Lemma 10.93.4 (05A0)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.93: Characterizing projective modules
Lemma 10.93.4 (cite)

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Lemma 10.93.4. Let $f: M \to N$ be universally injective map of $R$ -modules. Suppose $M$ is a direct sum of countably generated $R$ -modules, and suppose $N$ is flat and Mittag-Leffler. Then $M$ is projective.

Proof. By Lemmas 10.82.7 and 10.89.7, $M$ is flat and Mittag-Leffler, so the conclusion follows from Theorem 10.93.3. $\square$

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