Homological Behavior of Modules over Commutative Local Rings

本项目旨在帮助理解交换环上模的同调行为。模块是给定环作用的对象：在几何上，它们对应于空间上的束，更一般地说是捆。Sega将研究某些交换局部环的类别，它们在几个方向上扩展了已被充分研究的完全相交环的类别，并将研究完全相交的已知性质在多大程度上可以转化为更大的背景。PI的早期工作，与D. Jorgensen合作，已经证明了非完全交环上同调的消失可以具有相当非刚性的行为。所提出的工作的重点是在完整的交叉点和其他良好的环类之间寻找一个共同的桥梁。注意（co）同调的消失，模的最小自由分辨率的性质，Betti数和其他同调定义的不变量。一个重要的方面是努力表明环类和所考虑的模在各种几何兴趣的情况下大量出现。交换代数允许人们将多项式方程的解集信息编码到环和模块等对象中，并使用代数工具和最近从处理物理空间（如拓扑）的其他领域引入的技术，进一步理解它们的结构。这种编码信息的方法及其进一步的研究几乎与数学和理论物理的任何领域有关。特别地，本课题所考虑的环的种类和性质与代数几何领域有关，其中一些方法和结果与非交换代数领域有联系。

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.