On the Artinianness of Generalized Local Cohomology

Let R be a commutative Noetherian local ring, I a proper ideal of R, M, and N finitely generated R-modules. It is proved that f-depth(I + Ann(M), N) is the least integer r such that the generalized local cohomology module is not Artinian. Let r ≥ 0 be an integer. We also discuss the property that is Artinian for all i ≥ r.