The Topology, Geometry and Algebra of Projective Linear Groups

Azumaya代数是矩阵代数的一种扭曲形式。由于矩阵本身是无处不在的，所以这些对象在不同的数学上下文中存在并且值得注意。*在域上的代数的上下文中，Azumaya代数仅仅是当扩张到某种(可分的)代数扩张时同构于矩阵代数的代数。这里的典型例子是实数域上的哈密尔顿四元数，当扩展到复数域时，它变得同构于2x2复矩阵。域上的Azumaya代数是具有经典重要性的中心单代数。*在拓扑环境中，空间X上的Azumaya代数是由X参数化为局部同构于平凡的nxn复矩阵族的一族代数。*介于上述两个环境之间的是簇上的Azumaya代数，因为簇跨越了拓扑空间和域上的代数的世界。*这个项目的目的是将适用于拓扑空间上的Azumaya代数的拓扑工具用于涉及算术或代数问题的特定情况，从而提供一系列的例子和反例。这些例子将有助于通过反驳或支持现有猜想来改进代数上下文中的Azumaya代数的研究，并通过提供关于高维簇及其函数域的行为的指南，这些行为很难用代数技巧直接解决。*它还旨在通过上同调和A1或基元同伦理论，将在拓扑上下文中进行的计算转置到代数上下文中，并通过这种方法在代数理论中建立积极的结果。**

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.**