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

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 $R$.Comment: 16 pages. The proof of Lemma 3.8 is modified. It has been accepted for publication in Osaka Journal of Mathematic

τ

Let A be a finite dimensional algebra over an algebraic closed field k. In this note, we will show that if T is a separating and splitting tilting A-module, then τ-complexities of A and B are equal, where B=EndA(T)

Simple modules over Auslander regular rings

In this paper, we discuss the properties of simple modules over Auslander regular rings with global dimension at most 3. Using grade theory, we show the right projective dimension of ExtΛ1(S,Λ)$\begin{array}{} \text{Ext}_{{\it\Lambda}}^{1}(S,\ {\it\Lambda}) \end{array}$ is equal to 1 for any simple Λ-module S with gr S = 1. As a result, we give some equivalent characterization of diagonal Auslander regular rings