Octonion algebras over schemes and the equivalence of isotopes and isometric quadratic forms

Abstract

Octonion algebras are certain algebras with a multiplicative quadratic form. In [1], Alsaody and Gille show that for octonion algebras over unital commutative rings there is an equivalence between isotopes and isometric quadratic forms. The contravariant equivalence of unital commutative rings and affine schemes leads us to a question: can the equivalence of isometry and isotopy be generalized to octonion algebras over a not necessarily affine scheme? We present the basic definitions and properties of octonion algebras, both over rings and over schemes. Then we show that an isotope of an octonion algebra $C$ over a scheme is isomorphic to a twist by an $\mathrm{Aut}\left(C\right)$–torsor. We conclude the paper by giving an affirmative answer to our question.