arXiv subjects

Chonghui Huang

Publications and source records attributed to Chonghui Huang.

5 recordsLinked to original sources

arXiv2016

On the representation dimension of smash products

Let $A$ be a finite dimensional $G$-graded algebra with $G$ a finite group, and $A\# k[G]^{\ast}$ be the smash product of $A$ with the group $G$. Our results can be stated as follows: (1) If $A$ is a self-injective algebra and separably graded, then the dimensions of triangulated categories $\underline{\rm mod}A$ and $\underline{\rm mod}A\# k[G]^{\ast}$ are equal. In particular, we obtain that the representation dimension of $A\# k[G]^{\ast}$ is at least the dimension of triangulated category $\underline{\rm mod}A$ plus 2; (2) Generally, if $A$ is a $k$-algebra and separably graded, then the Oppermann dimensions of $A$ and $A\# k[G]^{\ast}$ are equal. In particular, we obtain that the representation dimension of $A\# k[G]^{\ast}$ is at least the Oppermann dimension of $A$ plus 2. In the end, we give two examples to illustrate our results.

math.RT↗

arXiv2013

Syzygy Modules and Injective Cogenerators for Noether Rings

In this paper, we focus on $n$-syzygy modules and the injective cogenerator determined by the minimal injective resolution of a noether ring. We study the properties of $n$-syzygy modules and a category $R_n(\mod R)$ which includes the category consisting of all $n$-syzygy modules and their applications on Auslander-type rings. Then, we investigate the injective cogenerators determined by the minimal injective resolution of $R$. We show that $R$ is Gorenstein with finite self-injective dimension at most $n$ if and only if $\id R\leq n$ and $\fd \bigoplus_{i=0}^n I_i(R)< \infty$. Some known results can be our corollaries.

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↗

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↗