Let R be a commutative Noetherian ring, a an ideal of R and M an R-module with dim M = d. We get equivalent conditions for top local cohomology module Hd a (M) to be Artinian and a-cofinite Artinian separately. In addition, we prove that if (R, m) is a local ring such that ExtiR (R/a, M ) is minimax, for each i ≤ d, then ExtiR(N, M ) is minimax R-module for each i ≥ 0 and for each finitely generated R-module N with dim N ≤ 2 and SuppR (N ) ⊆ V (a). As a consequence we prove that if dim R/a = 2 and Supp R (M ) ⊆ V(a), then M is a-cominimax if (and only if) Hom R (R/a, M ), Ext1 R (R/a, M ) and Ext R 2 (R/a, M ) are minimax. We also prove that if dim R/a = 2 and n ∈ N0 such that ExtiR (R/a, M ) is minimax for all i ≤ n+1, then Hi a (M) is a-cominimax for all i < n if (and only if) HomR (R/a, Hi a (M)) is minimax for all i ≤ n.
