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The Stacks project

Lemma 15.43.3. Let A be a Noetherian local ring. Then A is Cohen-Macaulay if and only if A^\wedge is so.

Proof. A local ring A is Cohen-Macaulay if and only if \dim (A) = \text{depth}(A). As both of these invariants are preserved under completion (Lemmas 15.43.1 and 15.43.2) the claim follows. \square

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