Lemma 80.11.7. Let $S$ be a scheme. Let $X \to B$ be a morphism of algebraic spaces over $S$. Let $G$ be a group algebraic space over $B$ and let $a : G \times _ B X \to X$ be an action of $G$ on $X$ over $B$. If

$a$ is a free action, and

$G \to B$ is flat and locally of finite presentation,

then $X/G$ (see Groupoids in Spaces, Definition 78.19.1) is an algebraic space, the morphism $X \to X/G$ is surjective, flat, and locally of finite presentation, and $X$ is an fppf $G$-torsor over $X/G$.

**Proof.**
The fact that $X/G$ is an algebraic space is immediate from Theorem 80.10.1 and the definitions. Namely, $X/G = X/R$ where $R = G \times _ B X$. The morphisms $s, t : G \times _ B X \to X$ are flat and locally of finite presentation (clear for $s$ as a base change of $G \to B$ and by symmetry using the inverse it follows for $t$) and the morphism $j : G \times _ B X \to X \times _ B X$ is a monomorphism by Groupoids in Spaces, Lemma 78.8.3 as the action is free. The morphism $X \to X/G$ is surjective, flat, and locally of finite presentation by Lemma 80.11.6. To see that $X \to X/G$ is an fppf $G$-torsor (Groupoids in Spaces, Definition 78.9.3) we have to show that $G \times _ S X \to X \times _{X/G} X$ is an isomorphism and that $X \to X/G$ fppf locally has sections. The second part is clear from the properties of $X \to X/G$ already shown. The map $G \times _ S X \to X \times _{X/G} X$ is injective (as a map of fppf sheaves) as the action is free. Finally, the map is also surjective as a map of sheaves by Groupoids in Spaces, Lemma 78.19.5. This finishes the proof.
$\square$

## Comments (1)

Comment #1323 by Kestutis Cesnavicius on