Syzygy modules for quasi $k$-Gorenstein rings

Abstract

Let $\Lambda$ be a quasi $k$-Gorenstein ring. For each $d$th syzygy module $M$ in mod $\Lambda$ (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 $\Lambda$ 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$_{\Lambda}^{d+1}(D(M), \Lambda)$ 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 $\Lambda ^{op}$. (3) We show that for any $0 \leq d \leq k-1$ each $d$th syzygy module in mod $\Lambda$ has an Evans-Griffith representation. As an immediate consequence of (3), we have that, if $\Lambda$ is a commutative noetherian ring with finite self-injective dimension, then for any non-negative integer $d$, each $d$th syzygy module in mod $\Lambda$ has an Evans-Griffith representation, which generalizes an Evans and Griffith's result to much more general setting.