Azumaya Algebras With Orthogonal Involution Admitting an Improper Isometry

Abstract

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 $R$.