Local cohomology, tight closure, and related questions

局部上同调、紧闭性及相关问题

• 批准号: 1162585
• 负责人: Anurag Singh
• 金额: $ 25.5万
• 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: 2012
• 资助国家: 美国
• 起止时间: 2012-06-01 至 2016-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1162585&HistoricalAwards=false
• 关键词: Local; cohomology; tight; closure; related

The PI will work on several problems in commutative algebra that are centered around local cohomology theory, the homological conjectures, and the theory of tight closure. The PI will continue work on the conjecture that local cohomology modules of regular rings have finitely many associated prime ideals. This is known in various cases due to the work of Huneke-Sharp and Lyubeznik, but remains unresolved for polynomial rings over the integers. Some critical cases have been settled recently in joint work with Lyubeznik and Walther; these in turn, lead to new vanishing theorems for local cohomology modules. Hochster's monomial conjecture remains unresolved for rings that do not contain a field, such as those arising in number theory; the PI will pursue an approach to this via local cohomology. The proposed work in tight closure includes investigating the question whether weak $F$-regularity---the property that all ideals of a ring are tightly closed---is preserved under localization; this will be attacked via the notion of splinter rings. 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 are questions 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 areas of strategic interest.

PI将研究交换代数中的几个问题，这些问题围绕着局部上同调理论，同调代数和紧闭包理论。PI将继续研究正则环的局部上同调模有1000个相关联的素理想的猜想。由于Huneke-Sharp和Lyubeznik的工作，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。一些关键的情况下已经解决最近在联合工作与Lyubeznik和瓦尔特;这些反过来，导致新的消失定理的局部上同调模。对于不包含域的环，例如数论中出现的环，Hochster的单项猜想仍然没有得到解决; PI将通过局部上同调来寻求解决方法。拟议的工作在紧封闭包括调查的问题是否弱F$-正则性--财产，所有理想的一个环是紧封闭的-是保存在本地化;这将是攻击通过的概念分裂环。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：代数几何专注于多项式方程的解集的几何，而交换代数的观点是研究解集上多项式函数的环。大多数的问题，将被调查的问题的性质的解决方案集。交换代数继续与数学的几个分支发展着迷人的相互作用，并正在成为工程，编码理论，密码学和其他战略利益领域越来越有价值的工具。