Timeline for Algebraic vector bundles on affine punctured plane
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Nov 15, 2017 at 10:31 | comment | added | user21574 | The conjecture of Serre: Is every algebraic vector bundle $E_{alg} \to\mathbb C^n$algebraically trivial , When $E $ is stabilized; see Bass, H.: Algebraic K-Theory. New York: W.A.Benjamin Inc. 1968. Now my conjecture is that serre conjecture holds on special affine variety in thesense of Griffiths see page 82 link.springer.com/article/10.1007/BF01389905 | |
| Feb 4, 2013 at 16:59 | comment | added | Allen Knutson | The topological statement is that $\pi_3(BU(n)) = 1$. It seems interesting to note that $\pi_3(BO(n)) = Z_2$ for $n$ large enough. Is there some algebraic avatar of this nontrivial class, maybe a vector bundle with nondegenerate symmetric form? | |
| Feb 4, 2013 at 16:23 | answer | added | anonymous | timeline score: 2 | |
| Feb 4, 2013 at 16:03 | comment | added | Martin Brandenburg | Using Cech cohomology, the claim is equivalent to $\mathrm{GL}_n(k[X,Y]_{X \cdot Y}) = \mathrm{GL}_n(k[X,Y]_X) \cdot \mathrm{GL}_n(k[X,Y]_Y)$. I wonder, is there a purely algebraic proof (similar to Hazewinkels Short and elementary proof of Grothendieck's Theorem on algebraic vector bundles over the projective line)? | |
| Feb 4, 2013 at 15:35 | comment | added | Hailong Dao | Or, one can quote algebraic Hartog's Lemma plus the fact mentioned above. | |
| Feb 4, 2013 at 15:30 | comment | added | Hailong Dao | Yes, use the Corollary after Theorem 4.1 in Horrocks' paper "Vector bundle on punctured..." plus the fact that any vector bundle on the whole affine plane is trivial. | |
| Feb 4, 2013 at 15:08 | history | asked | anonymous | CC BY-SA 3.0 |