Algebra & Number Theory

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

Aravind Asok, Marc Hoyois, and Matthias Wendt

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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.

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ant/1554775225

Digital Object Identifier
doi:10.2140/ant.2019.13.695

Mathematical Reviews number (MathSciNet)
MR3928340

Zentralblatt MATH identifier
07046300

Subjects
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

Keywords
$A^1$-homotopy obstruction theory octonion algebras

Citation

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. https://projecteuclid.org/euclid.ant/1554775225


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