The Stacks project

Lemma 32.19.2. Notation and assumptions as in Lemma 32.19.1. Let $f : X \to S$ correspond to $g : Y \to \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s})$ via the equivalence. Then $f$ is separated, proper, finite, ├ętale and add more here if and only if $g$ is so.

Proof. The property of being separated, proper, integral, finite, etc is stable under base change. See Schemes, Lemma 26.21.12 and Morphisms, Lemmas 29.41.5 and 29.44.6. Hence if $f$ has the property, then so does $g$. The converse follows from Lemma 32.18.4 but we also give a direct proof here. Namely, if $g$ has to property, then $f$ does in a neighbourhood of $s$ by Lemmas 32.8.6, 32.13.1, 32.8.3, and 32.8.10. Since $f$ clearly has the given property over $S \setminus \{ s\} $ we conclude as one can check the property locally on the base. $\square$


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Your email address will not be published.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