Multiplicity theory and related topics in commutative algebra

交换代数中的多重性理论及相关主题

• 批准号: 0901613
• 负责人: Bernd Ulrich
• 金额: $ 30.8万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901613&HistoricalAwards=false
• 关键词: Multiplicity; theory; related; topics; commutative

Linkage, or liaison, has been used as a tool for classifying ideals and projective varieties. Finding necessary and sufficient conditions for two ideals to belong to the same linkage class is the central, though wide open, problem in the theory. The investigator suggests an answer to this question for zero-dimensional monomial ideals. The local-algebraic and the projective-geometric branch of linkage theory have proceeded in parallel, with only few implications known between them. The proposer plans to investigate this subtle interplay by studying the differences between local linkage and homogeneous linkage. Since the equivalence relation generated by classical linkage might be too restrictive for some purposes, the more inclusive notion of Gorenstein linkage has become a welcome alternative. The investigator has the broader goal of showing that in a given regular local ring all Cohen-Macaulay ideals of a fixed codimension belong to the same Gorenstein linkage class. The proposer plans to exploit recent advances on the topic of j-multiplicities to address some long standing problems in equisingularity theory. Equisingularity theory strives to find numerical conditions for when the members of a family of analytic sets are `equivalent' to one another. The investigator seeks to provide such a criterion by proving a general `principle of specialization of integral dependence' based on j-multiplicities. The investigator also proposes an intersection theoretic expression for the j-multiplicity of a module that would yield a recursive formula for computing this multiplicity. The core of an ideal is a subideal that encodes information about all possible reductions of the ideal, while being closely related to Briancon-Skoda type theorems and a conjecture of Kawamata about sections of line bundles. The investigator wishes to explore the connection between cores and multiplier ideals, a notion from complex algebraic geometry. He expects that a better understanding of this interplay will shed new light on both concepts and may lead to a combinatorial description of the core in the monomial case. Having worked out an algorithm for computing the core of zero-dimensional monomial ideals, the investigator now wishes to find such an algorithm for monomial ideals of arbitrary dimension. The known results about cores of ideals require restrictions on the local numbers of generators of the ideal as well as residual or depth conditions, assumptions that are satisfied by any zero-dimensional ideal. The investigator proposes to advance the theory beyond the class of ideals satisfying these conditions, using the monomial case as a testing ground.The proposer works in Commutative Algebra, an area concerned with the qualitative study of systems of polynomial equations in several variables. Such systems arise in algebraic geometry, but also in numerous applications outside of mathematics. Over the past two decades commutative algebraists have become increasingly interested in computational aspects, thereby establishing connections to applied areas such as computer algebra, robotics, cryptography, coding theory, statistics, and biology. This investigator's research too has a strong computational component, and his work on equisingularity theory in particular encompasses concrete geometric applications.

链接或联络已被用作对理想和投射品种进行分类的工具。在理论中，找到两个理想属于同一链接类别的必要条件是属于同一连锁类别的必要条件。研究人员为零维单体理想提出了这个问题的答案。链接理论的局部代数和投射几何分支已经并行进行，它们之间只有很少的含义。提议者计划通过研究局部联系与均匀链接之间的差异来研究这种微妙的相互作用。由于经典链接产生的等效关系对于某些目的而言可能太过限制了，因此戈伦斯坦链接的更具包容性概念已成为一种受欢迎的选择。研究者的更广泛的目标是表明在给定的常规局环中，固定的固定构成的所有科恩 - 麦克劳莱理想属于同一戈伦斯坦链接类别。提议者计划利用有关J-宗教性主题的最新进展，以解决公平理论中一些长期存在的问题。方程式理论旨在找到分析集的家族成员彼此“等效”时找到数值条件。研究人员试图通过证明基于J-义务的一般“整体依赖的专业原则”来提供此类标准。研究者还提出了一个相交的理论表达，以实现模块的J-义务性，该图表将产生用于计算此多重性的递归公式。理想的核心是一个次要的，它编码有关理想的所有可能减少的信息，同时与Briancon-Skoda型定理密切相关，以及关于线束段的Kawamata的猜想。研究人员希望探索核心与乘数理想之间的联系，这是复杂代数几何形状的概念。他希望更好地理解这种相互作用会给这两个概念带来新的启示，并可能导致对单个案例中核心的组合描述。研究者已经制定了一种用于计算零维单体理想核心的算法，研究者现在希望找到一种用于任意维度的单一理想的算法。关于理想核心的已知结果需要限制理想和残留或深度条件的局部发电机的局部数量，即任何零维理想所满足的假设。研究者建议将理论推向满足这些条件的理想等级，并使用单一案例作为测试地面。提议者在交换代数方面起作用，该领域与几个变量中多项式方程系统的定性研究有关。这种系统出现在代数几何形状中，但在数学以外的许多应用中也出现。在过去的二十年中，可交换的代数主义者对计算方面变得越来越感兴趣，从而建立了与计算机代数，机器人技术，密码学，编码理论，统计和生物学等应用领域的联系。这项研究者的研究也具有强大的计算成分，他在公式理论方面的工作特别涵盖了具体的几何应用。