Lemma 15.45.9. Let $R$ be a Noetherian local ring. The following are equivalent: $R$ is Cohen-Macaulay, the henselization $R^ h$ of $R$ is Cohen-Macaulay, and the strict henselization $R^{sh}$ of $R$ is Cohen-Macaulay.

Proof. By Lemma 15.45.3 we know that $R^ h$ and $R^{sh}$ are Noetherian, hence the lemma makes sense. Since we have $\text{depth}(R) = \text{depth}(R^ h) = \text{depth}(R^{sh})$ and $\dim (R) = \dim (R^ h) = \dim (R^{sh})$ by Lemmas 15.45.8 and 15.45.7 we conclude. $\square$

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