Corollary 1

Let $A$ be a complete filtered ring and $E$ a filtered $A$-module whose filtration is exhaustive and separated. If $\mathrm{gr}(E)$ is a finitely generated (resp. Noetherian) $\mathrm{gr}(A)$-module, then $E$ is a finitely generated (resp. Noetherian) $A$-module.

If $\mathrm{gr}(E)$ is finitely generated, there is a finite system of homogeneous generators and Proposition 12 shows that $E$ is finitely generated. Suppose now that $\mathrm{gr}(E)$ is Noetherian and let $F$ be a submodule of $E$; the filtration induced on $F$ by that on $E$ is exhaustive and separated and $\mathrm{gr}(F)$ is identified with a sub-gr(A)-module of gr(E) (no. 4, Proposition 2) and hence is finitely generated by hypothesis; we conclude that F is a finitely generated A-module and hence E is Noetherian.