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Let $k$ be a field.

$G/k$ be a simply connected semisimple algebraic group.

Let $X/k$ be a smooth affine $k$-scheme.

Question: Is every principal $G$ bundle on $X\times {\mathbb A}^1$ a pull back from $X$?

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2 Answers 2

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In full generality, the answer is no. There are examples of Parimala of non-extended torsors for special orthogonal groups over $\mathbb{A}^2_{\mathbb{R}}$, see e.g. Amer J. Math. 100 (1978), 913-924, (admittedly the group is not simply-connected in this case). These examples are also discussed in Lam's book "Serre's problem on projective modules". Further examples of a dramatic failure of Serre's problem for anisotropic groups were given by Raghunathan, see Principal bundles on affine space and bundles on the projective line, Math. Ann. 285 (1989), 309-332.

The answer is yes, if you restrict to $k$ being an infinite field, $G$ being an isotropic group (i.e. $G$ should contain a split torus of rank $\geq 1$) and consider only rationally trivial torsors. This can be found as Theorem 3.1 in my paper: Rationally trivial torsors in $\mathbb{A}^1$-homotopy theory, J. K-Theory 7 (2011), 541-572 (essentially all the relevant ingredients were in the literature before, but quite scattered; the result is a combination of a general local-global principle for torsors, the results of Raghunathan for torsors on affine spaces, and the extension of Lindel's method by Colliot-Thélène and Ojanguren)

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  • $\begingroup$ I just checked that ragunathan's Theorem B in the paper you mentioned has some conditions on G. Does his paper gives rise to simply connected examples? $\endgroup$
    – Amit H
    May 8, 2015 at 9:11
  • $\begingroup$ Yes. A semisimple simply-connected group with irreducible root system is absolutely almost simple. There are anisotropic simply-connected groups (Spin groups for anisotropic quadratic forms). The second condition is satisfied for Spin groups, because the special orthogonal groups are rational as varieties. $\endgroup$ May 8, 2015 at 9:29
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You may also want to look at http://arxiv.org/abs/1308.3078. You will see that the answer is in general "no" even if you assume that $k$ is the field of complex numbers.

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