On the non-neutral component of outer forms of the orthogonal group

TL;DR

This paper investigates the properties of outer forms of orthogonal groups over various rings, extending known results from central simple algebras to Azumaya algebras and exploring the limitations of these generalizations.

Contribution

It generalizes the known criterion for elements of reduced norm -1 in orthogonal groups from central simple algebras to Azumaya algebras over semilocal rings and discusses the failure of the 'if' part in broader contexts.

Findings

The criterion for elements of reduced norm -1 extends to Azumaya algebras over semilocal rings.

The 'if' part of the criterion does not hold in general over arbitrary base rings.

The paper clarifies the limitations of extending classical results to more general algebraic structures.

Abstract

Let $(A,\sigma)$ be a central simple algebra with an orthogonal involution. It is well-known that $O(A,\sigma)$ contains elements of reduced norm $-1$ if and only if the Brauer class of $A$ is trivial. We generalize this statement to Azumaya algebras with orthogonal involution over semilocal rings, and show that the "if" part fails if one allows the base ring to be arbitrary.