The Stacks project

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 $S$.

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

  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 $\{ S_ i \to S\} _{i \in I}$ such that each $X_{S_ i} \to S_ i$ has a section.

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

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

[1] This means that the default type of torsor is a pseudo torsor which is trivial on an fpqc covering. This is the definition in [Exposé IV, 6.5, SGA3]. It is a little bit inconvenient for us as we most often work in the fppf topology.

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