Lemma 78.9.4 (04TZ)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 78: Groupoids in Algebraic Spaces
Section 78.9: Principal homogeneous spaces
Lemma 78.9.4 (cite)

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Lemma 78.9.4. 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 78.9.3 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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