On the vanishing and the finiteness of supports of generalized local cohomology modules

Let (R, m) be a Noetherian local ring, I an ideal of R and M, N two finitely generated R-modules. The first result of this paper is to prove a vanishing theorem for generalized local cohomology modules which says that H-I(j) (M, N) = 0 for all j > dim(R), provided M is of finite projective dimension. Next, we study and give characterizations for the least and the last integer r such that Supp( H-I(r) (M, N)) is infinite.