The Stacks project

Lemma 37.70.5. Let $X$ be a separated scheme which has an open covering by $n + 1$ affines. Then the affine stratification number of $X$ is at most $n$.

Proof. Say $X = U_0 \cup \ldots \cup U_ n$ is an affine open covering. Set

\[ X_ i = (U_ i \cup \ldots \cup U_ n) \setminus (U_{i + 1} \cup \ldots \cup U_ n) \]

Then $X_ i$ is affine as a closed subscheme of $U_ i$. The morphism $X_ i \to X$ is affine by Morphisms, Lemma 29.11.11. Finally, we have $\overline{X_ i} \subset X_ i \cup X_{i - 1} \cup \ldots X_0$. $\square$


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Your email address will not be published.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 0F2W. Beware of the difference between the letter 'O' and the digit '0'.