Azumaya Algebras and Obstructions to Quadratic Pairs over a Scheme

TL;DR

The paper develops a cohomological obstruction theory for quadratic pairs on Azumaya algebras with orthogonal involution over a base scheme $S$, introducing strong and weak obstructions $\Omega(\mathcal{A},\sigma)$ and $\omega(\mathcal{A},\sigma)$ that live in first cohomology groups. It shows these obstructions vanish when $2$ is invertible and provides explicit non-trivial affine- and non-affine-base examples where obstructions persist, thereby proving the obstructions capture genuinely global phenomena. The authors classify all quadratic triples extending a locally quadratic involution and extend KMRT-style results to tensor products, detailing when a quadratic pair exists on the tensor product and its uniqueness. They also construct several explicit non-trivial obstruction examples and analyze their behavior under base change, especially highlighting phenomena that arise in characteristic $2$ and in non-flat base settings, laying groundwork for broader connections to adjoint semisimple groups and torsor theory.

Abstract

We investigate quadratic pairs for Azumaya algebras with involutions over a base scheme S as defined by Calm{è}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.

Theorems & Definitions (84)

• Theorem A
• Theorem B
• Remark B
• Lemma D: M
• Lemma E: Gir
• Lemma H
• proof
• Remark I
• Example J
• Lemma B