A note on Gorenstein dimension and the Auslander-Buchsbaum formula
By \(R\) is denoted a commutative Noetherian ring. For an \(R\)-module \(M\), \(\text{Ext}^t (M,R)\) will abbreviate \(\text{Ext}^t_R (M,R)\). If \(M\) is a nonzero finitely generated \(R\)-module then its grade number \(\text{(grade} (M))\) is the smallest integer \(k\) such that \(\text{Ext}^k (M,R) \neq 0\). If \(R\) is local with the maximal ideal \({\mathfrak m}\), then the depth of a finitely generated \(R\)-module \(M\) is defined by the formula \(\text{depth} (M) = \inf \{k : \text{Ext}^k (R/{\mathfrak m}, M) \neq 0\}\). The author gives the following definitions for a finitely generated \(R\)-module \(M\): (1) It is defined the weak Gorenstein dimension of \(M\) (denoted by \(\text{w.g.d} (M))\) thus: (a) \(\text{w.g.d} (M) = 0\) if \(\text{Ext}^t (M,R) = 0\) for all \(t > 0\). (b) \(\text{w.g.d} (M) = k\), where \(k > 0\), if \(\text{Ext}^t (M,R) = 0\) for all \(t > k\) while \(\text{Ext}^k (M,R) \neq 0\). (c) \(\text{w.g.d} (M) = \infty\) if \(\text{Ext}^t (M,R) \neq 0\) for all \(k = 0,1,2, \ldots\). (2) If \(M\) is nonzero it is said that \(M\) is weakly perfect if \(\text{grade} (M) = \text{w.g.d} (M)\). The main results are: Theorem A. Let \(R\) be local. Assume that every finitely generated \(k\)- module with \(\text{w.g.d.} (M) = 0\) is reflexive. Then \(\text{w.g.d} (M) + \text{depth} (M) = \text{depth} (R)\) for every finitely generated \(R\)-module of finite weak Gorenstein dimension. Theorem B. Let \(R\) be as in the theorem A. We assume also that \(R\) is Cohen-Macaulay. Then a finitely generated \(R\)-module \(M\) is weakly perfect if and only if \(M\) is a Cohen-Macaulay module of finite weak Gorenstein dimension. Theorem C. Let \(R\) be such that every localization of \(R\) at a prime ideal satisfies the assumption of theorem A. Let \(M\) be a finitely generated \(R\)-module. If \(\text{w.g.d} (M_{\mathfrak p}) < \infty\) for every prime ideal \({\mathfrak p}\) then: \(\text{grade(Ext}^t(M,R)) \geq t\) for each \(t = 0,1,2, \ldots\). Consequently, if \(R\) is local and satisfies the hypotheses of the theorem A, then \(R\) is Gorenstein if and only if every finitely generated \(R\)-module has finite weak Gorenstein dimension.
scientific article; zbMATH DE number 729229