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

1. $R$ is Noetherian and $I$-adically complete,

2. $M$ is flat over $R$, and

3. $M/IM$ 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. $\square$

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