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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