Algebra & Number Theory

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

Aravind Asok, Marc Hoyois, and Matthias Wendt

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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.

Article information

Algebra Number Theory, Volume 13, Number 3 (2019), 695-747.

Received: 7 June 2018
Accepted: 7 January 2019
First available in Project Euclid: 9 April 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14F42: Motivic cohomology; motivic homotopy theory [See also 19E15]
Secondary: 14L30: Group actions on varieties or schemes (quotients) [See also 13A50, 14L24, 14M17] 20G41: Exceptional groups 57T20: Homotopy groups of topological groups and homogeneous spaces

$A^1$-homotopy obstruction theory octonion algebras


Asok, Aravind; Hoyois, Marc; Wendt, Matthias. Generically split octonion algebras and $\mathbb{A}^1$-homotopy theory. Algebra Number Theory 13 (2019), no. 3, 695--747. doi:10.2140/ant.2019.13.695.

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