Questions in commutative algebra

项目负责人：Anurag singh项目编号：0300600研究机构：Georgia Tech Research Corporation - GA Institute of technology摘要该研究员拟研究紧闭理论、局部上同论和交集多重性理论中出现的问题。紧闭理论是由Hochster和Huneke提出的，并导致了f正则环的概念。这类环包括被充分研究过的例子，如行列式环和正规单项式环，但是关于f正则环的几个问题仍然没有答案。局部上同调理论应用于一些基本问题，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质，研究者打算继续他关于正则环的局部上同调模具有有限多个关联素数理想的Lyubeznik猜想的研究。交叉多重度的工作将集中在最近发展的罗伯茨环理论上：这些环提供了一个框架，其中交叉多重度具有几个理想的性质。提案中描述的一些问题为学生提供了参与研究的可能性，通过执行计算机验证来提出答案和合理的方法。研究者还将参与与该提案相关领域的课程开发，并打算增加对深南方代数感兴趣的学生的机会。这个项目是关于交换代数的问题。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，而交换代数的主要研究对象是这些解集上的某些函数。它继续与其他几个数学分支发展迷人的互动，并正在成为工程、编码理论、密码学和其他战略应用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.