2023/09/04 by Victor Hildebrandsson, Hildebrandsson, Victor
Mathematics · #17D05 (Primary) 14L30 (Secondary) #Algebraic Geometry (math.AG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2309.01776
openalex publication_date 2023/09/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.