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Keywords:
torsionless module; reflexive module; Gorenstein dimension
Summary:
Let $R$ be a commutative Noetherian ring. It is shown that the finitely generated $R$-module $M$ with finite Gorenstein dimension is reflexive if and only if $M_{\mathfrak p}$ is reflexive for ${\mathfrak p} \in {\rm Spec}(R) $ with ${\rm depth}(R_{\mathfrak p}) \leq 1$, and ${\mbox {G-{\rm dim}}}_{R_{\mathfrak p}} (M_{\mathfrak p}) \leq {\rm depth}(R_{\mathfrak p})-2 $ for ${\mathfrak p}\in {\rm Spec} (R) $ with ${\rm depth}(R_{\mathfrak p})\geq 2 $. This gives a generalization of Serre and Samuel's results on reflexive modules over a regular local ring and a generalization of a recent result due to Belshoff. In addition, for $n\geq 2$ we give a characterization of $n$-Gorenstein rings via Gorenstein dimension of the dual of modules. Finally it is shown that every $R$-module has a $k$-torsionless cover provided $R$ is a $k$-Gorenstein ring.
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