The Topology, Geometry and Algebra of Projective Linear Groups

射影线性群的拓扑、几何和代数

• 批准号: RGPIN-2016-03780
• 负责人: Williams, Thomas
• 金额: $ 1.97万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=668616
• 关键词: Topology; Geometry; Algebra; Projective; Linear

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.**

Azumaya代数是矩阵代数的扭曲形式。由于矩阵本身是无处不在的，这些对象在不同的数学环境中存在并值得注意。在域上的代数的上下文中，Azumaya代数只是一个代数，它在扩展到某个（可分）代数扩展时同构于矩阵代数。这里的典型例子是真实的域上的汉密尔顿四元数，它在扩展到复数域时同构于2 × 2的复数矩阵。域上的Azumaya代数是具有经典重要性的中心单代数。在拓扑学中，空间X上的Azumaya代数是由X参数化的代数族，其局部同构于平凡的nxn复矩阵族。介于上述两种背景之间的是簇上的Azumaya代数，因为簇横跨拓扑空间和域上的代数的世界。这个项目的目的是使拓扑工具适用于Azumaya代数拓扑空间承担在特定情况下，涉及算术或代数问题，从而提供了一系列的例子和反例。这些例子将有助于改进Azumaya代数在代数背景下的研究，通过反驳或支持现有的假设，并通过提供一个指导，人们期望在高维簇及其函数域上的行为，这可能很难直接使用代数技术来解决。它还旨在通过Eale上同调和A1或motivic同伦理论，将在拓扑背景下进行的计算转置到代数背景下，并通过这种方法在代数理论中建立积极的结果。