Proposition 2

Let $A$ be a commutative Noetherian ring, $m$ an ideal of $A$ and $E, F$ two finitely generated $A$-modules. If $(F_n)$ is an $m$-good filtration on $F$, the submodules $\mathrm{Hom}_A(E, F_n)$ form an $m$-good filtration on the $A$-module $\mathrm{Hom}_A(E, F)$. As $m^k F_n \subset F_{n+k}$ for $n \in \mathbf{Z}, k \geq 0$, it is also true that

the family $(\mathrm{Hom}_A(E, F_n))_{n \in \mathbf{Z}}$ is then a filtration on $\mathrm{Hom}_A(E, F)$ compatible with its module structure over the ring $A$ filtered by the $m$-adic filtration. Since $E$ is finitely generated, there exists an integer $r > 0$ and a surjective $A$-homomorphism $u : A^r \to E$ which defines an injective $A$-homomorphism

clearly $v$ is compatible with the filtrations $(\mathrm{Hom}_A(E, F_n))$ and $(\mathrm{Hom}_A(A^r, F_n))$. As $\mathrm{Hom}_A(E, F)$ and $\mathrm{Hom}_A(A^r, F)$ are finitely generated (Lemma 2), it is sufficient by virtue of Proposition 1 to show that the filtration $(\mathrm{Hom}_A(A^r, F))$ is m-good; but this is immediate by virtue of the existence of the canonical isomorphism $\mathrm{Hom}_A(A^r, F_r) \to F_n^r$ and the fact that the relation $mF_n = F_n +$, implies $m(F_n^r) = (mF_n)^r = F_{n+1}^r$ (Algebra, Chapter II, § 3, no. 7, Remark).