Questions in commutative algebra

交换代数问题

• 批准号: 0300600
• 负责人: Anurag Singh
• 金额: $ 10.35万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-05-01 至 2006-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0300600&HistoricalAwards=false
• 关键词: Questions; commutative; algebra

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.

首席研究员：Anurag Singh 提案编号：0300600 机构：佐治亚理工学院研究公司 - GA 理工学院 摘要 研究者提议研究紧闭理论、局部上同调和交叉重数理论所产生的问题。 紧闭理论由 Hochster 和 Huneke 提出，并引出了 F 正则环的概念。这类环包括经过充分研究的例子，例如行列式环和正规单项式环，但有关 F 正则环的几个问题仍未得到解答。局部上同调理论可应用于基本问题，例如确定定义代数集所需的最小方程数。局部上同调模通常具有有用的有限性，研究人员建议继续研究柳别兹尼克猜想，该猜想指出正则环的局部上同调模具有有限多个相关的素理想。关于交叉多重性的工作将集中在最近发展的罗伯茨环理论：这些环提供了一个框架，其中交叉多重性具有几个理想的属性。提案中描述的一些问题通过执行计算机验证来建议答案和合理的方法，为学生参与研究创造了可能性。研究人员还将参与与该提案相关领域的课程开发，并打算增加南方腹地对代数感兴趣的学生的机会。该项目涉及交换代数问题。这是一个与代数几何密切相关的领域：代数几何关注多项式方程解集的几何形状，而交换代数的主要研究对象是这些解集上的某些函数。它继续与数学的其他几个分支发展出令人着迷的相互作用，并且正在成为工程、编码理论、密码学和其他具有战略意义的应用中越来越有价值的工具。