## Tag `00MH`

Chapter 10: Commutative Algebra > Section 10.98: Criteria for flatness

Lemma 10.98.4. Let $R \to S$ be a local homomorphism of Noetherian local rings. Let $\mathfrak m$ be the maximal ideal of $R$. Let $M$ be a finite $S$-module. Suppose that (a) $M/\mathfrak mM$ is a free $S/\mathfrak mS$-module, and (b) $M$ is flat over $R$. Then $M$ is free and $S$ is flat over $R$.

Proof.Let $\overline{x}_1, \ldots, \overline{x}_n$ be a basis for the free module $M/\mathfrak mM$. Choose $x_1, \ldots, x_n \in M$ with $x_i$ mapping to $\overline{x}_i$. Let $u : S^{\oplus n} \to M$ be the map which maps the $i$th standard basis vector to $x_i$. By Lemma 10.98.1 we see that $u$ is injective. On the other hand, by Nakayama's Lemma 10.19.1 the map is surjective. The lemma follows. $\square$

The code snippet corresponding to this tag is a part of the file `algebra.tex` and is located in lines 22347–22355 (see updates for more information).

```
\begin{lemma}
\label{lemma-free-fibre-flat-free}
Let $R \to S$ be a local homomorphism of Noetherian
local rings. Let $\mathfrak m$ be the maximal
ideal of $R$. Let $M$ be a finite $S$-module.
Suppose that (a) $M/\mathfrak mM$
is a free $S/\mathfrak mS$-module, and (b) $M$ is flat over $R$.
Then $M$ is free and $S$ is flat over $R$.
\end{lemma}
\begin{proof}
Let $\overline{x}_1, \ldots, \overline{x}_n$ be a basis
for the free module $M/\mathfrak mM$. Choose
$x_1, \ldots, x_n \in M$ with $x_i$ mapping to $\overline{x}_i$. Let
$u : S^{\oplus n} \to M$ be the map which maps the $i$th
standard basis vector to $x_i$. By Lemma \ref{lemma-mod-injective}
we see that $u$ is injective. On the other hand, by
Nakayama's Lemma \ref{lemma-NAK} the map is surjective. The
lemma follows.
\end{proof}
```

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