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Let $k$ be a field, let $G$ be an algebraic group scheme flat 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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T-Equivariant trivialization of a principal G-bundle

Let $k$ be a field, let $G$ be an algebraic group scheme flat 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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