The Stacks project

33.13 Change of fields and the Cohen-Macaulay property

The following lemma says that it does not make sense to define geometrically Cohen-Macaulay schemes, since these would be the same as Cohen-Macaulay schemes.

Lemma 33.13.1. Let $X$ be a locally Noetherian scheme over the field $k$. Let $k'/k$ be a finitely generated field extension. Let $x \in X$ be a point, and let $x' \in X_{k'}$ be a point lying over $x$. Then we have

\[ \mathcal{O}_{X, x}\text{ is Cohen-Macaulay} \Leftrightarrow \mathcal{O}_{X_{k'}, x'}\text{ is Cohen-Macaulay} \]

If $X$ is locally of finite type over $k$, the same holds for any field extension $k'/k$.

Proof. The first case of the lemma follows from Algebra, Lemma 10.167.2. The second case of the lemma is equivalent to Algebra, Lemma 10.130.6. $\square$

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