Questions in commutative algebra

主要研究者：Anurag Singh提案编号：0300600机构：格鲁吉亚技术研究公司- GA Institute of TechnologyAbstract研究人员提出了从紧闭理论，局部上同调和交叉多重性理论中产生的问题。 紧闭包理论由Hochster和Huneke发展，并导致F-正则环的概念。这类环包括一些研究得很好的例子，如行列式环和正规单项式环，但关于F-正则环的几个问题仍然没有答案。局部上同调理论可以应用于一些基本问题，例如确定定义一个代数集所需的最少数量的方程。局部上同调模通常具有有用的有限性性质，研究者建议继续他对Lyubeznik猜想的研究，该猜想指出正则环的局部上同调模有100个相关的素理想。交叉多重性的工作将集中在最近开发的罗伯茨环理论：这些环提供了一个框架，其中交叉多重性有几个理想的属性。提案中描述的一些问题为学生参与研究创造了可能性，通过计算机验证来提出答案和合理的方法。调查员还将参与与该提案有关的领域的课程开发，并打算增加对南方腹地代数感兴趣的学生的机会。这是一个领域密切相关的代数几何：而代数几何的重点是几何的解决方案集的多项式方程，在交换代数的主要对象的研究是某些功能，这些解决方案集。它继续与其他几个数学分支发展着迷人的相互作用，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.