## Tag `0584`

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

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 $$ \mathop{\rm Ker}(N \otimes_R M \to N' \otimes_R M) \otimes_S S' = \mathop{\rm Ker}(N \otimes_R M' \to N' \otimes_R M') $$ Hence the equivalence of the lemma follows from the second characterization of flatness in Lemma 10.38.5. $\square$

The code snippet corresponding to this tag is a part of the file `algebra.tex` and is located in lines 8572–8578 (see updates for more information).

```
\begin{lemma}
\label{lemma-flatness-descends-more-general}
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 $R$.
\end{lemma}
\begin{proof}
Let $N \to N'$ be an injection of $R$-modules. By the faithful flatness
of $S \to S'$ we have
$$
\Ker(N \otimes_R M \to N' \otimes_R M) \otimes_S S'
=
\Ker(N \otimes_R M' \to N' \otimes_R M')
$$
Hence the equivalence of the lemma follows from the second characterization
of flatness in
Lemma \ref{lemma-flat}.
\end{proof}
```

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