Corollary

Under the hypotheses of Proposition 9, if we assume also that $N$ is Hausdorf with the $m$-adic filtration, then $u$ is injective.

This follows from the fact that $\mathrm{gr}(u)$ is injective (Corollary 1 to Theorem 1).

\* Remark. Suppose that the hypotheses of Proposition 9 hold and also one of the following conditions: (1) $m$ is nilpotent; (2) $A$ is Noetherian and $P$ is ideally Hausdorff (cf. § 5, no. 1); then $P$ is a flat $A$-module. This follows from the fact that $\gamma_P$ is bijective and § 5, no. 2, Theorem 1 (iv), since $A/m$ is a field. \*