Lemma 77.9.4. Let $S$ be a scheme. Let $(G, m)$ be a group algebraic space over $S$. Let $X$ be an algebraic space over $S$, and let $a : G \times _ S X \to X$ be an action of $G$ on $X$. Then $X$ is a $G$-torsor in the $fppf$-topology in the sense of Definition 77.9.3 if and only if $X$ is a $G$-torsor on $(\mathit{Sch}/S)_{fppf}$ in the sense of Cohomology on Sites, Definition 21.4.1.

**Proof.**
Omitted.
$\square$

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