Proposition 10

Let $A, B$ be two Zariski rings, $\hat{A}, \hat{B}$ their completions, $f : A \to B$ a continuous ring homomorphism and $\hat{f} : \hat{A} \to \hat{B}$ the homomorphism obtained from $f$ by passing to the completions; if $\hat{f}$ is bijective, the $A$-module $B$ is faithfully flat.

As $A$ and $B$ are Hausdorff, the hypothesis that $\hat{f}$ is bijective implies first that $f$ is injective. Identifying (algebraically) $A$ with $f(A)$ by means off and $\hat{A}$ with $\hat{B}$ by means of $\hat{f}$, we then obtain the inclusions $A \subset B \subset \hat{A} = \mathbf{8}$; we know that $\hat{A}$ is a faithfully flat $A$-module and a faithfully flat $B$-module (Proposition 9); we conclude that $B$ is a faithfully flat $A$-module (Chapter I, § 3, no. 4, Remark 2).