Corollary

Let $A$ be a commutative Noetherian ring, $\mathfrak{J}$ an ideal of $A$, $\hat{A}$ the Hausdorff completion of $A$ with respect to the $\mathfrak{J}$-adic topology and $M$ an ideally Hausdorff $\hat{A}$-module with respect to $\mathfrak{J} \hat{A}$. For $M$ to be a flat $A$-module, it is necessary and sufficient that $M$ be a flat $\hat{A}$-module.

We know in fact that $\hat{A}$ is a Noetherian ring ($§ 3$, no. 4, Proposition 8) and a flat $A$-module ($§ 3$, no. 4, Theorem 3), that $\mathfrak{J} \hat{A} = \hat{\mathfrak{J}}$ ($§ 2$, no. 12, Proposition 16) and that the canonical homomorphism $A/\mathfrak{J} \to \hat{A}/\hat{\mathfrak{J}}$ is bijective ($§ 2$, no. 12, Proposition 15); Proposition 3 can therefore be applied.