Lemma 10.82.1. Let $M$ be an $R$-module. Then $M$ is finite projective if and only if $M$ is finitely presented and flat.

**Proof.**
This is part of Lemma 10.77.2. However, at this point we can give a more elegant proof of the implication (1) $\Rightarrow $ (2) of that lemma as follows. If $M$ is finitely presented and flat, then take a surjection $R^ n \to M$. By Lemma 10.80.3 applied to $P = M$, the map $R^ n \to M$ admits a section. So $M$ is a direct summand of a free module and hence projective.
$\square$

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