Lemma 10.39.10. Let $R \to S$ be a ring map. Let $M$ be an $S$-module. If $M$ is flat as an $R$-module and faithfully flat as an $S$-module, then $R \to S$ is flat.
Proof. Let $N_1 \to N_2 \to N_3$ be an exact sequence of $R$-modules. By assumption $N_1 \otimes _ R M \to N_2 \otimes _ R M \to N_3 \otimes _ R M$ is exact. We may write this as
\[ N_1 \otimes _ R S \otimes _ S M \to N_2 \otimes _ R S \otimes _ S M \to N_3 \otimes _ R S \otimes _ S M. \]
By faithful flatness of $M$ over $S$ we conclude that $N_1 \otimes _ R S \to N_2 \otimes _ R S \to N_3 \otimes _ R S$ is exact. Hence $R \to S$ is flat. $\square$
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