Lemma 72.10.4. 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 66.34.1. For $y \in |Y|$ the following are equivalent
$y$ is in the schematic locus of $Y$, and
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 66.34.1 that the morphism $X \to Y$ is surjective and étale. Of course we have $X \times _ Y X = X \times G$ hence the morphism $X \to Y$ is even finite étale. It is also surjective. Thus the lemma follows from Decent Spaces, Lemma 68.10.3. $\square$
Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)