Lemma 39.11.2 (0499)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 39: Groupoid Schemes
Section 39.11: Principal homogeneous spaces
Lemma 39.11.2 (cite)

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Lemma 39.11.2. In the situation of Definition 39.11.1.

The scheme $X$ is a pseudo $G$ -torsor if and only if for every scheme $T$ over $S$ the set $X(T)$ is either empty or the action of the group $G(T)$ on $X(T)$ is simply transitive.
A pseudo $G$ -torsor $X$ is trivial if and only if the morphism $X \to S$ has a section.

Proof. Omitted. $\square$

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