On Several Homological Problems in Commutative Algebra

关于交换代数中的几个同调问题

• 批准号: 9970263
• 负责人: Sankar Dutta
• 金额: $ 5万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-05-15 至 2003-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970263&HistoricalAwards=false
• 关键词: Several; Homological; Problems; Commutative; Algebra

The main goal of this project is to study the following problems incommutative algebra: 1) The Canonical Element Conjecture of Hochster,2) Serr's Conjecture on Intersection Multiplicity and the Chow group problem,3) the study of asymptotic behavior of homologies in positive characteristic and4) the Strong Intersection Conjecture due to Peskine and Szpiro. Each of these isa longstanding open problem and occupies a central position in the areas ofcommutative algebra and algebraic geometry.Commutative algebra and algebraic geometry may be thought of as studying solutions of many equations in many unknowns when, typically, the solution is not unique. The set of solutions can then be viewed geometrically, but one can use instead encode all the pertinent information about the equations in abstract algebraic objects called commutative rings. The two points of view interact in beautiful and productive ways. A breakthrough method in commutative algebra that has had explosive development in this decade is the theory of tight closure. Part of the underlying idea is to consider the system of equations, after sutiable modification, or the ring, modulo many different prime integers. This is a powerful technique that has solved many problems, while opening up vast new areas for research.

本项目的主要目标是研究交换代数中的下列问题：1）Hochster的标准元猜想，2）Serr关于交重数的猜想和Chow群问题，3）正特征同调的渐近性研究，4）Peskine和Szpiro的强交猜想。这些问题伊萨长期存在的问题，在交换代数和代数几何领域占据着中心地位。交换代数和代数几何可以被认为是研究许多未知数的许多方程的解，通常情况下，解不是唯一的。解的集合可以被几何地观察，但是我们可以使用将所有关于方程的相关信息编码在抽象的代数对象中，称为交换环。这两种观点以美丽而富有成效的方式相互作用。交换代数的一个突破性方法是紧闭包理论，它在这十年中得到了爆炸性的发展。部分基本思想是考虑方程组，经过适当的修改，或环，模许多不同的素数。这是一种强大的技术，解决了许多问题，同时开辟了广阔的新研究领域。