Vol. 13, No. 3, 2019

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Generically split octonion algebras and $\mathbb{A}^1$-homotopy theory

Aravind Asok, Marc Hoyois and Matthias Wendt

Vol. 13 (2019), No. 3, 695–747
Abstract

We study generically split octonion algebras over schemes using techniques of A1-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 3” 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.

Keywords
$A^1$-homotopy, obstruction theory, octonion algebras
Mathematical Subject Classification 2010
Primary: 14F42
Secondary: 14L30, 20G41, 57T20
Milestones
Received: 7 June 2018
Accepted: 7 January 2019
Published: 23 March 2019
Authors
Aravind Asok
Department of Mathematics
University of Southern California
Los Angeles, CA
United States
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 fĂĽr Mathematik
Universität Osnabrück
Germany