Lemma 27.12.3. Let $X$ be a locally Noetherian scheme. Then $X$ is Cohen-Macaulay if and only if $X$ has $(S_ k)$ for all $k \geq 0$.

**Proof.**
By Lemma 27.8.2 we reduce to looking at local rings. Hence the lemma is true because a Noetherian local ring is Cohen-Macaulay if and only if it has depth equal to its dimension.
$\square$

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