The Stacks project

Lemma 69.5.11. Notation and assumptions as in Situation 69.5.5. If $X$ is a scheme, then there exists an $i$ such that $X_ i$ is a scheme.

Proof. Choose a finite affine open covering $X = \bigcup W_ j$. By Lemma 69.5.7 we can find an $i \in I$ and open subspaces $W_{j, i} \subset X_ i$ whose base change to $X$ is $W_ j \to X$. By Lemma 69.5.10 we may assume that each $W_{j, i}$ is an affine scheme. This means that $X_ i$ is a scheme (see for example Properties of Spaces, Section 65.13). $\square$

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Your email address will not be published.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 07SR. Beware of the difference between the letter 'O' and the digit '0'.