Corollary 3

Let A be a complete filtered ring, (A_n) its filtration, E a Hausdorff filtered A-module, (E_n) its filtration and F a finitely generated submodule of E; suppose that A_∞ = A and E_0 = E.

(i) If, for all k ≥ 0, E_k = E_{k+1} + A_k F, then F = E. (ii) If it is further supposed that the filtration on E is derived from that on A (no. 1, Example 2), the relation E = E_1 + F implies F = E.

Let ξ_i (1 ≤ i ≤ n) be the classes mod. E_1 of a finite system of generators of F. It follows from the given hypothesis that for all k ≥ 0 every element of gr_k(E) can be expressed in the form $\sum_{i=1}^n \alpha_i \xi_i$ where $\alpha_i \in \mathrm{gr}(A)$; the $\xi_i$ therefore generate the gr(A)-module gr(E), which proves (i) by virtue of Proposition 12. If the filtration on E is derived from that on A, the relation E = E_1 + F implies

whence (ii).