Azumaya Algebras With Orthogonal Involution Admitting an Improper Isometry
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866912111966289920 |
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| author | First, Uriya A. |
| author_facet | First, Uriya A. |
| contents | Let $(A,σ)$ be an Azumaya algebra with orthogonal involution over a ring $R$ with $2\in R^\times$. We show that if $(A,σ)$ admits an improper isometry, i.e., an element $a\in A$ with $σ(a)a=1$ and $\mathrm{Nrd}_{A/R}(a)=-1$, then the Brauer class of $A$ is trivial. An analogue of this statement also holds for Azumaya algebras with quadratic pair when $2\notin R^\times$. We also show that at this level of generality, the hypotheses do not guarantee that $A$ is a matrix algebra over $R$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2201_04921 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Azumaya Algebras With Orthogonal Involution Admitting an Improper Isometry First, Uriya A. Rings and Algebras Algebraic Geometry Number Theory Let $(A,σ)$ be an Azumaya algebra with orthogonal involution over a ring $R$ with $2\in R^\times$. We show that if $(A,σ)$ admits an improper isometry, i.e., an element $a\in A$ with $σ(a)a=1$ and $\mathrm{Nrd}_{A/R}(a)=-1$, then the Brauer class of $A$ is trivial. An analogue of this statement also holds for Azumaya algebras with quadratic pair when $2\notin R^\times$. We also show that at this level of generality, the hypotheses do not guarantee that $A$ is a matrix algebra over $R$. |
| title | Azumaya Algebras With Orthogonal Involution Admitting an Improper Isometry |
| topic | Rings and Algebras Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2201.04921 |