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

这个研究项目涉及交换代数与代数几何的观点。通常，解决问题最有效的方法是建立数学模型。通常，这种模型涉及由几个方程相关的未知参数，这些方程往往无法精确求解。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，包括计算机科学、密码学、编码理论、机器人、模式识别和理论物理等多个领域。人们可以用几何或代数的方法来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理解描述几何物体（如曲线或曲面）所需的最少数量的方程。另一种是构造定义给定几何对象的方程组。第三个目标是研究一组给定多项式方程之间的关系，关系之间的关系，等等。该项目涉及本科生、研究生和博士后研究人员。这个研究项目有三个主题。第一个是发展的准则，在射影空间的变化是一个集合论的完全交。解决这个问题的一个基本工具是局部上同模理论。局部上同调模对代数变体的代数结构和拓扑结构进行编码。作为环上的模，局部上同模是巨大的（既不是有限生成的，也不是无穷大的），因此难以处理。然而，作为Weil代数上的模块，它们可以被简单对象过滤并变得易于管理。因此，了解局部上同模的d模结构是一个重要的任务。第二个主题是用距离的概念研究局部环。这个概念是研究者和合作者在最近的工作中引入的，以理解理想的整体封闭。这个想法是使用距离作为齐次分辨率和分级模块的Castelnuovo-Mumford规则的位移的替代品。主要目标是证明由评分案例中的陈述所启发的一般结果。最后一个主题是研究定义投影空间间有理映射的图和象的隐式方程。这是一个经典的问题，在消去理论，交换代数和代数几何的应用，例如，在几何建模。

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.