Two changes

Thanks to Remy and Laurent Moret-Bailly
https://stacks.math.columbia.edu/tag/07S7#comment-4253
https://stacks.math.columbia.edu/tag/07S7#comment-4254

spaces-properties.tex

@@ -1765,14 +1765,15 @@ \section{Obtaining a scheme}
 \item the action $a$ is free,
 \item $X \to S$ is affine, or quasi-affine, or projective, or
 quasi-projective, or $X$ is isomorphic to an open subscheme of an
-affine scheme or isomorphic to an open subscheme of $\text{Proj}(A)$
-for some graded ring $A$.
+affine scheme, or $X$ is isomorphic to an open subscheme of $\text{Proj}(A)$
+for some graded ring $A$, or $G \to S$ is radicial.
 \end{enumerate}
-Then the fppf quotient sheaf $X/G$ is a scheme.
+Then the fppf quotient sheaf $X/G$ is a scheme and $X \to X/G$
+is an fppf $G$-torsor.
 \end{lemma}
 
 \begin{proof}
-Since the action is free the morphism
+We first show that $X/G$ is a scheme. Since the action is free the morphism
 $j = (a, \text{pr}) : G \times_S X \to X \times_S X$
 is a monomorphism and hence an equivalence relation, see
 Groupoids, Lemma \ref{groupoids-lemma-free-action}. The maps
@@ -1786,8 +1787,22 @@ \section{Obtaining a scheme}
 affine scheme or isomorphic to an open subscheme of $\text{Proj}(A)$
 for some graded ring $A$ this follows from
 Properties, Lemma \ref{properties-lemma-ample-finite-set-in-affine}.
-In the remaining cases, we may replace $S$ by an affine open and we
-get back to the case we just dealt with. Some details omitted.
+If $X \to S$ is affine, or quasi-affine, or projective, or
+quasi-projective, we may replace $S$ by an affine open and we
+get back to the case we just dealt with. If $G \to S$ is radicial,
+then the orbits of points on $X$ under the action of $G$ are singletons
+and the condition trivially holds. Some details omitted.
+
+\medskip\noindent
+To see that $X \to X/G$ is an fppf $G$-torsor
+(Groupoids, Definition \ref{groupoids-definition-principal-homogeneous-space})
+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 fact that $X \to X/G$ is surjective
+as a map of fppf sheaves (by construction). The first part follows from
+the isomorphism $R = U \times_M U$ in the conclusion of
+Proposition \ref{proposition-finite-flat-equivalence-global}
+(note that $R = G \times_S X$ in our case).
 \end{proof}
 
 \begin{lemma}