Let R be a commutative Noetherian local ring, a an ideal of R, and M a ¯nitely generated generalized f-module. Let t be a positive integer such that Ht a(M) 6= 0 and t > dimM¡dimM=aM. In this paper, we prove that there exists an ideal b ¶ a such that (1) dimM ¡ dimM=bM = t; and (2) the natural homomorphism Hi b(M) ! Hi a (M) is an isomorphism for all i > t and it is surjective for i = t. Also, we show that if SuppR(Hi a(M)) is a ¯nite set for all i < t, then there exists an ideal b of R such that dimR=b · 1 and Hi b (M) »= Hi a (M) for all i < t.
