Cominimaxness with respect to ideals of dimension two and local cohomology

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 H-a(d)(M) to be Artinian and a-cofinite Artinian separately. In addition, we prove that if (R, m) is a local ring such that Ext(R)(i)(R/a, M) is minimax, for each i <= d, then Ext(R)(i)(N, M) is minimax R-module for each i >= 0 and for each finitely generated R-module N with dim N <= 2 and Supp(R)(N) subset of V (a). As a consequence we prove that if dim R/a = 2 and Supp(R)(M) subset of V(a), then M is a-cominimax if (and only if) Hom(R)(R/a, M), Ext(R)(1)(R/a, M) and Ext(R)(2)(R/a, M) are minimax. We also prove that if dim R/a = 2 and n is an element of N-0 such that Ext(R)(i)(R/a, M) is minimax for all i <= n + 1, then H-a(i)(M) is a-cominimax for all i < n if (and only if) Hom(R)(R/a, H-a(i)(M)) is minimax for all i <= n.