# Are principal bundles isotrivial?

Let $U$ be a $k$-scheme, where $k$ is a field. Let $G$ be a smooth affine $k$-group. Recall that a principal $G$-bundle over $U$ is a smooth surjective $U$-scheme $E$ with an action of $G$ on $E$ such that the action commutes with the projection to $U$ and the obvious morphism $G\times E\to E\times_UE$ is an isomorphism.

It is easy to see that every principal $G$-bundle is locally trivial in etale topology. However Serre in his Espaces fibres algebriques required a stronger condition called local isotriviality: for all $u\in U$ there is a a finite etale morphism $T'\to T$, where $T$ is a Zariski neighborhood of $u$ such that the pull-back of $E$ to $T'$ is trivial. I wonder if these definitions are known to coincide (I think I can prove it in some generality but the proof is not trivial).

Sketch of a proof. Let us embed $G$ into $GL(n)$. Consider the associated space $E'=E\times^GGL(n)$. It is a scheme because $GL(n)$ is affine and affine schemes can be glued in any reasonable topology. Moreover, it is a principal $GL(n)$-bundle, so passing to a Zariski cover we can assume it is trivial. The original bundle can be obtained from E' via reduction of the structure group, that is, it is a pull-back of the principal $G$-bundle $GL(n)\to GL(n)/G$. It remains to use the isotriviality of the latter bundle.

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I agree that the proof is not completely trivial, nor should it be. The obvious relative version is false: for a smooth, affine group scheme $G_U$ over $U$, there exist $G_U$-torsors over $U$ that are not "locally isotrivial" in the strong sense given. –  Jason Starr Jun 20 at 13:15
Thank you, Laurent, I edited the question. Jason, do you have a reference to such an example? –  Roman Fedorov Jun 20 at 14:20
The conditions are in fact equivalent, at least when $k$ is infinite. I don't know a reference, but I know a proof, which I can post, if you'd like (although you seem to say that you have your own proof). –  Angelo Jun 20 at 15:23
@Angelo: Also the proof I know uses Bertini. Since we are allowed to make an (etale) field extension, I think probably we can get around the hypothesis that $k$ is infinite. –  Jason Starr Jun 20 at 15:56
@Roman: Oops! Upon closer inspection, the examples I thought I had produced are not truly torsors: they are generically torsors, and the bad fibers are "set theoretically" torsors. But the bad fibers are non-reduced, so these are not in fact torsors. I will keep thinking about this. –  Jason Starr Jun 20 at 16:57
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