Lemma 38.8.1. Let $R$ be a ring. Let $I \subset R$ be an ideal. Let $A$ be a set. Assume $R$ is Noetherian and complete with respect to $I$. The completion $(\bigoplus \nolimits _{\alpha \in A} R)^\wedge $ is flat and Mittag-Leffler.
Proof. By More on Algebra, Lemma 15.27.1 the map $(\bigoplus \nolimits _{\alpha \in A} R)^\wedge \to \prod _{\alpha \in A} R$ is universally injective. Thus, by Algebra, Lemmas 10.82.7 and 10.89.7 it suffices to show that $\prod _{\alpha \in A} R$ is flat and Mittag-Leffler. By Algebra, Proposition 10.90.6 (and Algebra, Lemma 10.90.5) we see that $\prod _{\alpha \in A} R$ is flat. Thus we conclude because a product of copies of $R$ is Mittag-Leffler, see Algebra, Lemma 10.91.3. $\square$
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