Lemma 39.11.4. Let $S$ be a scheme. Let $(G, m)$ be a group scheme over $S$. Let $X$ be a scheme over $S$, and let $a : G \times _ S X \to X$ be an action of $G$ on $X$. Let $\tau \in \{ Zariski, {\acute{e}tale}, smooth, syntomic, fppf\}$. Then $X$ is a $G$-torsor in the $\tau$-topology if and only if $\underline{X}$ is a $\underline{G}$-torsor on $(\mathit{Sch}/S)_\tau$.

Proof. Omitted. $\square$

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