Fix a scheme $S$, a group scheme $G/S$ (let us say smooth, maybe even affine with some finiteness conditions if you like), and suppose I have some other $S$-scheme $P$ with a right $G$-action. We want to investigate the question of whether $P$ is a $G$-torsor (in the fppf topology).
If I understand correctly, then since $G$ is smooth it is equivalent for $P$ to be a torsor in the etale topology, which is a priori a relatively strong condition. On the other hand, an a priori weaker statement is that it be a torsor in the fpqc topology.
My question is: suppose I can find an fpqc (but not finitely presented) cover $\{S_i \rightarrow S\}$ with $P\times_S S_i \rightarrow S_i$ isomorphic to the trivial $G\times_S S_i$-torsor. Are there circumstances under which I may conclude that $P$ is in fact an fppf torsor?
One (perhaps silly) example is if I have $S$ equal to the spectrum of some small field $k$ with an interesting Galois group, and I am able to find some huge complicated transcendental extension $K/k$ over which $P$ becomes a trivial torsor. Under what conditions does this imply the existence of a finite separable extension over which $P$ becomes a trivial torsor?
Best Answer
One way to define an fppf (or etale, or...) torsor is to require that $G \times_S P \rightarrow P \times_S P$ given by $(g, p) \mapsto (gp, p)$ is an isomorphism and that $P$ has a section fppf (resp., etale, etc.) locally. The first condition can be checked even fpqc locally: use fpqc descent for the property 'an isomorphism'; in particular, it holds in your setup. The second condition is satisfied whenever $P \rightarrow S$ is itself fppf: this is one possible choice of an fppf cover over which $P$ acquires a section; this also holds in your setup, since $P$ is smooth. In fact, in your setup $P$ will be a torsor even for the etale topology due to the same argument: smooth morphisms have sections etale locally on the base, see [EGA IV, 17.16.3 (ii)].