Proposition 13

Let $S$ be a multiplicative subset of a ring $A$ and $M$ an $A$-module. If $M$ is jflat (resp. faithfully flat), $S^{-1}M$ is ajflat (resp. faithfully flat) $S^{-1}A$-module and a flat $A$-module.

As $S^{-1}M = M \otimes_A S^{-1}A$, the first assertion follows from Chapter I, § 2, no. 7, Corollary 2 to Proposition 8 (resp. Chapter I, § 3, no. 3, Proposition 5); moreover, $S^{-1}A$ is a flat $A$-module ($§ 2$, no. 4, Theorem 1); hence if $M$ is a flat $A$-module, so is $S^{-1}M$ by virtue of Chapter I, § 2, no. 7, Corollary 3 to Proposition 8.