$k$-torsionless modules with finite Gorenstein dimension

Salimi, Maryam, Tavasoli, Elham, and Yassemi, Siamak. "$k$-torsionless modules with finite Gorenstein dimension." Czechoslovak Mathematical Journal 62.3 (2012): 663-672. <http://eudml.org/doc/246577>.

@article{Salimi2012,
abstract = {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 $\{\rm depth\}(R_\{\mathfrak \{p\}\}) \le 1$, and $\{\mbox\{G-\{\rm dim\}\}\}_\{R_\{\mathfrak \{p\}\}\} (M_\{\mathfrak \{p\}\}) \le \{\rm depth\}(R_\{\mathfrak \{p\}\})-2 $ for $\{\mathfrak \{p\}\}\in \{\rm Spec\} (R) $ with $\{\rm depth\}(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.},
author = {Salimi, Maryam, Tavasoli, Elham, Yassemi, Siamak},
journal = {Czechoslovak Mathematical Journal},
keywords = {torsionless module; reflexive module; Gorenstein dimension; Gorenstein dimension; torsionless module},
language = {eng},
number = {3},
pages = {663-672},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {$k$-torsionless modules with finite Gorenstein dimension},
url = {http://eudml.org/doc/246577},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Salimi, Maryam
AU - Tavasoli, Elham
AU - Yassemi, Siamak
TI - $k$-torsionless modules with finite Gorenstein dimension
JO - Czechoslovak Mathematical Journal
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 3
SP - 663
EP - 672
AB - 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 ${\rm depth}(R_{\mathfrak {p}}) \le 1$, and ${\mbox{G-{\rm dim}}}_{R_{\mathfrak {p}}} (M_{\mathfrak {p}}) \le {\rm depth}(R_{\mathfrak {p}})-2 $ for ${\mathfrak {p}}\in {\rm Spec} (R) $ with ${\rm depth}(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.
LA - eng
KW - torsionless module; reflexive module; Gorenstein dimension; Gorenstein dimension; torsionless module
UR - http://eudml.org/doc/246577
ER -