Lemma 10.38.7. Suppose that $M$ is (faithfully) flat over $R$, and that $R \to R'$ is a ring map. Then $M \otimes _ R R'$ is (faithfully) flat over $R'$.

Proof. For any $R'$-module $N$ we have a canonical isomorphism $N \otimes _{R'} (R'\otimes _ R M) = N \otimes _ R M$. Hence the desired exactness properties of the functor $-\otimes _{R'}(R'\otimes _ R M)$ follow from the corresponding exactness properties of the functor $-\otimes _ R M$. $\square$

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