# Generically split octonion algebras and A^1-homotopy theory

## TL;DR

This paper applies ${
m A}^1$-homotopy theory to classify generically split octonion algebras over schemes, revealing their structure via characteristic classes and extending classical constructions to algebraic geometry.

## Contribution

It combines affine representability and obstruction theory to classify octonion algebras over low-dimensional schemes, generalizes Zorn's construction, and analyzes trivial norm forms.

## Key findings

- Classification of octonion algebras via characteristic classes.
- Generically split octonion algebras are obtained from Zorn's construction.
- Octonion algebras with trivial norm form are split in low dimensions.

## 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.