The Stacks project

Lemma 32.8.12. Notation and assumptions as in Situation 32.8.1. If

  1. $f$ is an open immersion, and

  2. $f_0$ is locally of finite presentation,

then $f_ i$ is an open immersion for some $i \geq 0$.

Proof. By Lemma 32.8.10 we can find an $i$ such that $f_ i$ is ├ętale. Then $V_ i = f_ i(X_ i)$ is a quasi-compact open subscheme of $Y_ i$ (Morphisms, Lemma 29.36.13). let $V$ and $V_{i'}$ for $i' \geq i$ be the inverse image of $V_ i$ in $Y$ and $Y_{i'}$. Then $f : X \to V$ is an isomorphism (namely it is a surjective open immersion). Hence by Lemma 32.8.11 we see that $X_{i'} \to V_{i'}$ is an isomorphism for some $i' \geq i$ as desired. $\square$

Comments (0)

There are also:

  • 3 comment(s) on Section 32.8: Descending properties of morphisms

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Your email address will not be published.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