Lemma 64.34.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $G$ be an abstract group with a free action on $X$. Then the quotient sheaf $X/G$ is an algebraic space.

**Proof.**
The statement means that the sheaf $F$ associated to the presheaf

is an algebraic space. To see this we will construct a presentation. Namely, choose a scheme $U$ and a surjective étale morphism $\varphi : U \to X$. Set $V = \coprod _{g \in G} U$ and set $\psi : V \to X$ equal to $a(g) \circ \varphi $ on the component corresponding to $g \in G$. Let $G$ act on $V$ by permuting the components, i.e., $g_0 \in G$ maps the component corresponding to $g$ to the component corresponding to $g_0g$ via the identity morphism of $U$. Then $\psi $ is a $G$-equivariant morphism, i.e., we reduce to the case dealt with in the next paragraph.

Assume that there exists a $G$-action on $U$ and that $U \to X$ is surjective, étale and $G$-equivariant. In this case there is an induced action of $G$ on $R = U \times _ X U$ compatible with the projection mappings $t, s : R \to U$. Now we claim that

where the map

is given by $(r, g) \mapsto (t(r), g(s(r)))$. Note that $j$ is a monomorphism: If $(t(r), g(s(r))) = (t(r'), g'(s(r')))$, then $t(r) = t(r')$, hence $r$ and $r'$ have the same image in $X$ under both $s$ and $t$, hence $g = g'$ (as $G$ acts freely on $X$), hence $s(r) = s(r')$, hence $r = r'$ (as $R$ is an equivalence relation on $U$). Moreover $j$ is an equivalence relation (details omitted). Both projections $\coprod \nolimits _{g \in G} R \to U$ are étale, as $s$ and $t$ are étale. Thus $j$ is an étale equivalence relation and $U/\coprod \nolimits _{g \in G} R$ is an algebraic space by Spaces, Theorem 63.10.5. There is a map

induced by the map $U \to X$. We omit the proof that it is an isomorphism of sheaves. $\square$

## 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Your email address will not be published.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.

## Comments (0)