The Stacks project

Lemma 64.14.5. Notation and assumptions as in Lemma 64.14.3. Assume $G$ is finite. Then

  1. if $U \to S$ is quasi-separated, then $U/G$ is quasi-separated over $S$, and

  2. if $U \to S$ is separated, then $U/G$ is separated over $S$.

Proof. In the proof of Lemma 64.13.1 we saw that it suffices to prove the corresponding properties for the morphism $j : R \to U \times _ S U$. If $U \to S$ is quasi-separated, then for every affine open $V \subset U$ which maps into an affine of $S$ the opens $g(V) \cap V$ are quasi-compact. It follows that $j$ is quasi-compact. If $U \to S$ is separated, the diagonal $\Delta _{U/S}$ is a closed immersion. Hence $j : R \to U \times _ S U$ is a finite coproduct of closed immersions with disjoint images. Hence $j$ is a closed immersion. $\square$

Comments (0)

There are also:

  • 9 comment(s) on Section 64.14: Examples of algebraic spaces

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Your email address will not be published.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 02Z4. Beware of the difference between the letter 'O' and the digit '0'.