Document Type : Original Manuscript


Department of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili, 56199-11367, Ardabil, Iran.


Let $(R, \mathfrak{m})$ be a Noetherian local ring and $M$, $N$ be two finitely generated $R$-modules. In this paper it is shown that $R$ is a Cohen-Macaulay ring if and only if $R$ admits a non-zero Artinian $R$-module $A$ of finite projective dimension; in addition, for all such Artinian $R$-modules $A$, it is shown that $\mathrm{pd}_R\, A=\dim R$. Furthermore, as an application of these results it is shown that
$$\pdd H^i_{{\frak p}R_{\frak p}}(M_{\frak p}, N_{\frak p})\leq \pd H^{i+\dim R/{\frak p}}_{\frak m}(M,N)$$
for each ${\frak p}\in \mathrm{Spec} R$ and each integer $i\geq 0$. This result answers affirmatively a question raised by the present authors in [13].


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