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.

该奖项支持了通勤赛车场当地戒指的代表理论的进步计划。 对此类环上最大的cohen-macaulay模块的研究已从Artin代数的代表理论中发展出来，Artin代数的表示理论已开发出了精致的理论技术，用于对非交流性Artinian Rings的模块进行分类和表征。 非共同理论中的经典问题通常可以直接在交换性的高维框架中说明，并且该框架配备了其独特的问题，专门的机械以及具有代数几何形状的深层，有力的连接。 这两个领域之间的相互作用对两者都非常有生产力。但是，Artinian武器库中许多最强大的工具本质上是非交通性的，因为它们即使是在交换性的输入中也会产生非交互性环，从而限制了它们在交换代数中的使用。 非共同代数几何形状的最新理论进步已经开始提供方法，包括与经典代数几何形状提供的奇异性的非交通分辨率的概念。用Artinian理论的工具来描述这些非交互性决议，将使该领域的强大工具依赖于有关最大的Cohen--Macaulay模块的基本猜想，这些模块对Cohen--Macaulay本地戒指。这些项目位于通勤和非commutative Alge Algebra and andrabra and anbra and anbra andbra andbra and comminator，Commintor，Commintry and Commintor，Commintor，Commintry and commintry and commintry and commintry。 交换代数的经典目的是通过将其与代数小工具相关联来描述多项式方程式系统的解决方案集。 另一方面，非共同的代数开发了将环与定向图相关联的理论，也称为Quivers。 这些主题之间的协同作用丰富了所有四个主题，并导致在机器人技术，统计学，密码学，尤其是理论物理学等各个领域的应用中，在这种领域中，非交通性方法对于诸如量子力学，弦乐理论和基本颗粒的研究之类的主题至关重要。