Type: Article
Publication Date: 2009-05-18
Citations: 24
DOI: https://doi.org/10.1216/jca-2009-1-2-283
The concept of Gorenstein dimension, defined by Auslander and Bridger for finitely generated modules over a Noetherian ring, is studied in the context of finitely presented modules over a coherent ring.A generalization of the Auslander-Bridger formula is established and is used as a cornerstone in the development of a theory of coherent Gorenstein rings. Introduction.In addressing a problem posed by Glaz ([11, 12]), Hamilton and the second author give a definition of Cohen-Macaulay for commutative rings which agrees with the usual notion for Noetherian rings with the property that every coherent regular ring is Cohen-Macaulay [13].(A quasi-local ring is defined to be regular if every finitely generated ideal has finite projective dimension.)A natural question is whether there is a reasonable concept of Gorenstein for commutative rings such that every coherent regular ring is Gorenstein and every coherent Gorenstein ring is Cohen-Macaulay.In this paper, we develop such a theory of coherent Gorenstein rings which mirrors much of the theory in the Noetherian case.Central to this development is the concept of Gorenstein dimension (G-dimension, for short), first introduced in the context of finitely generated modules over Noetherian rings by Auslander and Bridger [1].In particular, we prove the following generalization of the Auslander-Bridger formula for coherent rings using a notion of depth for arbitrary quasi-local rings developed by Barger [2], Hochster [14], and Northcott [18]: Theorem 1.1.Let R be a quasi-local coherent ring and M a finitely presented R-module of finite G-dimension.Then depth M + Gdim R M = depth R.