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arXiv · 2201.04921

Azumaya Algebras With Orthogonal Involution Admitting an Improper Isometry

Uriya A. First
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Abstract

Let $(A,\sigma)$ be an Azumaya algebra with orthogonal involution over a ring $R$ with $2\in R^\times$. We show that if $(A,\sigma)$ admits an improper isometry, i.e., an element $a\in A$ with $\sigma(a)a=1$ 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 $R$.

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Uriya A. First. 2022-01-13. Azumaya Algebras With Orthogonal Involution Admitting an Improper Isometry. https://arxiv.org/abs/2201.04921

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