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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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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 PDF
J. Amer. Math. Soc. 28 (2015), 1031-1062 Request permission

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.
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Additional 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