The Stacks project

Lemma 65.30.6. A flat morphism locally of finite presentation is universally open.

Proof. Let $f : X \to Y$ be a flat morphism locally of finite presentation of algebraic spaces over $S$. Choose a diagram

\[ \xymatrix{ U \ar[r]_\alpha \ar[d] & V \ar[d] \\ X \ar[r] & Y } \]

where $U$ and $V$ are schemes and the vertical arrows are surjective and ├ętale, see Spaces, Lemma 63.11.6. By Lemmas 65.30.5 and 65.28.4 the morphism $\alpha $ is flat and locally of finite presentation. Hence by Morphisms, Lemma 29.25.10 we see that $\alpha $ is universally open. Hence $X \to Y$ is universally open according to Lemma 65.6.5. $\square$


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Your email address will not be published.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 042S. Beware of the difference between the letter 'O' and the digit '0'.