Let a be an ideal of a commutative Noetherian local ring R, and let M and N be two ¯nitely generated R-modules. Let t be a positive integer. It is shown that if the support of the generalized local cohomology module Hi a(M;N) is ¯nite for all i < t, then the set of associated prime ideals of the generalized local cohomology module Ht a(M;N) is ¯nite. Also, if the support of the local cohomology module Hi a(N) is ¯nite for all i < t, then the set ¡S i2N AssR(Extt R(M=aiM;N)) ¢ \ fp 2 Spec(R) : dimR=p > 1g is ¯nite. Moreover, we prove that gdepth(a + Ann(M);N) is the least integer t such that the support of the generalized local cohomology module Ht a(M;N) is an in¯nite set
