Corollary

Let $\rho : A \to B$ be a homomorphism of noetherian commutative rings, $M$ a finitely generated $A$-module, $N$ a $B$-module. If $N$ is a finitely generated $B$-module (resp. of finite length), the same is true of the $B$-modules $\operatorname{Tor}^{A}_{n}(M,N)$ and $\operatorname{Ext}^{n}_{A}(M,N)$.