Let R be a commutative Noetherian ring, a an ideal of R, M and N be two finitely generated R-modules. Let t be a positive integer. We prove that if R is local with maximal ideal m and M ⊗R N is of finite length then Htm (M,N) is of finite length for all t 0 and lR(Htm (M,N)) t i=0 lR(Exti R(M,Ht−i m (N))). This yields, lR(Htm (M,N)) = lR(Extt R(M,N)). Additionally, we show that Exti R(R/a,N) is Artinian for all i t if and only if Hia (M,N) is Artinian for all i t. Moreover, we show that whenever dim(R/a) = 0 then Hta (M,N) is Artinian for all t 0.
