Studies in Commutative Algebra

交换代数研究

• 批准号: 1259142
• 负责人: Craig Huneke
• 金额: $ 23.01万
• 依托单位: University of Virginia Main Campus
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-09-01 至 2016-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1259142&HistoricalAwards=false
• 关键词: Studies; Commutative; Algebra

Huneke will continue investigating several open questions concerning the theory of Noetherian rings, especially local Noetherian rings or polynomial rings. There are several main thrusts to this work, including investigating singularities via reduction to prime characteristic. Other problems deal with the uniform behavior of symbolic powers on multiple levels, from points in projective space, to primes in local rings, to square-free monomial ideals. A major effort is proposed to understand non-commutative crepant resolutions in broader classes of rings, and to answer several questions concerning the concept of height.The proposed research concerns the theory of commutative rings, which are higher abstract systems where one can add and multiply. The rings arising in this proposal usually comes from a system of polynomial equations. The ring is a type of abstract model where solutions to the equations exist. By studying the properties of this model, one can then better understand the original system of equations. There are two main methods.One is to understand the theory of modules over such rings. Modules are a type of special representation of spaces where the equations hold. Studying these models has been an extremely effective way to study equations. The other main technical method is to study the same basic equations in rings which are reduced modulo a prime number. In such a system, arithmetic becomes easier. For instance modulo 2 means that every even number is thought of as 0, and all odd numbers as 1. This has a number of profound advantages which are used in this proposal.

Huneke将继续调查有关诺特环理论的几个开放性问题，特别是局部诺特环或多项式环。这项工作有几个主要的推动力，包括通过减少到主要特征来研究奇点。其他的问题涉及多个层次上的符号幂的统一行为，从射影空间中的点到局部环中的素数，再到无平方的单项理想。一个主要的努力，提出了理解非交换crepant决议在更广泛的类环，并回答几个问题的概念height.The拟议的研究涉及交换环，这是更高的抽象系统，其中一个可以添加和乘法的理论。在这个提议中产生的环通常来自多项式方程组。环是一种存在方程解的抽象模型。通过研究这个模型的性质，人们可以更好地理解原始方程组。主要有两种方法：一是理解这类环上的模理论;模是方程成立的空间的一种特殊表示。研究这些模型是研究方程的一种非常有效的方法。另一个主要的技术方法是研究环中相同的基本方程，这些方程以素数为模约化。在这样的系统中，算术变得更容易。例如，模2意味着每个偶数都被认为是0，所有奇数都被认为是1。这具有本提案中使用的许多深刻的优点。