ON THE PROJECTIVE DIMENSION OF ARTINIAN MODULES

Let (R, 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 pdR A = dim R. Furthermore, as an application of these results it is shown that pdRp H^i pRp (Mp, Np) ≤ pdR H^i+dim R/p m (M, N) for each p ∈ SpecR and each integer i ≥ 0. This result answers affirmatively a question raised by the present authors in [13].