Homological Behavior of Modules over Commutative Local Rings

这个项目的目的是为了理解交换环上模的同调行为。模是给定环作用于的对象：在几何上，它们对应于空间上的丛，更一般地，它们对应于空间上的束。世嘉将研究某些类型的交换局部环，这些环在几个方向上延伸到研究得很好的完全交环的几个方向上，并将调查完全交集的已知性质在多大程度上可以被翻译到更大的背景下。PI与D.Jorgensen合作的早期工作表明，在非完全交环上上同调的消失可以具有相当非刚性的行为。建议中的工作重点是在完全交集和其他好的环类之间找到一座公共桥梁。将注意(余)同调的消失，模的最小自由分解的性质，Betti数和其他同调定义的不变量。一个重要的方面是努力证明所考虑的环类和所考虑的模在各种几何感兴趣的情况下大量出现。交换代数允许人们将关于多项式方程的解集的信息编码到对象中，例如环和模，并使用代数工具和最近从处理物理空间的其他领域(如拓扑)中注入的技术来进一步了解它们的结构。这种对信息进行编码的方法及其进一步研究与几乎所有数学领域和理论物理都有关。特别地，本项目中所考虑的环的类和性质在代数几何领域中是相关的，并且一些方法和结果与非交换代数领域有关。

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.