Splitting vector bundles outside the stable range and ${\mathbb A}^1$-homotopy sheaves of punctured affine spaces
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- by Aravind Asok and Jean Fasel
- J. Amer. Math. Soc. 28 (2015), 1031-1062
- DOI: https://doi.org/10.1090/S0894-0347-2014-00818-3
- Published electronically: August 7, 2014
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Abstract:
We discuss the relationship between the ${\mathbb A}^1$-homotopy sheaves of ${\mathbb A}^n {\setminus } 0$ and the problem of splitting off a trivial rank $1$ summand from a rank $n$ vector bundle. We begin by computing $\boldsymbol {\pi }_3^{{\mathbb A}^1}({\mathbb A}^3 {\setminus } 0)$ and providing a host of related computations of “non-stable” ${\mathbb A}^1$-homotopy sheaves. We then use our computation to deduce that a rank $3$ vector bundle on a smooth affine $4$-fold over an algebraically closed field having characteristic unequal to $2$ splits off a trivial rank $1$ summand if and only if its third Chern class (in Chow theory) is trivial. This result provides a positive answer to a case of a conjecture of M.P. Murthy.References
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Bibliographic Information
- Aravind Asok
- Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089-2532
- MR Author ID: 802326
- Email: asok@usc.edu
- Jean Fasel
- Affiliation: Fakultät Mathematik, Universität Duisburg-Essen, Campus Essen, Thea-Leymann-Strasse 9, D-45127 Essen, Germany
- MR Author ID: 824144
- Email: jean.fasel@gmail.com
- Received by editor(s): June 11, 2013
- Received by editor(s) in revised form: February 18, 2014, April 15, 2014, June 3, 2014, and June 10, 2014
- Published electronically: August 7, 2014
- Additional Notes: The first author was supported in part by NSF Awards DMS-0900813 and DMS-1966589.
The second author was supported by DFG Grant SFB Transregio 45. - © Copyright 2014 American Mathematical Society
- Journal: J. Amer. Math. Soc. 28 (2015), 1031-1062
- MSC (2010): Primary 14F42, 55S35, 13C10; Secondary 19A13, 19D45
- DOI: https://doi.org/10.1090/S0894-0347-2014-00818-3
- MathSciNet review: 3369908