The Stacks project

Lemma 32.4.10. In Situation 32.4.5. Suppose we are given an $i$ and a locally constructible subset $E \subset S_ i$ such that $f_ i(S) \subset E$. Then $f_{i'i}(S_{i'}) \subset E$ for all sufficiently large $i'$.

Proof. Writing $S_ i$ as a finite union of open affine subschemes reduces the question to the case that $S_ i$ is affine and $E$ is constructible, see Lemma 32.2.2 and Properties, Lemma 28.2.1. In this case the complement $S_ i \setminus E$ is constructible too. Hence there exists an affine scheme $T$ and a morphism $T \to S_ i$ whose image is $S_ i \setminus E$, see Algebra, Lemma 10.29.4. By Lemma 32.4.9 we see that $T \times _{S_ i} S_{i'}$ is empty for all sufficiently large $i'$, and hence $f_{i'i}(S_{i'}) \subset E$ for all sufficiently large $i'$. $\square$

Comments (2)

Comment #5377 by Matthieu Romagny on

In the statement of the lemma should be .

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Your email address will not be published.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 05F4. Beware of the difference between the letter 'O' and the digit '0'.