The Stacks project

Lemma 28.29.3. Let $X$ be a quasi-compact scheme. There exists a dense open $V \subset X$ which is separated.

Proof. Say $X = \bigcup _{i = 1, \ldots , n} U_ i$ is a union of $n$ affine open subschemes. We will prove the lemma by induction on $n$. It is trivial for $n = 1$. Let $V' \subset \bigcup _{i = 1, \ldots , n - 1} U_ i$ be a separated dense open subscheme, which exists by induction hypothesis. Consider

\[ V = V' \amalg (U_ n \setminus \overline{V'}). \]

It is clear that $V$ is separated and a dense open subscheme of $X$. $\square$

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Your email address will not be published.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 03J1. Beware of the difference between the letter 'O' and the digit '0'.