The Topology, Geometry and Algebra of Projective Linear Groups

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An Azumaya algebra is a twisted form of a matrix algebra. Since matrices themselves are ubiquitous, these objects exist and are noteworthy in different mathematical contexts. In the context of algebras over a field, an Azumaya algebra is simply an algebra that becomes isomorphic to a matrix algebra upon extension to some (separable) algebraic extension. The prototypical example here is Hamilton's Quaternions over the real field, which becomes isomorphic to 2x2 complex matrices upon extension to the complex field. Azumaya algebras over fields are the Central Simple Algebras of classical importance. In the topological context, an Azumaya algebra over a space X is a family of algebras parametrized by X that is locally isomorphic to the trivial family of nxn complex matrices. Intermediate between the two contexts above is that of Azumaya algebras over varieties, since varieties straddle the worlds of topological spaces and of algebras over fields. This project aims to bring topological tools applicable to Azumaya algebras over topological spaces to bear in specific cases that relate to arithmetic or algebraic questions, and consequently to furnish a stream of examples and counterexamples. These examples will help to refine the study of Azumaya algebras in the algebraic context by either disproving or supporting existing conjectures, and by providing a guide to the behaviour one expects over high-dimensional varieties and their function fields, which can be hard to address directly using algebraic techniques. It also aims to transpose the calculations made in the topological context to an algebraic context, via etale cohomology and A1 or motivic homotopy theories, and to establish positive results in the algebraic theory by this method.