Let R be a commutative Noetherian ring, I and J be two ideals of R and M be an R-module (not necessary I-torsion). In this paper among other things, it is shown that if dim M ≤ 1, then the R-module ExtiR (R/I, M ) is finitely generated, for all i ≥ 0, if and only if the R-module ExtiR (R/I, M ) is finitely generated, for i = 0, 1. As a consequence, we prove that if M is finitely generated and t ∈ N such that the R-module HiI,J (M ) is FD≤1 (or weakly Laskerian) for all i < t, then HiI,J (M ) is (I, J)-cofinite for all i < t and for any FD≤0 (or minimax) submodule N of HtI,J (M ), the R-modules HomR (R/I, HtI,J (M )/N ) and Ext1 R (R/I, HtI,J (M )/N ) are finitely generated. Also it is shown that if dim M/aM ≤ 1 (e.g. dim R/a ≤ 1) for all a ∈ W ̃ (I, J), then the local cohomology module HiI,J (M ) is (I, J)-cofinite for all i ≥ 0.
