Corollary 4

Let $A, B$ be two rings and $\rho$ a homomorphism from $A$ to $B$. (i) Suppose that there exists a $B$-module $E$ such that $\rho_*(E)$ is a faithfully flat $A$-module. Then, for every prime ideal $p$ of $A$, there exists a prime ideal $p'$ of $B$ such that $\bar{\rho}^{-1}(p') = p$. (ii) Conversely, suppose that $B$ is a flat $A$-module. Then, if, for every prime ideal $p$ of $A$, there exists an ideal $p'$ of $B$ such that $\bar{\rho}^{-1}(p') = p$, $B$ is a faithfully flat $A$-module.

(i) The hypothesis implies that, for every ideal $a$ of $A$, $\bar{\rho}^{-1}(B_{\rho}(a)) = a$ (Chapter I, § 3, no. 5, Proposition 8 (ii)) and it is sufficient to apply Corollary 3. (ii) It is sufficient to show that, for every maximal ideal $m$ of $A$, there exists a maximal ideal $m'$ of $B$ such that $\bar{\rho}^{-1}(m') = m$ (Chapter I, § 2, no. 5, Proposition 9 (e)). Now there exists by hypothesis an ideal $q$ of $B$ such that $\bar{\rho}^{-1}(q) = m$; as $q \neq B$, there exists a maximal ideal $m'$ of $B$ containing $q$ and consequently $\bar{\rho}^{-1}(m') \supseteq m$; but, as $\bar{\rho}^{-1}(m')$ cannot contain 1, $\bar{\rho}^{-1}(m') = m$.