Quadratic Pairs on Azumaya Algebras over a Scheme

Philippe Gille (1) , Erhard Neher (2) , Cameron Roy Ruether (2)

1 ICJ - Institut Camille Jordan
2 Department of Mathematics and Statistics [Ottawa]

We investigate quadratic pairs for Azumaya algebras with involutions over a base scheme S as defined by Calmès 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.

Mots clés: Azumaya algebras; involutions; quadratic forms; quadratic pairs. MSC 2000: 16H05; 20G35

Domaines: Anneaux et algèbres [math.RA]; Géométrie algébrique [math.AG]

Philippe Gille, Erhard Neher, Cameron Roy Ruether. Quadratic Pairs on Azumaya Algebras over a Scheme. 2022. ⟨hal-03776674v1⟩