ON THE PROJECTIVE DIMENSION OF ARTINIAN MODULES

Abstract

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 $igeq 0$. This result answers affirmatively a question raised by the present authors in [13].