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Generically split
octonion algebras and $\mathbb{A}^1$-homotopy theory
Aravind Asok, Marc Hoyois and Matthias Wendt |
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Vol. 13 (2019), No. 3, 695–747
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Abstract |
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We study generically split octonion algebras over schemes using techniques of -homotopy theory. By combining affine representability results with techniques of obstruction theory, we establish classification results over smooth affine schemes of small dimension. In particular, for smooth affine schemes over algebraically closed fields, we show that generically split octonion algebras may be classified by characteristic classes including the second Chern class and another “mod ” invariant. We review Zorn’s “vector matrix” construction of octonion algebras, generalized to rings by various authors, and show that generically split octonion algebras are always obtained from this construction over smooth affine schemes of low dimension. Finally, generalizing P. Gille’s analysis of octonion algebras with trivial norm form, we observe that generically split octonion algebras with trivial associated spinor bundle are automatically split in low dimensions. |
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Keywords
$A^1$-homotopy, obstruction theory, octonion algebras
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Mathematical Subject Classification 2010
Primary: 14F42
Secondary: 14L30, 20G41, 57T20
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Milestones
Received: 7 June 2018
Accepted: 7 January 2019
Published: 23 March 2019
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Authors |
| Aravind Asok | |
| Department of Mathematics University of Southern California Los Angeles, CA United States |
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| Marc Hoyois | |
| Department of Mathematics University of Southern California Los Angeles United States |
Department of Mathematics Massachusetts Institute of Technology Cambridge, MA United States |
| Matthias Wendt | |
| Institut für Mathematik Universität Osnabrück Germany |
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