Lemma 10.38.9. Let $R$ be a ring. Let $S \to S'$ be a faithfully flat map of $R$-algebras. Let $M$ be a module over $S$, and set $M' = S' \otimes _ S M$. Then $M$ is flat over $R$ if and only if $M'$ is flat over $R$.
Proof. Let $N \to N'$ be an injection of $R$-modules. By the faithful flatness of $S \to S'$ we have
Hence the equivalence of the lemma follows from the second characterization of flatness in Lemma 10.38.5. $\square$
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