Questions in commutative algebra

主要研究人员：Anurag Singh Proposal编号：0300600机构：佐治亚理工学院研究公司-GA理工学院摘要研究人员建议研究紧闭包理论、局部上同调理论和交集重数理论产生的问题。紧闭包理论是由Hochster和Huneke发展起来的，并由此引出了F-正则环的概念。这类环包括研究得很好的例子，如行列式环和正规单项环，但关于F-正则环的几个问题仍未得到回答。局部上同调理论在一些基本问题上有应用，例如确定定义一个代数集合所需的最小方程数。局部上同调模通常具有有用的有限性质，研究者建议继续他关于Lyubeznik猜想的工作，该猜想指出正则环的局部上同调模有有限多个相关素理想。关于交重数的工作将集中在最近发展起来的Roberts环理论上：这些环提供了一个框架，在这个框架中交重数具有几个理想的性质。提案中描述的一些问题为学生参与研究创造了可能性，通过执行计算机验证来建议答案和合理的方法。调查员还将参与与这项提议相关领域的课程开发，并打算为对南方腹地代数感兴趣的学生提供更多机会。这个项目涉及交换代数的问题。这是一个与代数几何密切相关的领域：虽然代数几何关注的是多项式方程解集的几何，但在交换代数中，主要研究对象是这些解集上的某些函数。它继续与其他几个数学分支发展着令人着迷的交互作用，并正在成为工程、编码理论、密码学和其他具有战略意义的应用程序中越来越有价值的工具。

Principal Investigator: Anurag SinghProposal Number: 0300600Institution: Georgia Tech Research Corporation - GA Institute of TechnologyAbstract The investigator proposes to work on questions that arise from tight closure theory, local cohomology, and the theory of intersection multiplicities. Tight closure theory was developed by Hochster and Huneke, and leads to the notion of F-regular rings. This class of rings includes well-studied examples such as determinantal rings and normal monomial rings, but several questions about F-regular rings remain unanswered. The theory of local cohomology has applications to fundamental questions such as determining the minimal number of equations needed to define an algebraic set. Local cohomology modules often have useful finiteness properties, and the investigator proposes to continue his work on Lyubeznik's conjecture, which states that local cohomology modules of regular rings have finitely many associated prime ideals. The work on intersection multiplicities will focus on the recently developed theory of Roberts rings: these rings provide a framework in which intersection multiplicities have several desirable properties. Some of the questions described in the proposal create possibilities for students to be involved in research by performing computer verifications to suggest answers and plausible approaches. The investigator will also be involved in curriculum development in areas related to this proposal, and intends to enhance opportunities for students interested in algebra in the Deep South.This project is concerned with questions in commutative algebra. This is a field closely related to algebraic geometry: while algebraic geometry focuses on the geometry of solution sets of polynomial equations, in commutative algebra the main objects of study are certain functions on these solution sets. It continues to develop a fascinating interaction with several other branches of mathematics, and is becoming an increasingly valuable tool in engineering, coding theory, cryptography, and other applications of strategic interest.