$G$-bundles on affine spaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T15:50:50Z http://mathoverflow.net/feeds/question/46127 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/46127/g-bundles-on-affine-spaces $G$-bundles on affine spaces Alexander Braverman 2010-11-15T17:43:11Z 2010-11-15T22:21:27Z <p>Let $G$ be a connected algebraic group. Is it true that every $G$-bundle on ${\mathbb A}^n$ is trivial? What is the reference? I am actually only interested in the case $n=2$.</p> http://mathoverflow.net/questions/46127/g-bundles-on-affine-spaces/46132#46132 Answer by Francesco Polizzi for $G$-bundles on affine spaces Francesco Polizzi 2010-11-15T18:24:43Z 2010-11-15T18:29:49Z <p>This is true for $G=GL(r)$, as shown by Quillen and Suslin.</p> <p>For arbitrary $G$ there are counterexamples. Quite surprisingly, even $G$-bundles over $\textrm{Spec }k$ may not be trivial. See the paper</p> <p>M. S. Raghunathan</p> <p>"Principal bundles on affine space and bundles on the projective line",</p> <p>Mathematische Annalen Volume 285, Number 2, 309-332</p> http://mathoverflow.net/questions/46127/g-bundles-on-affine-spaces/46134#46134 Answer by Dave Anderson for $G$-bundles on affine spaces Dave Anderson 2010-11-15T18:58:02Z 2010-11-15T18:58:02Z <p>Over an algebraically closed field, for $G$ connected and reductive, every principal $G$-bundle on ${\Bbb A}^n$ is trivial, also by a theorem of Raghunathan:</p> <p>"Principal bundles on affine space", in <em>C. P. Ramanujam—a tribute</em>, pp. 187–206, Tata Inst. Fund. Res. Studies in Math. <strong>8</strong> (1978).</p> <p>(Unfortunately I can't find this reference free online.)</p> http://mathoverflow.net/questions/46127/g-bundles-on-affine-spaces/46159#46159 Answer by Tom Goodwillie for $G$-bundles on affine spaces Tom Goodwillie 2010-11-15T22:21:27Z 2010-11-15T22:21:27Z <p>In characteristic $p$ you can make an easy counterexample with $n=1$, right? An exact sequence of commutative algebraic groups $0\to E\to X\to \mathbb A\to 0$ with $E$ an elliptic curve.</p>