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.

该研究项目涉及交换性代数，以对代数几何形状进行观点。通常，解决问题的最有效方法是创建数学模型。通常，此类模型涉及通过几个方程相关的未知参数，这些参数通常无法准确求解。交换代数是对这种多项式方程系统的定性研究。它的应用是深远的，包括计算机科学，密码学，编码理论，机器人技术，模式识别和理论物理等不同领域。可以通过几何或代数研究这些系统的解决方案集。该项目涉及代数方法。该项目的目标之一是了解描述曲线或表面等几何对象所需的最少数量方程。另一个是构建定义给定几何对象的方程系统。第三个目标是研究给定的一组多项式方程，关系之间的关系等之间的关系。该项目涉及研究生，研究生和博士后研究员。该研究项目具有三个主要主题。首先是制定投影空间中各种的标准，成为一个固定的理论完整交集。解决此问题的基本工具是局部共同学模块的理论。局部同种学模块编码代数品种的代数和拓扑结构。作为环上的模块，局部的共同体学模块是巨大的（既不有限的也不生成Artinian），因此很难。但是，随着Weil代数上的模块可以通过简单对象过滤并变得易于管理。因此，一个重要的任务是了解局部共同体模块的D模块结构。第二个主题是使用距离概念研究本地环。该概念是在研究者和合作者的最新工作中引入的，以了解理想的整体关闭。这个想法是将距离用作均匀分辨率的转移以及坡度模块的Castelnuovo-Mumford的规律性。主要目标是证明在分级案例中启发的一般结果。最后一个主题是研究定义图表的隐式方程和投影空间之间的理性图的图像。这是消除理论，交换代数和代数几何形状的经典问题，例如在几何建模中。