Lemma 10.39.7 (00HI)—The Stacks project

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Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.39: Flat modules and flat ring maps
Lemma 10.39.7 (cite)

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Lemma 10.39.7. 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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