Commutative Algebra: Set-Theoretic Complete Intersections, Local Cohomology, Free Resolutions, and Rees Rings

交换代数：集合论完全交集、局部上同调、自由解析和里斯环

• 批准号: 1601865
• 负责人: Claudia Polini
• 金额: $ 26.2万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-09-15 至 2020-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1601865&HistoricalAwards=false
• 关键词: Commutative; Algebra; Set; Theoretic; Complete

This research project concerns commutative algebra with a view towards algebraic geometry. Often, the most effective method to solve a problem is to create a mathematical model. Frequently, such models involve unknown parameters related by several equations that are often impossible to solve exactly. Commutative algebra is the qualitative study of such systems of polynomial equations. Its applications are far-reaching and include diverse fields such as computer science, cryptography, coding theory, robotics, pattern recognition, and theoretical physics. One can study the sets of solutions of these systems either geometrically or algebraically. This project deals with the algebraic approach. One of the goals of the project is to understand the smallest number of equations needed to describe a geometric object like a curve or surface. Another is to construct the system of equations defining a given geometric object. A third goal is the study of the relations among a given set of polynomial equations, the relations among the relations, and so on. The project involves undergraduate students, graduate students, and postdoctoral fellows in the research. This research project has three main themes. The first is to develop criteria for a variety in projective space to be a set-theoretic complete intersection. A fundamental tool to solve this problem is the theory of local cohomology modules. Local cohomology modules encode the algebraic and topological structure of an algebraic variety. As modules over the ring, local cohomology modules are huge (neither finitely generated nor Artinian), hence intractable. However, as modules over the Weil algebra they can be filtered by simple objects and become manageable. Hence an important task is to understand the D-module structure of local cohomology modules. The second theme is the study of local rings using the notion of distance. This notion was introduced in recent work of the investigator and collaborators to understand the integral closure of ideals. The idea is to use distance as a substitute for shifts in homogeneous resolutions and for the Castelnuovo-Mumford regularity of graded modules. The main goal is to prove general results that are inspired by statements in the graded case. The last theme is to study the implicit equations defining the graph and the image of rational maps between projective spaces. This is a classical problem in elimination theory, commutative algebra, and algebraic geometry with applications, for instance, in geometric modeling.

本研究计画是以代数几何的观点来探讨交换代数。通常，解决问题最有效的方法是建立数学模型。通常，这样的模型涉及未知参数相关的几个方程，往往是不可能精确求解。交换代数是对这种多项式方程组的定性研究。它的应用是深远的，包括不同的领域，如计算机科学，密码学，编码理论，机器人，模式识别和理论物理。人们可以用几何方法或代数方法研究这些系统的解集。这个项目涉及代数方法。该项目的目标之一是了解描述曲线或曲面等几何对象所需的最小数量的方程。另一种是构造定义给定几何对象的方程组。第三个目标是研究给定的多项式方程组之间的关系，关系之间的关系等。该项目涉及本科生，研究生和博士后研究员。这个研究项目有三个主题。第一个是发展标准的各种在射影空间是一个集论完全相交。局部上同调模理论是解决这一问题的基本工具。局部上同调模编码了代数簇的代数和拓扑结构。作为环上的模，局部上同调模是巨大的（既不是代数生成的，也不是阿廷的），因此是难以处理的。然而，作为Weil代数上的模块，它们可以被简单的对象过滤，变得易于管理。因此，一个重要的任务是了解局部上同调模的D-模结构。第二个主题是使用距离的概念研究局部环。这个概念在最近的研究者和合作者的工作中引入，以理解理想的积分闭包。我们的想法是使用距离作为替代齐次分辨率的移位和分次模的Castelnuovo-Mumford正则性。主要目标是证明一般的结果，启发语句的分级情况下。最后一个主题是研究定义射影空间之间有理映射的图形和图像的隐式方程。这是一个经典的问题，在消除理论，交换代数，代数几何与应用，例如，在几何建模。