Lemma 10.97.4 (031C)—The Stacks project

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Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.97: Completion for Noetherian rings
Lemma 10.97.4 (cite)

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Lemma 10.97.4. Let $R$ be a Noetherian ring. Let $I$ be an ideal of $R$ . Let $M$ be an $R$ -module. Then the completion $M^\wedge$ of $M$ with respect to $I$ is $I$ -adically complete, $I^ n M^\wedge = (I^ nM)^\wedge$ , and $M^\wedge /I^ nM^\wedge = M/I^ nM$ .

Proof. This is a special case of Lemma 10.96.3 because $I$ is a finitely generated ideal. $\square$

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