Octonion algebras are certain algebras with a multiplicative quadratic form. In 2019, Alsaody and Gille showed 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 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.
| Identifer | oai:union.ndltd.org:UPSALLA1/oai:DiVA.org:uu-507863 |
| Date | January 2023 |
| Creators | Hildebrandsson, Victor |
| Publisher | Uppsala universitet, Matematiska institutionen |
| Source Sets | DiVA Archive at Upsalla University |
| Language | English |
| Detected Language | English |
| Type | Student thesis, info:eu-repo/semantics/bachelorThesis, text |
| Format | application/pdf |
| Rights | info:eu-repo/semantics/openAccess |
| Relation | U.U.D.M. project report ; 2023:24 |
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