Lemma 38.9.1 (05DP)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 38: More on Flatness
Section 38.9: Projective modules
Lemma 38.9.1 (cite)

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Lemma 38.9.1. Let $R$ be a ring. Let $I \subset R$ be an ideal. Let $R \to S$ be a ring map, and $N$ an $S$ -module. Assume

$R$ is Noetherian and $I$ -adically complete,
$R \to S$ is of finite type,
$N$ is a finite $S$ -module,
$N$ is flat over $R$ ,
$N/IN$ is projective as a $R/I$ -module, and
for any prime $\mathfrak q \subset S$ which is an associated prime of $N \otimes _ R \kappa (\mathfrak p)$ where $\mathfrak p = R \cap \mathfrak q$ we have $IS + \mathfrak q \not= S$ .

Then $N$ is projective as an $R$ -module.

Proof. By Lemma 38.8.4 the map $N \to N^\wedge$ is universally injective. By Lemma 38.8.2 the module $N^\wedge$ is Mittag-Leffler. By Algebra, Lemma 10.89.7 we conclude that $N$ is Mittag-Leffler. Hence $N$ is countably generated, flat and Mittag-Leffler as an $R$ -module, whence projective by Algebra, Lemma 10.93.1. $\square$

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