Corollary 4

Let $A$ be a commutative Noetherian ring, $m$ an ideal of $A$, $E$ ajnitely generated $A$-module and $(E_n)$ and $(E'_n)$ two filtrations consisting of submodules of $E$. Suppose that the filtrations $(E_n)$ and $(E'_n)$ are compatible with the $A$-module structure on $E$ when $A$ is given the $m$-adic filtration. If the filtration $(E_n)$ is $m$-good and $E'_n \subset E_n$ for all $n \in \mathbf{Z}$, the filtration $(E'_n)$ is $m$-good. This a special case of Proposition 1.