Azumaya Algebras and Obstructions to Quadratic Pairs over a Scheme

TL;DR

This paper explores the existence of quadratic pairs on Azumaya algebras with involutions over schemes, identifying a cohomological obstruction that can prevent their existence, especially in non-affine cases.

Contribution

It introduces a cohomological obstruction for Azumaya algebras with involutions to admit quadratic pairs, extending previous results from fields to schemes.

Findings

Obstruction vanishes over affine schemes

Explicit examples with non-trivial obstructions are constructed

Obstruction is cohomological and non-trivial in general

Abstract

We investigate quadratic pairs for Azumaya algebras with involutions over a base scheme S as defined by Calm{\`e}s and Fasel, generalizing the case of quadratic pairs on central simple algebras over a field (Knus, Merkurjev, Rost, Tignol). We describe a cohomological obstruction for an Azumaya algebra over S with orthogonal involution to admit a quadratic pair. When S is affine this obstruction vanishes, however it is non-trivial in general. In particular, we construct explicit examples with non-trivial obstructions.