Azumaya algebras without involution

Abstract

Generalizing a theorem of Albert, Saltman showed that an Azumaya algebra $A$ over a ring represents a 2-torsion class in the Brauer group if and only if there is an algebra $A’$ in the Brauer class of $A$ admitting an involution of the first kind. Knus, Parimala, and Srinivas later showed that one can choose $A’$ such that deg $A’=2$ deg $A$. We show that 2 deg $A$ is the lowest degree one can expect in general. Specifically, we construct an Azumaya algebra A of degree 4 and period 2 such that the degree of any algebra $A’$ in the Brauer class of $A$ admitting an involution is divisible by 8.

Separately, we provide examples of split and nonsplit Azumaya algebras of degree 2 admitting symplectic involutions, but no orthogonal involutions. These stand in contrast to the case of central simple algebras of even degree over fields, where the presence of a symplectic involution implies the existence of an orthogonal involution and vice versa.