We study generically split octonion algebras over schemes using techniques of
-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
”
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.
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