# Degree project: Octonion algebras over schemes and the equivalence of isotopes and isometric quadratic forms

- Date: 23 May 2023, 15:15–17:00
- Location: Ångström Laboratory, Å64119 + Zoom
- Type: Course
- Lecturer: Victor Hildebrandsson
- Organiser: Matematiska institutionen
- Contact person: Volodymyr Mazorchuk

Welcome to Victor Hildebrandsson´s presentation of his master degree project with the title "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 from unital commutative rings to affine schemes, sending a ring to its spectrum, leads us to a question: can the equivalence of isometry and isotopy be generalized to octonion algebras over a (not necessarily affine) scheme? We will begin the presentation with some necessary background and discuss octonion algebras over rings. Then we will give the basic definitions and properties of octonion algebras over schemes. We show that an isotope of an octonion algebra C over a scheme is isomorphic to a twist by the Aut(C)–torsor. Then we conclude the thesis by giving an affirmative answer to our question.

References: [1] S. Alsaody and P. Gille. Isotopes of octonion algebras, g2-torsors and triality. Advances in Mathematics, 343:864-909, 2019.

This degree project presentation is part of our algebra seminar series.