The Stacks project

Lemma 70.10.4. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$ and let $G$ be a finite group acting freely on $X$. Set $Y = X/G$ as in Properties of Spaces, Lemma 64.34.1. For $y \in |Y|$ the following are equivalent

  1. $y$ is in the schematic locus of $Y$, and

  2. there exists an affine open $U \subset X$ containing the preimage of $y$.

Proof. It follows from the construction of $Y = X/G$ in Properties of Spaces, Lemma 64.34.1 that the morphism $X \to Y$ is surjective and ├ętale. Of course we have $X \times _ Y X = X \times G$ hence the morphism $X \to Y$ is even finite ├ętale. It is also surjective. Thus the lemma follows from Decent Spaces, Lemma 66.10.3. $\square$


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Your email address will not be published.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 0B87. Beware of the difference between the letter 'O' and the digit '0'.