Methods in the Representation Theory of Local Rings

局部环表示论中的方法

• 批准号: 0902119
• 负责人: Graham Leuschke
• 金额: $ 18.11万
• 依托单位: Syracuse University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0902119&HistoricalAwards=false
• 关键词: Methods; Representation; Theory; Local; Rings

This award supports a program for advancement in the representation theory of commutative Noetherian local rings. The study of maximal Cohen--Macaulay modules over such rings has grown out of the theory of representations of Artin algebras, which has developed sophisticated theoretical techniques for classifying and characterizing module theories of non-commutative Artinian rings. Classical problems from the non-commutative theory can often be stated directly in the commutative higher-dimensional framework, and this framework comes equipped with its own unique problems, specialized machinery, and deep, powerful connections with algebraic geometry. The interplay between the two areas has been exceptionally productive for both. However, many of the most powerful tools of the Artinian arsenal are essentially noncommutative, in that they produce noncommutative rings even from commutative input, restricting their use in commutative algebra. Recent theoretical advances in non-commutative algebraic geometry have begun to provide methods, including the notion of a non-commutative resolution of singularities, similar to those provided by classical algebraic geometry. Describing these non-commutative resolutions in terms of tools from the Artinian theory will allow the powerful tools of that area to be brought to bear on fundamental conjectures about the maximal Cohen--Macaulay modules over Cohen--Macaulay local rings.This project lies at the intersection of the areas of commutative and non-commutative algebra, combinatorics, and algebraic geometry. The classical aim of commutative algebra is to describe the solution sets of systems of polynomial equations by associating to them algebraic gadgets known as rings. Non-commutative algebra, on the other hand, has developed theory for associating rings to directed graphs, also called quivers. The synergy among these topics has enriched all four subjects, and has led to applications in such varied fields as robotics, statistics, cryptography, and particularly theoretical physics, where non-commutative methods are central to such subjects as quantum mechanics, string theory and the study of fundamental particles.

该奖项支持促进交换诺特局部环表示理论发展的计划。 对此类环上的最大 Cohen-Macaulay 模的研究源于 Artin 代数的表示理论，该理论开发了用于分类和表征非交换 Artinian 环的模理论的复杂理论技术。 非交换理论中的经典问题通常可以直接在可交换的高维框架中表述，并且该框架配备了自己独特的问题、专门的机制以及与代数几何的深刻而强大的联系。 这两个领域之间的相互作用对双方来说都非常富有成效。然而，Artinian 武器库中许多最强大的工具本质上都是非交换的，因为它们甚至从交换输入中产生非交换环，限制了它们在交换代数中的使用。 非交换代数几何的最新理论进展已经开始提供方法，包括奇点的非交换解决的概念，类似于经典代数几何提供的方法。用 Artinian 理论中的工具描述这些非交换性解决方案将允许该领域的强大工具应用于关于 Cohen-Macaulay 局部环上的最大 Cohen-Macaulay 模的基本猜想。该项目位于交换和非交换代数、组合学和代数几何领域的交叉点。 交换代数的经典目标是通过与称为环的代数小工具相关联来描述多项式方程组的解集。 另一方面，非交换代数发展了将环与有向图（也称为箭袋）相关联的理论。 这些主题之间的协同作用丰富了所有四个学科，并导致了机器人学、统计学、密码学等不同领域的应用，特别是理论物理学，其中非交换方法是量子力学、弦理论和量子力学等学科的核心。基本粒子的研究。