The Stacks project

Lemma 29.7.9. Let $X$ be a scheme and let $U \subset X$ be a reduced open subscheme. Then the following are equivalent

  1. the scheme theoretic closure of $U$ in $X$ is $X$, and

  2. $U$ is scheme theoretically dense in $X$.

If this holds then $X$ is a reduced scheme.

Proof. This follows from Lemma 29.7.7 and the fact that the scheme theoretic closure of $U$ in $X$ is reduced by Lemma 29.6.7. $\square$

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Your email address will not be published.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 056E. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 29.7.9 as pdf