Generically split octonion algebras and 𝔸1-homotopy theory

Asok, Aravind; Hoyois, Marc; Wendt, Matthias

doi:10.2140/ant.2019.13.695

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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

DOI: 10.2140/ant.2019.13.695

Abstract

We study generically split octonion algebras over schemes using techniques of $A^{1}$ -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 für Mathematik Universität Osnabrück Germany

Vol. 13, No. 3, 2019