The Stacks project

Lemma 31.11.3. Let $X$ be a scheme which is set theoretically the union of finitely many affine closed subschemes. Then $X$ is affine.

Proof. Let $Z_ i \subset X$, $i = 1, \ldots , n$ be affine closed subschemes such that $X = \bigcup Z_ i$ set theoretically. Then $\coprod Z_ i \to X$ is surjective and integral with affine source. Hence $X$ is affine by Proposition 31.11.2. $\square$


Comments (2)

Comment #3026 by Brian Lawrence on

Suggested slogan: A set-theoretic union of finitely many affine closed subschemes is affine.

Comment #3140 by on

This slogan is ignored as it is the same as the statement of the lemma.


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Your email address will not be published.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 09NL. Beware of the difference between the letter 'O' and the digit '0'.