Definition 77.9.1. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $(G, m)$ be a group algebraic space over $B$. Let $X$ be an algebraic space over $B$, and let $a : G \times _ B X \to X$ be an action of $G$ on $X$.

1. We say $X$ is a pseudo $G$-torsor or that $X$ is formally principally homogeneous under $G$ if the induced morphism $G \times _ B X \to X \times _ B X$, $(g, x) \mapsto (a(g, x), x)$ is an isomorphism.

2. A pseudo $G$-torsor $X$ is called trivial if there exists an $G$-equivariant isomorphism $G \to X$ over $B$ where $G$ acts on $G$ by left multiplication.

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