Lemma 2

Let $A, B$ be two commutative Noetherian rings, $\phi : A \to B$ a ring homomorphism, $E$ a finitely generated $A$-module and $F$ a finitely generated $B$-module. Then $\mathrm{Hom}_A(E, \phi_*(F))$ is a finitely generated $B$-module.

There exists by hypothesis a surjective $A$-homomorphism $v : A^n \to E$; the mapping $u \mapsto u \circ v$ of $\mathrm{Hom}_A(E, \phi_*(F))$ to $\mathrm{Hom}_A(A^n, \phi_*(F))$ is therefore injective and, as $B$ is Noetherian, it is sufficient to prove that $\mathrm{Hom}_A(A^n, \phi_*(F))$ is a finitely generated $B$-module; which is immediate since it is isomorphic to $F^n$.