Octonion Algebras over Schemes and the Equivalence of Isotopes and Isometric Quadratic Forms

TL;DR

This paper extends the known equivalence between isotopes and isometric quadratic forms of octonion algebras from rings to more general schemes, broadening the algebraic geometric context.

Contribution

It generalizes the isotopy-isometry equivalence for octonion algebras from rings to arbitrary schemes, using torsor twists.

Findings

Isotopes of octonion algebras over schemes are isomorphic to twists by Aut(C)-torsors.

The equivalence between isotopy and isometry holds over general schemes, not just rings.

Provides foundational definitions and properties of octonion algebras over schemes.

Abstract

Octonion algebras are certain algebras with a multiplicative quadratic form. In their 2019 article, 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 from unital commutative rings to affine schemes, sending a ring to its spectrum, leads us to a question: can the equivalence of isotopy and isometry 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 Aut(C)-torsor. We conclude the thesis by giving an affirmative answer to our question.