On deformations of Azumaya algebras with quadratic pair

Abstract

We construct a tangent-obstruction theory for Azumaya algebras equipped with a quadratic pair. Under the assumption that either 2 is a global unit or the algebra is of degree 2, we show how the deformation theory of these objects reduces to the deformation theory of the underlying Azumaya algebra. Namely, if the underlying Azumaya algebra has unobstructed deformations then so does the quadratic pair. On the other hand, in the purely characteristic 2 setting, we construct an Azumaya algebra with unobstructed deformations which can be equipped with a quadratic pair such that the associated triple has obstructed deformations. Our example is a biquaternion Azumaya algebra on an Igusa surface. Independently from the above results, we also introduce a new obstruction for quadratic pairs, existing only in characteristic 2, which is intermediate to both the strong and weak obstructions that were recently introduced by Gille, Neher, and the second named author. This intermediate obstruction characterizes when a canonical extension of the Lie algebra sheaf of the automorphism group scheme of some quadratic triple is split.