On the finiteness property of generalized local cohomology modules

Let a be an ideal of a commutative Noetherian ring R, and let M and N be finitely generated R-modules. Let f(a) (N) = min {j >= 0 vertical bar H(a)(j) (N) not finitely generated} be the a-finiteness dimension of N. In this paper, among other things, we show that for each i <= f(a)(N), (i) the set of associated prime ideals of generalized local cohomology module H(a)(i)(M, N) is finite, and (ii) H(a)(i)(M, N) is a-cofinite if and only if H(a)(o)(Hom(R)(M, H(a)(i)(N))) is so. Moreover, we show that whenever a is a principal ideal, then H(a)(n)(M, N) is a-cofinite for all n.