The Stacks project

Lemma 77.9.5. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $G$ be a group algebraic space over $B$. Let $X$ be a pseudo $G$-torsor over $B$. Assume $G$ and $X$ locally of finite type over $B$.

  1. If $G \to B$ is unramified, then $X \to B$ is unramified.

  2. If $G \to B$ is locally quasi-finite, then $X \to B$ is locally quasi-finite.

Proof. Proof of (1). By Morphisms of Spaces, Lemma 66.38.10 we reduce to the case where $B$ is the spectrum of a field. If $X$ is empty, then the result holds. If $X$ is nonempty, then after increasing the field, we may assume $X$ has a point. Then $G \cong X$ and the result holds.

The proof of (2) works in exactly the same way using Morphisms of Spaces, Lemma 66.27.2. $\square$


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Your email address will not be published.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