Lemma 78.9.2 (04TX)—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 78.9.2 (cite)

Lemma 78.9.2. In the situation of Definition 78.9.1.

The algebraic space $X$ is a pseudo $G$ -torsor if and only if for every scheme $T$ over $B$ 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 $X \to B$ has a section.

Proof. Omitted. $\square$

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