Tight Closure, Local Cohomology, and Related Questions

紧闭、局部上同调及相关问题

• 批准号: 0600819
• 负责人: Anurag Singh
• 金额: $ 12万
• 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: 2006
• 资助国家: 美国
• 起止时间: 2006-08-01 至 2009-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0600819&HistoricalAwards=false
• 关键词: Tight; Closure; Local; Cohomology; Related

The proposed research problems stem from long-standing questions and conjectures in commutative algebra. These are related to the tight closure theory of Hochster and Huneke, to the homological conjectures, and to the theory of local cohomology. The PI will pursue an approach to Hochster's monomial conjecture which lies at the intersection of these three topics. This conjecture is unresolved for rings which do not contain a field, such as those which arise in number theory. The proposed approach involves annihilating the elements of obstruction local cohomology modules by elements of arbitrarily low valuation. This idea has proved remarkably strong in the work of Heitmann, where he settled the monomial conjecture for rings of dimension up to three. Obtaining a description of such annihilators of local cohomology is a vast program, and the proposed research will focus on some concrete initial cases. In joint work with Uli Walther, the PI will work on Lyubeznik's conjecture that local cohomology modules of regular rings have finitely many associated prime ideals. This is now known in various cases due to the work of Huneke-Sharp and Lyubeznik, but remains unresolved for polynomial rings over the integers.Commutative algebra is a field closely related to algebraic geometry: while algebraic geometry focuses on the geometry of solutions sets of polynomial equations, in commutative algebra the main objects of study are functions on these solution sets. Most of the questions which will be investigated in the proposed research are 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.

拟议的研究问题源于可交换代数的长期问题和猜想。这些与Hochster和Huneke的紧密封闭理论有关，与同源猜想以及局部协同理论有关。 PI将追求一种方法，以实现这三个主题的交汇处。对于不包含一个磁场的环，例如数字理论中出现的环，该猜想尚未解决。提出的方法涉及通过任意低估的要素消灭障碍物局部共同体模块的要素。事实证明，这个想法在Heitmann的工作中非常强烈，在那里他解决了多达三个尺寸的单一猜想。获取对当地共同学的这种歼灭者的描述是一项庞大的计划，拟议的研究将重点放在一些具体的初始案例上。在与Uli Walther的联合合作中，PI将在Lyubeznik的猜想中工作，即常规环的当地同居模块具有有限的许多相关理想。 This is now known in various cases due to the work of Huneke-Sharp and Lyubeznik, but remains unresolved for polynomial rings over the integers.Commutative algebra is a field closely related to algebraic geometry: while algebraic geometry focuses on the geometry of solutions sets of polynomial equations, in commutative algebra the main objects of study are functions on these solution 套。拟议的研究中将研究的大多数问题是有关方程家族的解决方案以及解决方案集合的性质的问题。交换代数继续与数学的多个分支发展出令人着迷的互动，并已成为工程学，编码理论，密码学和其他战略兴趣应用中越来越有价值的工具。