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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.

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The answer is yes if G is commutative. See III.Theorem 3.9 in Milne, 'Etale Cohomology'. An affirmative answer in general would come down to this: is every fppf G-torsor over a strictly Henselian local k-algebra trivial? In remark 3.11 following his proof, Milne shows that isomorphism classes of fppf G-torsors over a strictly Henselian ring are in bijection with those over its residue field. So we reduce the problem to: Suppose G is a smooth group scheme over a separably closed field K; are there non-trivial fppf G-torsors? I would imagine the answer is no, but I don't know a proof. – Keerthi Madapusi Pera Jan 3 '11 at 19:30
up vote 16 down vote accepted

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.

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Hi--Does this work if we don't assume that the torsor is representable? Maybe representability always holds over a field? – Keerthi Madapusi Pera Jan 3 '11 at 20:59
Thanks for the clarification. Could you point me to a reference for the result of Artin? – Keerthi Madapusi Pera Jan 3 '11 at 21:34
"Versal Deformations and Algebraic Stacks", Inv. Math. 27 (1974), 65-189, theorem 6.1. But the proof is sketchy. There is a more detialed proof in chapoter 10 of Laumon-LMB, Champs algébriques. – Laurent Moret-Bailly Jan 3 '11 at 21:44
Thanks again. I should have known to check your book first! – Keerthi Madapusi Pera Jan 3 '11 at 21:47
Very good, it is even true in more general settings. Thanks a lot for your answer! – TonyS Jan 4 '11 at 11:13

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