The Stacks project

Lemma 15.42.4. Let $\varphi : R \to S$ be a ring map. Assume

  1. $\varphi $ is regular,

  2. $S$ is Noetherian, and

  3. $R$ is Noetherian and Cohen-Macaulay.

Then $S$ is Cohen-Macaulay.

Proof. For Noetherian rings being Cohen-Macaulay is the same as having properties $(S_ k)$ for all $k$. Hence we may apply Algebra, Lemma 10.163.4. $\square$

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