The Stacks project

Definition 78.9.3. 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.