Uniformity in Commutative Algebra

交换代数的一致性

• 批准号: 1460638
• 负责人: Craig Huneke
• 金额: $ 24.6万
• 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: 2015
• 资助国家: 美国
• 起止时间: 2015-05-01 至 2019-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1460638&HistoricalAwards=false
• 关键词: Uniformity; Commutative; Algebra

Commutative algebra and algebraic geometry are the study of the algebraic and geometric properties of systems of non-linear equations. By contrast, linear algebra, a fundamental tool of the sciences, is the study of linear systems of equations. Often linear equations do not suffice to describe complex systems, and higher degree polynomials must be used. Commutative algebras provide models, called rings, of systems of polynomial equations where one can add and multiply. The rings studied in this proposal usually come from a system of polynomial equations. 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 throughout this project. This project has a substantial educational component. The PI has served as PhD advisor and postdoc mentor to many researchers, and will continue this level of activity.The PI will investigate several aspects of commutative algebras which all fall under the rubric of uniformity. Three fundamental questions on symbolic powers of ideals and their relationship to regular powers are proposed. Significant partial results have been obtained. The PI will also investigate a fundamental question of Stillman concerning the complexity of equations. A variety of other problems will be studied, including classification of Golod rings, existence of rigid modules, the structure of iterated socles, and the development of Frobenius Betti numbers and their properties. Several of the projects are intended for collaboration with graduate students, postdocs, and young researchers.

交换代数和代数几何形状是对非线性方程系统的代数和几何特性的研究。相比之下，科学的基本工具线性代数是对方程式的线性系统的研究。通常，线性方程不足以描述复杂的系统，并且必须使用更高的多项式。换向代数提供了多项式方程系统的模型，其中一个可以添加和繁殖的模型。该提案中研究的环通常来自多项式方程系统。 通过研究该模型的属性，可以更好地了解原始方程系统。有两种主要方法。一种是理解此类环上模块的理论。模块是方程保持的空间的一种特殊表示。研究这些模型一直是研究方程式的一种极其有效的方法。另一个主要的技术方法是研究环中相同的基本方程，这些方程式降低了模块数量。在这样的系统中，算术变得更加容易。例如，Modulo 2意味着每个偶数数字都被认为是0，所有奇数为1。这在整个项目中都有许多深刻的优势。该项目具有大量的教育组成部分。 PI曾担任许多研究人员的博士顾问和博士后导师，并将继续进行这一水平的活动。PI将研究通勤代数的几个方面，这些方面都属于统一性的标题。 提出了有关理想象征能力及其与常规权力的关系的三个基本问题。已经获得了重要的部分结果。 PI还将研究Stillman关于方程式复杂性的基本问题。 还将研究其他各种问题，包括戈洛德环的分类，刚性模块的存在，迭代的SOCLES结构以及Frobenius Betti数字及其特性的发展。其中一些项目旨在与研究生，博士后和年轻研究人员合作。