ar X iv : m at h / 04 09 17 4 v 1 [ m at h . R A ] 1 0 Se p 20 04 Syzygy modules for quasi k - Gorenstein rings ∗ †

Let Λ be a quasi k-Gorenstein ring. For each dth syzygy module M in mod Λ (where 0 ≤ d ≤ k − 1), we obtain an exact sequence 0 → B → M ⊕ P → C → 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 dth syzygy of Ext Λ (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. (3) We show that for any 0 ≤ d ≤ k − 1 each dth 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 dth syzygy module in mod Λ has an Evans-Griffith representation, which generalizes an Evans and Griffith’s result to much more general setting.