Twists of nonsingular quadrics over a field are in correspondence with quadratic pairs on central simple algebras. Calmes and Fasel have given a globalized version of this correspondence, introducing the notion of a quadratic pair on an Azumaya algebra. Gille, Neher, and Ruether asked whether every orthogonal involution on an Azumaya algebra was part of an associated quadratic pair; they showed that affine locally this is the case, they gave strong and weak cohomological obstruction classes for the existence of global quadratic pairs, and they produced examples illustrating that these obstructions can be nontrivial in general.

In this talk, we introduce an intermediate obstruction to the strong and weak obstructions of Gille, Neher, and Ruether which exists only in the characteristic 2 setting. We then revisit the examples of Gille, Neher, and Ruether and show how these obstructions convey information on the underlying module structure of the associated Azumaya algebras. This talk is based on joint work in-progress with Cameron Ruether.