Studies in Commutative Algebra

交换代数研究

• 批准号: 0856044
• 负责人: Anurag Singh
• 金额: $ 20.79万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-01 至 2014-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0856044&HistoricalAwards=false
• 关键词: Studies; Commutative; Algebra

The PI will work on several problems in commutative algebra that are centered around the homological conjectures, tight closure, and local cohomology. Some of these, such as the homological conjectures, are old problems for which recent advances have provided the hope of a solution; the problems on tight closure and finiteness properties of local cohomology modules are a continuation of the PI's long-term projects. Hochster's monomial conjecture is unresolved for rings that do not contain a field, such as those arising in number theory. Two approaches to this will be pursued: the first is a natural extension of Heitmann's work; another is via local cohomology theory. Brenner and Monsky recently proved that tight closure need not commute with localization. However, it appears likely that weak F-regularity---the property that all ideals of a ring are tightly closed---does localize. This will be approached via the notion of splinter rings. It is also proposed to attack Lyubeznik's conjecture that local cohomology modules of regular rings have finitely many associated prime ideals. This is now known in various cases, but remains unresolved for polynomial rings over the integers.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, the point of view in commutative algebra is to study the ring of polynomial functions on a solution set. Most of the questions that will be investigated may be viewed as questions about the existence of solutions for families of equations, and about the nature of the solution sets. Commutative algebra continues to develop a fascinating interaction with several branches of mathematics, and is becoming an increasingly valuable tool in engineering, coding theory, cryptography, and other applications of strategic interest.

PI将致力于交换代数中的几个问题，这些问题围绕着同调猜想、紧闭包和局部上同调。其中一些是老问题，例如同调猜想，最近的进展为解决这些问题提供了希望；关于局部上同调模的紧闭包和有限性质的问题是PI长期项目的继续。Hochster的单项猜想对于不包含域的环是未解决的，例如数论中出现的那些环。我们将寻求两种方法：第一种是对海特曼工作的自然扩展；另一种是通过局部上同调理论。布伦纳和蒙斯基最近证明，紧闭不需要与本地化通勤。然而，弱F-正则性-环的所有理想都是紧闭的性质-似乎确实是局部化的。这将通过分裂环的概念来实现。反驳了Lyubeznik关于正则环的局部上同调模有有限多个相关素理想的猜想。这在各种情况下都是已知的，但对于整数上的多项式环仍然没有解决。这个项目涉及交换代数中的问题。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，但交换代数的观点是研究解集上的多项式函数环。将被研究的大多数问题可以被看作是关于一族方程的解的存在性以及关于解集的性质的问题。交换代数继续与数学的几个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。