Homological Methods and Ideal Closures in Commutative Algebra

交换代数中的同调方法和理想闭包

• 批准号: 0244405
• 负责人: Craig Huneke
• 金额: $ 30.57万
• 依托单位: University of Kansas Center for Research Inc
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-05-15 至 2009-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0244405&HistoricalAwards=false
• 关键词: Homological; Methods; Ideal; Closures; Commutative

DMS-0244405Huneke, Craig L.Abstract:This project is in the field of commutative algebra, especially in thehomological theory of Noetherian rings. The lever being used to study thisarea is in developing the understanding of maximal Cohen-Macaulay modulesand extensions of modules. In addition, the project also studies several areas central to commutative algebra, including the tight closure ofideals, integral closures of ideals, the core of an ideal, evolutions,symbolic powers, and rational singularities. The focus of this proposalis on several open conjectures, especially regarding rings of finite andcountable Cohen-Macaulay type, conjectures of Auslander on the vanishingof extension modules, and questions of Schreyer, among others. Amongthe methods being used is the study of maximal Cohen-Macaulaymodules not only through study of infinite resolutions, but with techniquescoming from tight closure and reduction to characteristic p. The methods arein part classical methods as well as those being developed by the proposer.Commutative algebra arose from the 19th century study of polynomial equations inmany variables, and their solutions. The relationship between polynomial equations andgeometry goes back at least to Descartes and the idea of coordinatizing the plane.Commutative algebra studies the solutions of such polynomial or power series equationsby forming an algebraic object, called a ring, which consists of the 'generic' solutions.The algebraic properties of these generic solutions then give insight into thegeometric and algebraic nature of the equations. An important technique in thisfield has been to study such equations by reducing the coefficients modulo prime numbersfor all large primes. A particular example of this has been the explosive developmentof the theory of tight closure over the last fifteen years. Commutative algebra combinestechniques from a number of other areas including combinatorics, topology, and analysis.

DMS-0244405Huneke，Craig L.Abstract：该项目位于通勤代数的领域，尤其是在Noetherian Rings的本学理论中。用于研究本区的杠杆是在发展模块的最大Cohen-Macaulay模块和扩展的理解。 此外，该项目还研究了交换代数的主要领域，包括密闭的封闭，理想的整体封闭，理想的核心，演变，符号力量和理性的奇异性。该提议的重点是几个开放式猜想，尤其是关于有限且可分配的Cohen-Macaulay类型的环，Auslander在消失的扩展模块上的猜想以及Schreyer等问题。 在所使用的方法中，不仅通过研究无限分辨率，而且从紧密闭合和还原到特征性p的技术来研究最大Cohen-Macaulaymodules。 这些方法是部分经典方法，也是提议者开发的方法。共同代数来自19世纪的多项式方程Inmany变量及其解决方案的研究。多项式方程式和几何之间的关系至少可以追溯到笛卡尔和坐标的概念。代数研究这种多项式或功率序列方程的解决方案是通过形成一个代数对象，称为一个环，称为“通用溶液”的戒指。然后，这些通用的溶液和这些通用的溶液构成了深处的解决方案，并深入探讨了深处的溶液，以实现深处的解决方案。方程式。 Thisfield的一种重要技术是通过减少所有大型素数的系数模型数字来研究此类方程。一个特殊的例子是过去15年来紧密封闭理论的爆炸性发展。来自其他许多领域（包括组合学，拓扑和分析）的交换代数组合。