$k$-torsionless modules with finite Gorenstein dimension

Salimi, Maryam ; Tavasoli, Elham ; Yassemi, Siamak

Czechoslovak Mathematical Journal, Tome 62 (2012) no. 3, pp. 663-672.

Résumé

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 $\mathfrak{p}\in \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(R)$ with $\mathrm{d}\mathrm{e}\mathrm{p}\mathrm{t}\mathrm{h}(R_{\mathfrak{p}})\le 1$, and ${\text{G-}\text{dim}}_{R_{\mathfrak{p}}}(M_{\mathfrak{p}})\le \mathrm{d}\mathrm{e}\mathrm{p}\mathrm{t}\mathrm{h}(R_{\mathfrak{p}})-2$ for $\mathfrak{p}\in \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(R)$ with $\mathrm{d}\mathrm{e}\mathrm{p}\mathrm{t}\mathrm{h}(R_{\mathfrak{p}})\ge 2$. 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 $n\ge 2$ 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.

Détail

DOI : 10.1007/s10587-012-0058-x

Classification : 13C13, 13C15, 13D05
Mots-clés : torsionless module; reflexive module; Gorenstein dimension