Mathematical Sciences: On the Brauer Group of a Variety

本课题研究代数中关于簇上的Azumaya代数理论的问题。如果X是代数闭域k上的簇，则Azumaya X-代数被Brauer群B(X)参数化为Brauer等价。主要研究人员将使用代数几何中的上同调技巧来研究B(X)。然后，他将应用这些性质来获得关于变种X的几何信息。这项研究是在一般的代数领域，是环论、数论和代数几何的有趣组合。给定一个曲面，就有可能把一个坐标环和一个群联系在一起，以求确定该曲面的几何形状。

This project deals with questions in algebra concerning the theory of Azumaya algebras on a variety. If X is a variety over an algebraically closed field k, the Azumaya X-algebras are parametrized up to Brauer equivalence by the Brauer group B(X). The principal investigator will study B(X) using cohomological techniques from algebraic geometry. Then he will apply these properties to gain information about the geometry of the variety X. This research is in the general area of algebra and is an interesting combination of ring theory, number theory and algebraic geometry. Given a surface it is possible to associate with it a coordinate ring and with this ring a group in an effort to determine the geometry of the surface.