Lemma 38.8.2 (05DK)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 38: More on Flatness
Section 38.8: Completion and Mittag-Leffler modules
Lemma 38.8.2 (cite)

Lemma 38.8.2. Let $R$ be a ring. Let $I \subset R$ be an ideal. Let $M$ be an $R$ -module. Assume

$R$ is Noetherian and $I$ -adically complete,
$M$ is flat over $R$ , and
$M/IM$ is a projective $R/I$ -module.

Then the $I$ -adic completion $M^\wedge$ is a flat Mittag-Leffler $R$ -module.

Proof. Choose a surjection $F \to M$ where $F$ is a free $R$ -module. By Algebra, Lemma 10.97.9 the module $M^\wedge$ is a direct summand of the module $F^\wedge$ . Hence it suffices to prove the lemma for $F$ . In this case the lemma follows from Lemma 38.8.1. $\square$

The Stacks project

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