$ modulo the ideal generated by $\{β_{i}β_{i+1}| 1\leq i\leq n-1 \}$. As a consequence, we get that a finite-dimensional algebra over an algebraically closed field $K$ is an $(n-1)$-Auslander algebra with global dimension $n(\geq 2)$ admitting a trivial maximal $(n-1)$-orthogonal subcategory if and only if it is a finite direct product of $K$ and $Λ$ as above. Moreover, we give some necessary condition for an Artinian Auslander algebra admitting a non-trivial maximal 1-orthogonal subcategory.

math.RT↗

arXiv2009

The Existence of Maximal $n$-Orthogonal Subcategories

For an $(n-1)$-Auslander algebra $Λ$ with global dimension $n$, we give some necessary conditions for $Λ$ admitting a maximal $(n-1)$-orthogonal subcategory in terms of the properties of simple $Λ$-modules with projective dimension $n-1$ or $n$. For an almost hereditary algebra $Λ$ with global dimension 2, we prove that $Λ$ admits a maximal 1-orthogonal subcategory if and only if for any non-projective indecomposable $Λ$-module $M$, $M$ is injective is equivalent to that the reduced grade of $M$ is equal to 2. We give a connection between the Gorenstein Symmetric Conjecture and the existence of maximal $n$-orthogonal subcategories of $^{\bot}T$ for a cotilting module $T$. For a Gorenstein algebra, we prove that all non-projective direct summands of a maximal $n$-orthogonal module are $Ω^nτ$-periodic. In addition, we study the relation between the complexity of modules and the existence of maximal $n$-orthogonal subcategories for the tensor product of two finite-dimensional algebras.

math.RT↗

arXiv2007

When Are Torsionless Modules Projective?

In this paper, we study the problem when a finitely generated torsionless module is projective. Let $Λ$ be an Artinian local algebra with radical square zero. Then a finitely generated torsionless $Λ$-module $M$ is projective if ${\rm Ext^1_Λ}(M,M)=0$. For a commutative Artinian ring $Λ$, a finitely generated torsionless $Λ$-module $M$ is projective if the following conditions are satisfied: (1) ${\rm Ext}^i_Λ(M,Λ)=0$ for $i=1,2,3$; and (2) ${\rm Ext}^i_Λ(M,M)=0$ for $i=1,2$. As a consequence of this result, we have that for a commutative Artinian ring $Λ$, a finitely generated Gorenstein projective $Λ$-module is projective if and only if it is selforthogonal.

math.RA↗

arXiv2007

On the grade of modules over Noetherian rings

Let $Λ$ be a left and right noetherian ring and $\mod Λ$ the category of finitely generated left $Λ$-modules. In this paper we show the following results: (1) For a positive integer $k$, the condition that the subcategory of $\mod Λ$ consisting of $i$-torsionfree modules coincides with the subcategory of $\mod Λ$ consisting of $i$-syzygy modules for any $1\leq i \leq k$ is left-right symmetric. (2) If $Λ$ is an Auslander ring and $N$ is in $\mod Λ^{op}$ with $\grade N=k<\infty$, then $N$ is pure of grade $k$ if and only if $N$ can be embedded into a finite direct sum of copies of the $(k+1)$st term in a minimal injective resolution of $Λ$ as a right $Λ$-module. (3) Assume that both the left and right self-injective dimensions of $Λ$ are $k$. If $\grade {\rm Ext}_Λ^k(M, Λ)\geq k$ for any $M\in\mod Λ$ and $\grade {\rm Ext}_Λ^i(N, Λ)\geq i$ for any $N\in\mod Λ^{op}$ and $1\leq i \leq k-1$, then the socle of the last term in a minimal injective resolution of $Λ$ as a right $Λ$-module is non-zero.

math.RA↗

arXiv2007

Auslander-type conditions and cotorsion pairs

We study the properties of rings satisfying Auslander-type conditions. If an artin algebra $Λ$ satisfies the Auslander condition (that is, $Λ$ is an $\infty$-Gorenstein artin algebra), then we construct two kinds of subcategories which form functorially finite cotorsion pairs.

math.RA↗

arXiv2007

Approximation presentations of modules and homological conjectures

In this paper we give a sufficient condition of the existence of ${\rm \mathbb{W}}^{t}$-approximation presentations. We also introduce property (W$^{k}$). As an application of the existence of ${\rm \mathbb{W}}^{t}$-approximation presentations we give a connection between the finitistic dimension conjecture, the Auslander-Reiten conjecture and property (W$^{k}$).

math.RA↗

arXiv2006

Generalized tilting modules with finite injective dimension

Let $R$ be a left noetherian ring, $S$ a right noetherian ring and $_RU$ a generalized tilting module with $S={\rm End}(_RU)$. The injective dimensions of $_RU$ and $U_S$ are identical provided both of them are finite. Under the assumption that the injective dimensions of $_RU$ and $U_S$ are finite, we describe when the subcategory $\{{\rm Ext}_S^n(N, U)|N$ is a finitely generated right $S$-module$\}$ is closed under submodules. As a consequence, we obtain a negative answer to a question posed by Auslander in 1969. Finally, some partial answers to Wakamatsu Tilting Conjecture are given.

math.RA↗

arXiv2006

Wakamatsu Tilting Modules, $U$-Dominant Dimension and $k$-Gorenstein Modules

Let $Λ$ and $Γ$ be left and right noetherian rings and $_ΛU$ a Wakamatsu tilting module with $Γ={\rm End}(_ΛT)$. We introduce a new definition of $U$-dominant dimensions and show that the $U$-dominant dimensions of $_ΛU$ and $U_Γ$ are identical. We characterize $k$-Gorenstein modules in terms of homological dimensions and the property of double homological functors preserving monomorphisms. We also study a generalization of $k$-Gorenstein modules, and characterize it in terms of some similar properties of $k$-Gorenstein modules.

math.RA↗

arXiv2005

$k$-Gorenstein Modules

Let $Λ$ and $Γ$ be artin algebras and $_ΛU_Γ$ a faithfully balanced selforthogonal bimodule. In this paper, we first introduce the notion of $k$-Gorenstein modules with respect to $_ΛU_Γ$ and then characterize it in terms of the $U$-resolution dimension of some special injective modules and the property of the functors ${\rm Ext}^{i}({\rm Ext}^{i}(-, U), U)$ preserving monomorphisms, which develops a classical result of Auslander. As an application, we study the properties of dual modules relative to Gorenstein bimodules. In addition, we give some properties of $_ΛU_Γ$ with finite left or right injective dimension.

math.RA↗