Lemma 10.101.2. Let $R$ be a local ring with nilpotent maximal ideal. Let $M$ be an $R$-module. The following are equivalent

$M$ is flat over $R$,

$M$ is a free $R$-module, and

$M$ is a projective $R$-module.

Lemma 10.101.2. Let $R$ be a local ring with nilpotent maximal ideal. Let $M$ be an $R$-module. The following are equivalent

$M$ is flat over $R$,

$M$ is a free $R$-module, and

$M$ is a projective $R$-module.

**Proof.**
Since any projective module is flat (as a direct summand of a free module) and every free module is projective, it suffices to prove that a flat module is free. Let $M$ be a flat module. Let $A$ be a set and let $x_\alpha \in M$, $\alpha \in A$ be elements such that $\overline{x_\alpha } \in M/\mathfrak m M$ forms a basis over the residue field of $R$. By Lemma 10.101.1 the $x_\alpha $ are a basis for $M$ over $R$ and we win.
$\square$

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