Homological Behavior of Modules over Commutative Local Rings

交换本地环上模的同调行为

• 批准号: 1101131
• 负责人: Liana Sega
• 金额: $ 8.58万
• 依托单位: University of Missouri-Kansas City
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-01 至 2014-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101131&HistoricalAwards=false
• 关键词: Homological; Behavior; Modules; over; Commutative

This project aims to contribute towards understanding the homological behavior of modules over commutative rings. Modules are the objects on which a given ring acts: geometrically, they correspond to bundles and, more generally, sheaves on a space. Sega will study certain classes of commutative local rings which extend in several directions the well-studied class of complete intersection rings, and will investigate to what extent the known properties of complete intersections can be translated to a larger context. Earlier work of the PI, in collaboration with D. Jorgensen, has shown that vanishing of cohomology over non-complete intersection rings can have rather non-rigid behavior. The accent in the proposed work is shifted towards finding a common bridge between complete intersections and other good classes of rings. Attention will be paid to vanishing of (co)homology, properties of the minimal free resolutions of modules, Betti numbers and other homologically defined invariants. An important aspect is an effort to show that the classes of rings and the modules considered occur abundantly in a variety of situations of geometric interest.Commutative algebra allows one to encode information regarding solution sets of polynomial equations into objects such as rings and modules, and further understand their structure, using algebraic tools and a recent infusion of techniques from other fields that deal with physical spaces, such as topology. This method of encoding information and its further study is relevant to almost any area of mathematics, and to theoretical physics. In particular, the classes of rings and the properties considered in this project are relevant in the field of algebraic geometry, and some of the methods and outcomes make connections with the field of non-commutative algebra.

这个项目的目的是对理解交换环上模的同调行为做出贡献。模是给定环作用于其上的对象：在几何上，它们对应于丛，更一般地，对应于空间上的层。 世嘉将研究某些类的交换局部环，这些环在几个方向上扩展了研究充分的完全相交环类，并将研究在多大程度上可以将完全相交的已知属性转化为更大的上下文。PI的早期工作，与D。Jorgensen证明了非完全交环上的上同调消失可以具有相当非刚性的行为。口音在拟议的工作转向寻找一个共同的桥梁之间的完整的交叉点和其他良好的类环。注意将支付消失的（上）同调，性质的最小自由决议的模块，贝蒂数和其他同调定义的不变量。一个重要的方面是努力表明，类的环和考虑的模块大量出现在各种情况下的几何兴趣。交换代数允许一个编码的信息有关的解决方案集的多项式方程到对象，如环和模块，并进一步了解他们的结构，使用代数工具和最近注入的技术，从其他领域处理物理空间，例如拓扑。这种编码信息的方法及其进一步的研究与几乎所有的数学领域和理论物理学有关。特别是，在这个项目中考虑的环类和性质在代数几何领域是相关的，一些方法和结果与非交换代数领域建立了联系。