The Stacks project

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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