设$Λ$是拟$k$ -Gorenstein环.对于mod $Λ$中的每个$d$次合模$M$（其中$0 \leq d \leq k-1$），我们得到了mod $Λ$中的正合序列$0 \to B \to M \bigoplus P \to C \to 0$，它具有对偶正合，$P$是投射的，$C$是$(d+1)$次合模，$B$是Ext $_Λ^{d+1}(D(M), Λ)$的$d$次合模，且$B^*$的右投射维数小于或等于$d-1$ .然后我们给出这样一个正合序列的一些应用如下。(1)我们得到了一个关于$M$的满射链，通过对偶化它，我们得到了Auslander和Bridger关于$M^*$的球面滤子. (2)对于mod $Λ^{op}$中的每一个自反模，我们得到了Auslander和Bridger的逼近定理. (3)我们证明了对于任何$0 \leq d \leq k-1$，模$Λ$中的每个$d$次合模都有一个Evans-Griffith表示.作为（3）的直接结果，我们有，如果$Λ$是一个具有有限自内射维数的交换Noether环，则对任何非负整数$d$，mod $Λ$中的每个$d$次合合模都有Evans-Griffith表示，这将Evans和Griffith的结果推广到更一般的情形.
英文摘要：
Let $Λ$ be a quasi $k$-Gorenstein ring. For each $d$th syzygy module $M$ in mod $Λ$ (where $0 \leq d \leq k-1$), we obtain an exact sequence $0 \to B \to M \bigoplus P \to C \to 0$ in mod $Λ$ with the properties that it is dual exact, $P$ is projective, $C$ is a $(d+1)$st syzygy module, $B$ is a $d$th syzygy of Ext$_Λ^{d+1}(D(M), Λ)$ and the right projective dimension of $B^*$ is less than or equal to $d-1$. We then give some applications of such an exact sequence as follows. (1) We obtain a chain of epimorphisms concerning $M$, and by dualizing it we then get the spherical filtration of Auslander and Bridger for $M^*$. (2) We get Auslander and Bridger's Approximation Theorem for each reflexive module in mod $Λ^{op}$. (3) We show that for any $0 \leq d \leq k-1$ each $d$th syzygy module in mod $Λ$ has an Evans-Griffith representation. As an immediate consequence of (3), we have that, if $Λ$ is a commutative noetherian ring with finite self-injective dimension, then for any non-negative integer $d$, each $d$th syzygy module in mod $Λ$ has an Evans-Griffith representation, which generalizes an Evans and Griffith's result to much more general setting.
Gorenstein homological dimensions of modules over triangular matrix rings
设$A$和$B$是环，$U$是$(B, A)$ -双模，$T=\left(\begin{smallmatrix} A & 0 \\ U & B \\\end{smallmatrix}\right)$是三角形矩阵环.本文刻画了$T$上模的Gorenstein同调维数，并讨论了左$T$ -模何时是强Gorenstein投射模或强Gorenstein内射模.
