On Flat and Gorenstein Flat Dimensions of Local Cohomology Modules

Zargar, Majid Rahro ; Zakeri, Hossein

Canadian mathematical bulletin, Tome 59 (2016) no. 2, pp. 403-416

Résumé

Let $\mathfrak{a}$ be an ideal of a Noetherian local ring $R$ and let $C$ be a semidualizing $R$ -module. For an $R$ -module $X$ , we denote any of the quantities $\text{f}{{\text{d}}_{R}}X,\,\text{Gf}{{\text{d}}_{R}}X$ and ${{\text{G}}_{\text{C}}}-\text{f}{{\text{d}}_{R}}\,X\,\text{by}\,\text{T}\left( X \right)$ . Let $M$ be an $R$ -module such that $\text{H}_{\mathfrak{a}}^{i}\left( M \right)\,=\,0$ for all $i\,\ne \,n$ . It is proved that if $T\left( M \right)\,<\,\infty$ , then $\text{T}\left( \text{H}_{\mathfrak{a}}^{n}\left( M \right) \right)\,\le \,\text{T}\left( M \right)\,+\,n$ , and the equality holds whenever $M$ is finitely generated. With the aid of these results, among other things, we characterize Cohen–Macaulay modules, dualizing modules, and Gorenstein rings.

Détail

DOI : 10.4153/CMB-2015-080-x

Mots-clés : 13D05, 13D45, 18G20, flat dimension, Gorenstein injective dimension, Gorenstein flat dimension, local cohomology, relative Cohen–Macaulay module, semidualizing module