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Prime Characteristic Rings, Birational Morphisms, and Valuations

素特征环、双有理态射和估值

基本信息

批准号：
2101890
负责人：
Ken Ono
金额：
$ 10.5万
依托单位：
University of Virginia Main Campus
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-06-01 至 2025-05-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2101890&HistoricalAwards=false
关键词：
Prime Characteristic Rings Birational Morphisms

项目摘要

Commutative algebra is the local theory of algebraic geometry, and algebraic geometry is the study of geometric objects defined by polynomial equations. The study of algebraic geometry stems from Descartes’ introduction of the coordinate plane to better understand the geometry of objects defined by polynomial equations. For example, the polynomial equation y=x^2 defines a parabola in the coordinate plane and the polynomial equation x^2+y^2=1 defines a circle. In contrast, the graph of the equation y^2 = x^3 has a sharp point, a type of singularity. Geometric objects defined by polynomial equations may admit singularities. The local study of these singularities is a necessity to understanding the geometry of the object. The main research goals of this project concern itself with the study of singularities when the defining equations have coefficients in a positive characteristic numbering system. Singularities of prime characteristic rings are studied, classified, and understood through descriptive behavior of the Frobenius endomorphism. This project focuses on the study of F-pure and F-regular singularities, numerical invariants designed to relate rings among these singularity classes, and test ideals. A particular emphasis is to determine sufficient conditions that imply equality of the finitistic and big test ideal of a local F-pure ring. The purpose of doing so is to investigate the weak-implies-strong conjecture from tight closure theory, to establish unifying behavior of F-regular rings, and to relate the numerical invariant F-signature with the behavior of valuation rings centered over an F-regular ring.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

交换代数是代数几何的局部理论，而代数几何是研究由多项式方程定义的几何对象。代数几何的研究起源于笛卡尔为了更好地理解由多项式方程定义的物体的几何而引入的坐标平面。例如，多项式方程y=x^2定义了坐标平面中的抛物线，多项式方程x^2+y^2=1定义了圆。相比之下，方程y^2 = x^3的图形有一个尖锐的点，这是一种奇点。由多项式方程定义的几何对象可以有奇点。对这些奇点的局部研究是理解物体几何的必要条件。该项目的主要研究目标是研究当定义方程具有正特征编号系统中的系数时的奇异性。通过Frobenius自同态的描述性行为，对素特征环的奇异性进行了研究、分类和理解。这个项目的重点是研究F-纯和F-正则奇异性，数值不变量，旨在将这些奇异类之间的环，和测试理想。特别强调的是，确定充分条件，意味着平等的有限性和大测试理想的局部F-纯环。这样做的目的是调查弱蕴涵强猜想从紧闭包理论，建立统一的行为的F-正则环，并与数值不变的F-签名与行为的估值环中心的F-正则环。这个奖项反映了NSF的法定使命，并已被认为是值得支持的评估使用基金会的智力价值和更广泛的影响审查标准。