Lemma 10.81.2 (058E)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.81: Characterizing flatness
Lemma 10.81.2 (cite)

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Lemma 10.81.2. Let $M$ be an $R$ -module. Then $M$ is flat if and only if the following condition holds: if $P$ is a finitely presented $R$ -module and $f: P \to M$ a module map, then there is a free finite $R$ -module $F$ and module maps $h: P \to F$ and $g: F \to M$ such that $f = g \circ h$ .

Proof. This is just a reformulation of condition (4) from Lemma 10.81.1. $\square$

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