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.

联系，或联系，一直被用作对理想和投射变种进行分类的工具。寻找两个理想属于同一联结类的充要条件是理论中的中心问题，但也是一个广泛开放的问题。研究人员为零维单项理想给出了这个问题的答案。连杆理论的局部代数分支和射影几何分支是并行进行的，它们之间只有很少的含义。提出者计划通过研究本地链接和同质链接之间的差异来研究这种微妙的相互作用。由于经典连锁产生的等值关系对于某些目的可能过于严格，更具包容性的Gorenstein连锁概念已成为一种受欢迎的替代方案。研究者有一个更广泛的目标，即证明在一个给定的正则局部环中，所有具有固定余维的Cohen-Macaulay理想都属于同一个Gorenstein链类。提出者计划利用在j-多重性主题上的最新进展来解决等价性理论中的一些长期存在的问题。均衡性理论致力于找到解析集族成员彼此“等价”的数值条件。调查者试图通过证明以j-多重性为基础的一般“积分依赖的专门化原则”来提供这样的标准。研究人员还提出了模的j-重数的交集理论表达式，该表达式将给出计算该重数的递归公式。理想的核心是次理想，它编码关于理想的所有可能约化的信息，同时与Briancon-Skoda型定理和Kawamata关于线丛的截面的猜想密切相关。研究人员希望探索核心和乘子理想之间的联系，这是复代数几何中的一个概念。他预计，对这种相互作用的更好理解将为这两个概念带来新的曙光，并可能导致对单项情况下核心的组合描述。在设计出计算零维单项式理想的核的算法之后，研究者现在希望找到任意维单项式理想的这种算法。已知的关于理想核的结果需要对理想的生成元的局部个数以及剩余或深度条件进行限制，这些条件都是由任何零维理想所满足的。作者建议将这一理论发展到满足这些条件的理想类之外，并将单项式情形作为测试基础。这样的系统出现在代数几何中，但也在数学以外的许多应用中。在过去的二十年里，交换代数学家对计算方面越来越感兴趣，从而建立了与计算机代数、机器人学、密码学、编码理论、统计学和生物学等应用领域的联系。这位研究人员的研究也有很强的计算成分，他在均衡性理论方面的工作尤其包括具体的几何应用。