Proposition 1

Let A be a commutative Noetherian ring and m an ideal of A; let A be given the m-adic filtration. Let E, F be twofiltered A-modules and $j : F \to E$ an injective homomorphism compatible with the filtrations. If E is finitely generated and its filtration is m-good, then F is finitely generated and its filtration is m-good.

As F is isomorphic to a submodule of E, it is finitely generated since A is Noetherian and E is finitely generated. Let $(E_n)$, $(F_n)$ be the respective filtrations on E and F, which consist of finitely generated submodules; preserving the notation of Lemma 1, we set $E' = \sum_{n \geq 0} E_n \otimes_A A X^n, F' = \sum F_n \otimes_A A X^n$; as by hypothesis $F_n$ is isomorphic to a submodule of $E_n$, we see that F' is isomorphic to a submodule of $E'$. By Theorem 1, $E'$ is a finitely generated $A'$-module and hence so is F' since A' is Noetherian (Lemma 1). Hence the conclusion by virtue of Theorem 1.