The Stacks project

Lemma 27.8.2. Let $X$ be a scheme. The following are equivalent:

  1. $X$ is Cohen-Macaulay,

  2. $X$ is locally Noetherian and all of its local rings are Cohen-Macaulay, and

  3. $X$ is locally Noetherian and for any closed point $x \in X$ the local ring $\mathcal{O}_{X, x}$ is Cohen-Macaulay.

Proof. Algebra, Lemma 10.103.5 says that the localization of a Cohen-Macaulay local ring is Cohen-Macaulay. The lemma follows by combining this with Lemma 27.5.2, with the existence of closed points on locally Noetherian schemes (Lemma 27.5.9), and the definitions. $\square$


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