Reflexive modules with finite Gorenstein dimension with respect to a semidualizing module

Abstract

Let R be a commutative Noetherian ring and let C be a semidualizing R-module. It is shown that a finitely generated R-module M with finite G _C -dimension is C-reflexive if and only if \(M_{\mathfrak {p}}\) is \(C_{\mathfrak {p}}\)-reflexive for \(\mathfrak {p} \in \text {Spec}\,(R) \) with \(\text {depth}\,(R_{\mathfrak {p}}) \leq 1\), and \(G_{C_{\mathfrak {p}}}-\dim _{R_{\mathfrak {p}}} (M_{\mathfrak {p}}) \leq \text {depth}\,(R_{\mathfrak {p}})-2 \) for \(\mathfrak {p} \in \text {Spec}\, (R) \) with \(\text {depth}\,(R_{\mathfrak {p}})\geq 2 \). As the ring R itself is a semidualizing module, this result gives a generalization of a natural setting for extension of results due to Serre and Samuel (see Czech. Math. J. 62(3) (9) 663–672 and Beiträge Algebra Geom. 50(2) (3) 353–362). In addition, it is shown that over ring R with \(\dim R \leq n\), where n≥2 is an integer, \(G_{D}-\dim _{R} (\text {Hom}\,_{R} (M,D)) \leq n-2\) for every finitely generated R-module M and a dualizing R-module D.