Azumaya Algebras With Orthogonal Involution Admitting an Improper Isometry

TL;DR

This paper proves that Azumaya algebras with orthogonal involution admitting an improper isometry have trivial Brauer class, extending to quadratic pairs and highlighting limitations in algebra structure over the base ring.

Contribution

It establishes a link between improper isometries and trivial Brauer class for Azumaya algebras with orthogonal involution, including cases with quadratic pairs.

Findings

Improper isometries imply trivial Brauer class for Azumaya algebras.

Extension of results to quadratic pairs when 2 is not invertible.

Limitations in guaranteeing matrix algebra structure over the base ring.

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 $R$.