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# A Note on Homological Dimensions of Artinian Local Cohomology Modules

Published:2012-01-27
Printed: Sep 2013
• Kamal Bahmanpour,
Department of Mathematics, Ardabil branch Islamic Azad University, P.O. Box 5614633167, Ardabil, Iran
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## Abstract

Let $(R,{\frak m})$ be a non-zero commutative Noetherian local ring (with identity), $M$ be a non-zero finitely generated $R$-module. In this paper for any ${\frak p}\in {\rm Spec}(R)$ we show that $\operatorname{{\rm injdim_{_{R_{\frak p}}}}} H^{i-\dim(R/{\frak p})}_{{\frak p}R_{\frak p}}(M_{\frak p})$ and ${\rm fd}_{R_{\p}} H^{i-\dim(R/{\frak p})}_{{\frak p}R_{\frak p}}(M_{\frak p})$ are bounded from above by $\operatorname{{\rm injdim_{_{R}}}} H^i_{\frak m}(M)$ and ${\rm fd}_R H^i_{\frak m}(M)$ respectively, for all integers $i\geq \dim(R/{\frak p})$.
 Keywords: cofinite modules, flat dimension, injective dimension, Krull dimension, local cohomology