A Note on Homological Dimensions of Artinian Local Cohomology Modules

Bahmanpour, Kamal

Canadian mathematical bulletin, Tome 56 (2013) no. 3, pp. 491-499

Résumé

Let $\left( R,\,m \right)$ be a non-zero commutative Noetherian local ring (with identity) and let $M$ be a non-zero finitely generated $R$ -module. In this paper for any $\mathfrak{p}\,\in \,\text{Spec}\left( R \right)$ we show that 1 $$\text{injdi}{{\text{m}}_{{{R}_{\mathfrak{p}}}}}\,H_{\mathfrak{p}{{R}_{\mathfrak{p}}}}^{i-\dim\left( {R}/{\mathfrak{p}}\; \right)}\left( {{M}_{\mathfrak{p}}} \right)$$ and 1 $$\text{f}{{\text{d}}_{{{R}_{\mathfrak{p}}}}}H_{\mathfrak{p}}^{i-\dim\left( {R}/{\mathfrak{p}}\; \right)}\left( {{M}_{\mathfrak{p}}} \right)$$ are bounded from above by $\text{injdi}{{\text{m}}_{R}}\,H_{\text{m}}^{i}\left( M \right)$ and $\text{f}{{\text{d}}_{R}}\,H_{\text{m}}^{i}\left( M \right)$ respectively, for all integers $i\,\ge \,\dim\left( {R}/{\mathfrak{p}}\; \right)$ .

Détail

DOI : 10.4153/CMB-2011-197-9

Mots-clés : 13D45, cofinite modules, flat dimension, injective dimension, Krull dimension, local cohomology