Definition 39.11.3. Let $S$ be a scheme. Let $(G, m)$ be a group scheme over $S$. Let $X$ be a pseudo $G$-torsor over $S$.

We say $X$ is a

*principal homogeneous space*or a*$G$-torsor*if there exists a fpqc covering^{1}$\{ S_ i \to S\} _{i \in I}$ such that each $X_{S_ i} \to S_ i$ has a section (i.e., is a trivial pseudo $G_{S_ i}$-torsor).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 $\{ S_ i \to S\} _{i \in I}$ such that each $X_{S_ i} \to S_ i$ has a section.If $X$ is a $G$-torsor, then we say that it is

*quasi-isotrivial*if it is a torsor for the étale topology.If $X$ is a $G$-torsor, then we say that it is

*locally trivial*if it is a torsor for the Zariski topology.

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