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. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 045O. Beware of the difference between the letter 'O' and the digit '0'.