Problems in Commutative Algebra

交换代数问题

• 批准号: 0098654
• 负责人: Craig Huneke
• 金额: $ 31万
• 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: 2001
• 资助国家: 美国
• 起止时间: 2001-06-01 至 2007-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0098654&HistoricalAwards=false
• 关键词: Problems; Commutative; Algebra

This award supports research in commutative algebra. The investigator together with students and collaborators will study problems in four connected areas: the theory of tight closure, rings of F-finite Cohen-Macaulay type, the theory of integrally closed ideals with applications to evolutions, and the study of infinite free resolutions. At its core, tight closure theory rests on reduction to positive characteristic. The principal investigator will study two main problems using tight closure. The first deals with recent work concerning the behavior of symbolic powers in regular local rings, and the second concerns the question of whether tight closure commutes with localization. To study rings of finite (or F-finite) Cohen-Macaulay type, this proposal places the study in the broader context of understanding the decomposition of purely inseparable extensions of a fixed Cohen-Macaulay domain into direct sums of indecomposable Cohen-Macaulay modules. Integral closures of ideals are a basic object in commutative algebra. This proposal concentrates on questions pertaining to the existence of evolutions over the complex numbers. The study of infinite free resolutions and vanishing to Tors is the focus of the last part of the proposal, especially over Gorenstein rings of dimension zero.Commutative algebra studies the relationship between algebraic equations, such as polynomial equations, and geometry. This idea goes back to Descartes and the idea of coordinatizing the plane, and has proved to be a powerful tool. A wide range of problems can be put into the context of solving systems of equations. For example, linear algebra studies systems of linear (degree one) equations. Commutative algebra studies the solutions of polynomial or power series equations of higher order by forming an algebraic object consisting of the 'generic' solutions. The algebraic properties of these generic solutions then give insight into the geometric and algebraic nature of the equations.

该奖项支持交换代数的研究。 调查员与学生和合作者一起将研究四个相关领域的问题：紧封闭理论，F-有限科恩-麦考利型环，整体封闭理想理论及其在演化中的应用，以及无限自由决议的研究。 紧闭理论的核心在于还原为正特征。 主要研究者将使用紧密闭合研究两个主要问题。 第一个涉及最近的工作有关的行为，象征性的权力，在经常的地方环，第二个关注的问题，是否紧封闭与本地化。 为了研究有限（或F-有限）Cohen-Macaulay型的环，这个建议将研究放在更广泛的背景下，理解固定Cohen-Macaulay域的纯不可分扩张分解为不可分解的Cohen-Macaulay模的直和。 理想的整闭包是交换代数中的一个基本对象。 这个建议集中在与复数上的演化的存在有关的问题上。 研究无限自由决议和消失的托尔是重点的最后一部分的建议，特别是在Gorenstein环的维数为零。交换代数研究代数方程之间的关系，如多项式方程，和几何。 这个想法可以追溯到笛卡尔和协调平面的想法，并已被证明是一个强大的工具。 许多问题都可以放在解方程组的范围内。例如，线性代数研究线性（一次）方程组。 交换代数通过形成由“一般”解组成的代数对象来研究高阶多项式或幂级数方程的解。 这些通用的解决方案的代数性质，然后洞察到几何和代数性质的方程。