FINITENESS RESULT FOR GENERALIZED LOCAL COHOMOLOGY MODULES

Let R be a Noetherian ring, let M and N be finitely generated modules and let a and b be ideals of R. Let s be an integer such that b(p) subset of root Ann H(ap)(i) (M(p), N(p)) for all i <= s and all prime ideal p of R. Then we show the following statements hold: (1) If H(b)(i) (N) = 0 for all i < s, then H(a)(i)(M, N) is finitely generated for all i <= s. (2) b subset of root Ann H(a)(2)(M, N). These statements generalize the corresponding results which are shown in [6] and [1] for standard local cohomology module.