Mathematical Sciences: On Azumaya Algebras Over a Variety

本课题研究代数簇函数域上的Azumaya代数理论问题。设K是曲面S的函数域，A是K上的中心除法代数.主要研究人员将利用代数几何中的上同调技巧来研究代数A，以了解A在S上的分支因子.本课题将K的Brauer群的性质与S上的因子几何联系起来，并考虑关于A的Brauer类分解为符号的问题。这项研究属于代数的一般领域，是代数、数论和代数几何的有趣结合。给定一个曲面，就有可能把它和一个坐标环联系在一起，并把这个环和一个群联系起来。本项目将检查这一组的属性，以努力确定曲面的几何形状。

This project deals with questions on the theory of Azumaya algebras over the function field of an algebraic variety. Let K be the function field of the surface S and A a central division algebra over K. The principal investigator will study the algebra A by using cohomological techniques from algebraic geometry in order to understand the ramification divisor of A on S. The project relates properties of the Brauer group of K to the geometry of divisors on S and considers questions about the factorization of the Brauer class of A into symbols. This research is in the general area of algebra and is an interesting combination of algebra, number theory and algebraic geometry. Given a surface it is possible to associate with it a coordinate ring and with this ring a group. This project will examine properties of this group in an effort to determine the geometry of the surface.