Definition 77.9.3. 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 a pseudo $G$-torsor over $B$.

1. We say $X$ is a principal homogeneous space, or more precisely a principal homogeneous $G$-space over $B$ if there exists a fpqc covering1 $\{ B_ i \to B\} _{i \in I}$ such that each $X_{B_ i} \to B_ i$ has a section (i.e., is a trivial pseudo $G_{B_ i}$-torsor).

2. Let $\tau \in \{ Zariski, {\acute{e}tale}, smooth, syntomic, fppf\}$. We say $X$ is a $G$-torsor in the $\tau$ topology, or a $\tau$ $G$-torsor, or simply a $\tau$ torsor if there exists a $\tau$ covering $\{ B_ i \to B\} _{i \in I}$ such that each $X_{B_ i} \to B_ i$ has a section.

3. If $X$ is a principal homogeneous $G$-space over $B$, then we say that it is quasi-isotrivial if it is a torsor for the étale topology.

4. If $X$ is a principal homogeneous $G$-space over $B$, then we say that it is locally trivial if it is a torsor for the Zariski topology.

[1] The default type of torsor in Groupoids, Definition 39.11.3 is a pseudo torsor which is trivial on an fpqc covering. Since $G$, as an algebraic space, can be seen a sheaf of groups there already is a notion of a $G$-torsor which corresponds to fppf-torsor, see Lemma 77.9.4. Hence we use “principal homogeneous space” for a pseudo torsor which is fpqc locally trivial, and we try to avoid using the word torsor in this situation.

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