Lemma 29.7.9 (056E)—The Stacks project

Loading web-font TeX/Math/Italic

Table of contents
Part 2: Schemes
Chapter 29: Morphisms of Schemes
Section 29.7: Scheme theoretic closure and density
Lemma 29.7.9 (cite)

previous lemma
next lemma

numberstags

history
statistics
1 tag refers to this tag

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

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

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).

Comments (0)

Post a comment