## Tag `00HI`

Chapter 10: Commutative Algebra > Section 10.38: Flat modules and flat ring maps

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 $-\otimes_R M$. $\square$

The code snippet corresponding to this tag is a part of the file `algebra.tex` and is located in lines 8529–8533 (see updates for more information).

```
\begin{lemma}
\label{lemma-flat-base-change}
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 $R'$.
\end{lemma}
\begin{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 $-\otimes_R M$.
\end{proof}
```

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