Syzygy modules for quasi k-Gorenstein rings

Let Λ be a right quasi k-Gorenstein ring. For each dth syzygy module M in modΛ (where 0 d k−1), we obtain an exact sequence 0→B→M P→C→0 in modΛ 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 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 Bridgerʹs Approximation Theorem for each reflexive module in modΛop. (3) We show that for any 0 d k−1 each dth syzygy module in modΛ has an Evans–Griffith presentation. 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 modΛ has an Evans–Griffith presentation, which generalizes an Evans and Griffithʹs result to much more general setting.