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Problems in Commutative Algebra: Free Resolutions, Multiplicities, and Blowup Rings

交换代数问题：自由解析、重数和爆炸环

基本信息

批准号：
1503605
负责人：
Bernd Ulrich
金额：
$ 18.5万
依托单位：
Purdue University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2015
资助国家：
美国
起止时间：
2015-06-01 至 2020-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1503605&HistoricalAwards=false
关键词：
Problems Commutative Algebra Free Resolutions

项目摘要

This project is in commutative algebra, an area of pure mathematics with close ties to algebraic geometry. Commutative algebra deals, in a wider sense, with the qualitative study of systems of polynomial equations in several unknowns. Solutions of such systems play an important role in many areas of science and engineering, hence the proposed research has applied aspects as well. One of its objectives is to devise numerical criteria for when the solution sets of different systems of equations are "alike," another one aims at constructing systems of equations that have a given geometric object, like a curve or a surface, as its solution set. This second problem is also relevant for applications in geometric modeling and computer-aided design. A third goal of the project is to further develop a fundamental tool in algebra, called "free resolutions." Free resolutions are a method to study complex algebraic structures by means of a possibly infinite sequence of simpler objects, namely, matrices.In more technical terms, the topics of this project are equisingularity theory, the implicitization problem for Rees algebras, algebraic properties of multiplier ideals, the extension to local rings of results and techniques that are inherent to the graded setting. To be more specific, a general goal in equisingularity theory is to devise fiberwise numerical criteria for when a family of analytic spaces is topologically trivial, is Whitney equisingular, or satisfies other equisingularity conditions. With Kleiman and Validashti the PI proved that a family of isolated singularities is Whitney equisingular, and hence topologically trivial, if a newly defined generalized multiplicity, the epsilon multiplicity, is constant across the family. Now the PI plans to examine which stronger equisingularity conditions the constancy of the epsilon multiplicity implies. A related, fundamental problem is to understand singularities whose Jacobian module has epsilon multiplicity zero. Equisingularity conditions are closely related to numerical criteria for integral dependence of modules, and the PI wishes to devise such a criterion based on intersection numbers. A classical problem in elimination theory is to determine the defining ideals of Rees algebras, which gives, in particular, the implicit equations of graphs and images of rational maps between projective spaces. The PI plans to work on this problem for regular maps parametrizing a surface or, more generally, for rational maps with codimension three base locus. In recent joint work with Corso, Huneke, and Polini, the PI introduced the notion of distance in free resolutions over local rings and used it to study integral closures of ideals. Now the PI proposes this notion as a substitute for the shifts in graded free resolutions, in order to extend known results from the graded to the local case -- such as criteria for the Cohen-Macaulay and Gorenstein properties of associated graded rings and bounds for the Loewy length of modules having finite projective dimension. Another of the PI's goals is tostudy algebraic properties of (Mather-Jacobian) multiplier ideals on singular varieties and to determine which ideals can be realized as multiplier ideals.

这个项目是在交换代数，纯数学领域与代数几何密切相关。交换代数涉及，在更广泛的意义上，与定性研究系统的多项式方程在几个未知数。这类系统的解在科学和工程的许多领域中起着重要的作用，因此所提出的研究也具有应用方面。它的一个目标是设计不同方程组的解集“相似”的数值准则，另一个目标是构造具有给定几何对象（如曲线或曲面）作为其解集的方程组。第二个问题也与几何建模和计算机辅助设计中的应用相关。该项目的第三个目标是进一步开发一个基本的代数工具，称为“自由决议”。“自由分解是一种研究复杂代数结构的方法，通过一个可能无限序列的更简单的对象，即矩阵。在更技术性的术语，这个项目的主题是equisingularity理论，Rees代数的隐式化问题，乘子理想的代数性质，结果的局部环的扩展和技术是固有的分次设置。更具体地说，等奇异性理论的一个一般目标是设计出一个解析空间族是拓扑平凡的，是惠特尼等奇异的，或者满足其他等奇异性条件的数值判据。PI与Kleiman和Validashti一起证明了一个孤立奇点族是Whitney等奇异的，因此拓扑平凡，如果一个新定义的广义重数，即多重数，在整个族中是常数。现在，PI计划研究哪些更强的等奇异性条件意味着多重性的恒定性。一个相关的基本问题是理解其雅可比模的重数为零的奇点。等奇异性条件与模块积分依赖性的数值准则密切相关，PI希望基于交集数设计这样的准则。消除论中的一个经典问题是确定Rees代数的定义理想，特别是给出了射影空间之间的有理映射的图和像的隐方程。PI计划针对参数化曲面的正则映射，或者更一般地，针对具有余维三个基轨迹的有理映射，来解决这个问题。在最近与Corso、Huneke和Polini的合作中，PI在局部环上的自由解析中引入了距离的概念，并用它来研究理想的积分闭包。现在PI提出了这个概念作为替代的移位分级自由决议，以延长已知的结果从分级的地方情况下-如标准的科恩-麦考利和Gorenstein性质的相关分级环和界限的Loewy长度的模块具有有限的投影维数。PI的另一个目标是研究奇异簇上的（Mather-Jacobian）乘子理想的代数性质，并确定哪些理想可以实现为乘子理想。