The Stacks project

Lemma 29.16.4. The following types of schemes are universally catenary.

  1. Any scheme locally of finite type over a field.

  2. Any scheme locally of finite type over a Cohen-Macaulay scheme.

  3. Any scheme locally of finite type over $\mathbf{Z}$.

  4. Any scheme locally of finite type over a $1$-dimensional Noetherian domain.

  5. And so on.

Proof. All of these follow from the fact that a Cohen-Macaulay ring is universally catenary, see Algebra, Lemma 10.104.9. Also, use the last assertion of Lemma 29.16.2. Some details omitted. $\square$


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Your email address will not be published.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 02JB. Beware of the difference between the letter 'O' and the digit '0'.