Definition 28.8.1. Let $X$ be a scheme. We say $X$ is Cohen-Macaulay if for every $x \in X$ there exists an affine open neighbourhood $U \subset X$ of $x$ such that the ring $\mathcal{O}_ X(U)$ is Noetherian and Cohen-Macaulay.
28.8 Cohen-Macaulay schemes
Recall, see Algebra, Definition 10.104.1, that a local Noetherian ring $(R, \mathfrak m)$ is said to be Cohen-Macaulay if $\text{depth}_{\mathfrak m}(R) = \dim (R)$. Recall that a Noetherian ring $R$ is said to be Cohen-Macaulay if every local ring $R_{\mathfrak p}$ of $R$ is Cohen-Macaulay, see Algebra, Definition 10.104.6.
Lemma 28.8.2. Let $X$ be a scheme. The following are equivalent:
$X$ is Cohen-Macaulay,
$X$ is locally Noetherian and all of its local rings are Cohen-Macaulay, and
$X$ is locally Noetherian and for any closed point $x \in X$ the local ring $\mathcal{O}_{X, x}$ is Cohen-Macaulay.
Proof. Algebra, Lemma 10.104.5 says that the localization of a Cohen-Macaulay local ring is Cohen-Macaulay. The lemma follows by combining this with Lemma 28.5.2, with the existence of closed points on locally Noetherian schemes (Lemma 28.5.9), and the definitions. $\square$
Lemma 28.8.3. Let $X$ be a scheme. The following are equivalent:
The scheme $X$ is Cohen-Macaulay.
For every affine open $U \subset X$ the ring $\mathcal{O}_ X(U)$ is Noetherian and Cohen-Macaulay.
There exists an affine open covering $X = \bigcup U_ i$ such that each $\mathcal{O}_ X(U_ i)$ is Noetherian and Cohen-Macaulay.
There exists an open covering $X = \bigcup X_ j$ such that each open subscheme $X_ j$ is Cohen-Macaulay.
Moreover, if $X$ is Cohen-Macaulay then every open subscheme is Cohen-Macaulay.
More information on Cohen-Macaulay schemes and depth can be found in Cohomology of Schemes, Section 30.11.
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