On the cofiniteness of generalized local cohomology modules

Let R be a commutative Noetherian ring, let I be an ideal of R, and let M, N be two finitely generated R-modules. The aim of this paper is to investigate the I-cofiniteness of generalized local cohomology modules HIj(M,N)=lim⟶nExtRj(M/InM,N) of M and N with respect to I. We first prove that if I is a principal ideal, then HIj(M,N) is I-cofinite for all M, N and all j. Secondly, let t be a nonnegative integer such that dimSupp(HIj(M,N))≤1 for all j<t. Then HIj(M,N) is I-cofinite for all j<t and Hom(R/I,HIt(M,N)) is finitely generated. Finally, we show that if dim(M)≤2 or dim(N)≤2, then HIj(M,N) is I-cofinite for all j.