Given an algebraically closed field $k$, a smooth group scheme $G$ over $k$
and a principal $G$-bundle $X \rightarrow Y$, which is locally trivial in the fppf topology.
Is this bundle also locally trivial in the etale topology?
Or do we need some extra conditions for this to be true? Literature references about this subject are very welcome.
Best Answer
The answer is yes. Since smoothness is preserved by flat descent, $X$ is smooth over $Y$. This implies that it has sections locally for the étale topology.
No ground field is needed: $Y$ could be any scheme and $G$ any smooth $Y$-group scheme.
[EDIT] to answer a comment by Keerthi Madapusi Sampath: if you don't assume that $X$ is a scheme but only a sheaf on the fppf site of $Y$, the answer is still yes. In fact, since $X$ is fppf-locally isomorphic to $G$, a theorem of Artin implies that $X$ is an algebraic space. In other words, there is a scheme $X'$ and a morphism $X'\to X$ which is representable, surjective and étale. Now $X'$ must be smooth surjective over $Y$, hence has sections étale-locally, and so does $X$.
If $Y$ is the spectrum of a field, group schemes and torsors are always quasiprojective. For any $Y$, if $G$ is affine over $Y$, then so are all $G$-torsors (in particular they are schemes). On the other hand, Raynaud has constructed a regular two-dimensional (local?) scheme $Y$, an elliptic curve $E\to Y$ and an $E$-torsor which is not a scheme.