Proposition 10

The A-module $\bigoplus_{m \in \Omega} \mathbf{A}_m$, the direct sum of the $\mathbf{A}_m$ where $m \in \Omega$, is faithfully flat.

Each of the $\mathbf{A}_m$ is a flat A-module (§ 2, no. 4, Theorem 1), hence $E = \bigoplus_{m \in \Omega} \mathbf{A}_m$ is flat (Chapter I, § 2, no. 3, Proposition 2). Moreover, for every maximal ideal $m$ of $A$, $mA_m$ is the unique maximal ideal of $\mathbf{A}_m$, hence $mA_m \neq A_m$, whence it follows that $mE \neq E$ and consequently $E$ is faithfully flat (Chapter I, § 3, no. 1, Proposition 1 (d)).