The Stacks project

Lemma 28.8.3. Let $X$ be a scheme. The following are equivalent:

1. The scheme $X$ is Cohen-Macaulay.

2. For every affine open $U \subset X$ the ring $\mathcal{O}_ X(U)$ is Noetherian and Cohen-Macaulay.

3. There exists an affine open covering $X = \bigcup U_ i$ such that each $\mathcal{O}_ X(U_ i)$ is Noetherian and Cohen-Macaulay.

4. 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.