Lemma 10.82.5 (058M)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.82: Universally injective module maps
Lemma 10.82.5 (cite)

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Lemma 10.82.5. Let $M$ be an $R$ -module. Then $M$ is flat if and only if any exact sequence of $R$ -modules

$0 \to M_1 \to M_2 \to M \to 0$

is universally exact.

Proof. This follows from Lemma 10.81.3 and Theorem 10.82.3 (5). $\square$

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