On the Finiteness Results of the Generalized Local Cohomology Modules

Let a be an ideal of a commutative Noetherian local ring R, and let M and N be two finitely generated R-modules. Let t be a positive integer. It is shown that if the support of the generalized local cohomology module H(a)(i) (M, N) is finite for all i < t, then the set of associated prime ideals of the generalized local cohomology module H(a)(t) (M, N) is finite. Also, if the support of the local cohomology module H(a)(i)(N) is finite for all i < t, then the set (U(i is an element of N) Ass(R) (Ext(R)(t)(M/a(i) M, N))) boolean AND {p is an element of Spec (R) : dim R/p > 1} is finite. Moreover, we prove that gdepth(a + Ann(M), N) is the least integer t such that the support of the generalized local cohomology module H(a)(t)(M, N) is an infinite set.