Azumaya algebras over schemes

Abstract

This dissertation investigates the theory of Azumaya algebras in the context of schemes, with a particular focus on quadratic pairs and involutions. We start by giving an overview of the theory of Azumaya algebras and the Brauer group as defined by Grothendieck [Gro68a]. We discuss the injection of the Brauer group into the cohomological Brauer group H2´et(Gm, S)tors by using an interpretation of H2 (Gm, S) as the set of Gmgerbes, and constructing the Gm-gerbe of trivialisations of an Azumaya algebra. Quadratic pairs were originally introduced in the context of central simple algebras by Knus, Merkurjev, Rost, and Tignol [BoI] as a generalisation of regular quadratic forms, and are particularly effective in characteristic 2. This notion was later generalised to Azumaya algebras over schemes by Calmes and Fasel [ ` CF15]. We study the properties of quadratic pairs in this context, as well as their existence and classifications. We provide a cohomological obstruction for a specified orthogonal involution taking part in a quadratic pair, using recent results by Gille, Neher, and Ruether [GNR23]. Further, we address the existence of involutions of the first kind on Azumaya algebras. Using the proof of Knus, Parimala, and Srinivas [KPS90], we adapt the classical result by Saltman showing that a Brauer class [A] supports an involution if and only if 2[A] = 0 into the language of schemes. In contrast with the classical case of central simple algebras over a field, this is not equivalent to every representative of the Brauer class [A] carrying an involution, with the exception of degree 2 algebras. We adapt an old proof by Saltman to show that every degree 2 Azumaya algebra carries an involution.