Lemma 15.42.3. 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 15.42.1 and 15.42.2) the claim follows.
$\square$

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