Lemma 10.83.1 (058R)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.83: Descent for finite projective modules
Lemma 10.83.1 (cite)

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Lemma 10.83.1. 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 10.78.2. 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 10.81.3 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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