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Let $k$ be a field, let $G$ be an algebraic group scheme over $k$ and let $T = \textrm{Spec } k[x, x^{-1}]$ be a one-dimensional torus. Does there exist

  • a scheme $X$ over $k$,
  • an algebraic $T$-action on $X$,
  • and a principal $G$-bundle on $X$ which cannot be $T$-equivariantly trivalized in the fppf topology?

In other words, a principal $G$-bundle $P \to X$ such that there does not exist a scheme $U$ over $k$ with a $T$-action and a $T$-equivariant fppf morphism $U \to X$ such that the pull-back bundle $P \times_X U \to U$ is trivial? Here "fppf" means faithfully flat and finite presentation.

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Are there ever nontrivial principal $G$-bundles on $T$ (in the Zariski topology)? –  Tom Goodwillie May 13 '11 at 3:09
    
@Tom, the simplest one is the double covering of $T$ by itself, a nontrivial ${\Bbb Z}/2{\Bbb Z}$-bundle. –  Dave Anderson May 16 '11 at 21:52
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