Y. Irani - Department of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili, ۵۶۱۹۹-۱۱۳۶۷, Ardabil, Iran.
K. Bahmanpour - Department of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili, ۵۶۱۹۹-۱۱۳۶۷, Ardabil, Iran.
Gh. Ghasemi - Department of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili, ۵۶۱۹۹-۱۱۳۶۷, 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 $i\geq ۰$. This result answers affirmatively a question raised by the present authors in [۱۳].

projective dimension, flat dimension, injective dimension, generalized localcohomology module, local cohomology module