Abstract. Let R be a commutative Noetherian ring, a an ideal of R, M an arbitrary R-module and X a finite R-module. We prove a characterization for H i a (M) and H i a (X, M) to be a-weakly cofinite for all i, whenever one of the following cases holds: (a) ara(a) ≤ 1, (b) dim R/a ≤ 1 or (c) dim R ≤ 2. We also prove that, if M is a weakly Laskerian R-module, then H i a (X, M) is a-weakly cofinite for all i, whenever dim X ≤ 2 or dim M ≤ 2 (resp. (R, m) a local ring and dim X ≤ 3 or dim M ≤ 3). Let d = dim M < ∞, we prove an equivalent condition for top local cohomology module H d a (M) to be weakly Artinian.
