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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 cofinite modules, flat dimension, injective dimension, Krull dimension, local cohomology
MSC Classifications: 13D45 show english descriptions Local cohomology [See also 14B15] 13D45 - Local cohomology [See also 14B15]
 

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