Proposition 4

Let $A$ be a commutative Noetherian ring, $m$ an ideal of $A$ and $E$ a finitely generated $A$-module. All the m-good filtrations on $E$ define the same topology (namely the m-adic topology).

Let $(E_n)$ be an m-good filtration on $E$. As this filtration is exhaustive, every element of $E$ belongs to one of the $E_n$ and, as $E$ is finitely generated and the $E_n$ are $A$-modules, there exists an integer $n_1$ such that $E_{n_1} = E$. On the other hand let $n_0$ be such that $mE_n = E_{n+1}$ for $n \geq n_0$; then, for $n > n_0 - n_1$, $m^n E \subset E_{n+n_1} = m^{n+n_1-n_0} E_{n_0} \subset m^{n+n_1-n_0} E$, which proves the proposition.