Proposition 9

Let $A$ be a commutative Noetherian ring and $m$ an ideal of $A$; let $A$ be given the $m$-adic topology. For $\hat{A}$ to be a faithfully flat $A$-module, it is necessary and sufficient that $A$ be a Zariski ring.

For every finitely generated $A$-module $M$, the canonical mapping $M \to M \otimes_A \hat{A}$ is identified with the canonical mapping $M \to \hat{M}$ from $M$ to its Hausdorff completion with respect to the $m$-adic topology (no. 4, Theorem 3) and the kernel of this mapping is then the closure of $\{0\}$ in $M$ with respect to this topology. As we already know that $\hat{A}$ is a flat $A$-module (no. 4, Theorem 3), the proposition follows from the characterization of faithfully flat modules (Chapter I, § 3, no. 1, Proposition 1 (b)) and the characterization of Zariski rings (no. 3, Proposition 6).

If $A$ is a Zariski ring and $E$ is a finitely generated $A$-module, we may (by virtue of Proposition 9) identify $E$ with a subset of $\hat{E}$ by means of the canonical mapping $j_E : E \to \hat{E}$. With this identification: