Azumaya Algebras With Orthogonal Involution Admitting an Improper Isometry

Abstract

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

5 pages. Comments are welcome. Changes from last version: Main result extended to Azumaya algebras with quadratic pairs. Some results on torsors added