The Stacks project

Lemma 70.8.3. Let $S$ be a scheme. Let $X$ be a regular Noetherian separated algebraic space over $S$. Let $U \subset X$ be a dense affine open. Then there exists an effective Cartier divisor $D \subset X$ with $U = X \setminus D$.

Proof. We claim that the reduced induced algebraic space structure $D$ on $X \setminus U$ (Properties of Spaces, Definition 65.12.5) is the desired effective Cartier divisor. The construction of $D$ commutes with ├ętale localization, see proof of Properties of Spaces, Lemma 65.12.3. Let $X' \to X$ be a surjective ├ętale morphism with $X'$ affine. Since $X$ is separated, we see that $U' = X' \times _ X U$ is affine. Since $|X'| \to |X|$ is open, we see that $U'$ is dense in $X'$. Since $D' = X' \times _ X D$ is the reduced induced scheme structure on $X' \setminus U'$, we conclude that $D'$ is an effective Cartier divisor by Divisors, Lemma 31.16.6 and its proof. This is what we had to show. $\square$


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Your email address will not be published.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