#### Vol. 13, No. 3, 2019

 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print) Author Index To Appear Other MSP Journals
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 ${\mathbb{A}}^{1}$-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