Cofiniteness of local cohomology modules for a pair of ideals for small dimensions

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 Ext(R)(i) (R/I, M) is finitely generated, for all i >= 0, if and only if the R-module Ext(R)(i) (R/I, M) is finitely generated, for i = 0, 1. As a consequence, we prove that if M is finitely generated and t is an element of N such that the R-module H-I, J(i) (M) is FD <= 1 (or weakly Laskerian) for all i < t, then H-I, J(i) (M) is (I, J)-cofinite for all i < t and for any FD <= 0 (or minimax) submodule N of H-I, J(t) (M), the R-modules Hom(R) (R/I, H-I, J(t) (M)/N) and Ext(R)(1) (R/I, H-I, J(t) (M)/N) are finitely generated. Also it is shown that if dim M/aM <= 1 (e.g. dim R/a <= 1) for all a is an element of (W) over tilde (I, J), then the local cohomology module H-I, J(i) (M) is (I, J)-cofinite for all i >= 0.