On Several Homological Problems in Commutative Algebra

本课题主要研究交换代数中的几个问题：1)Hochster的正则元猜想，2)Serr关于交多重性的猜想和Chow群问题，3)同调正特征的渐近行为研究，4)基于Peskine和Szpiro的强交猜想。这些都是长期存在的开放问题，在交换代数和代数几何领域占据中心位置。交换代数和代数几何可以被认为是研究许多方程在许多未知数中的解，通常情况下，解不是唯一的。然后可以从几何角度来看待解的集合，但是我们可以将所有关于方程的相关信息编码为抽象的代数对象，称为交换环。这两种观点以美丽而富有成效的方式相互作用。紧闭理论是交换代数中近十年来取得爆炸性发展的一种突破性方法。部分潜在的思想是考虑方程系统，经过适当的修改，或环，模许多不同的素数整数。这是一项强大的技术，解决了许多问题，同时开辟了广阔的新研究领域。

The main goal of this project is to study the following problems incommutative algebra: 1) The Canonical Element Conjecture of Hochster,2) Serr's Conjecture on Intersection Multiplicity and the Chow group problem,3) the study of asymptotic behavior of homologies in positive characteristic and4) the Strong Intersection Conjecture due to Peskine and Szpiro. Each of these isa longstanding open problem and occupies a central position in the areas ofcommutative algebra and algebraic geometry.Commutative algebra and algebraic geometry may be thought of as studying solutions of many equations in many unknowns when, typically, the solution is not unique. The set of solutions can then be viewed geometrically, but one can use instead encode all the pertinent information about the equations in abstract algebraic objects called commutative rings. The two points of view interact in beautiful and productive ways. A breakthrough method in commutative algebra that has had explosive development in this decade is the theory of tight closure. Part of the underlying idea is to consider the system of equations, after sutiable modification, or the ring, modulo many different prime integers. This is a powerful technique that has solved many problems, while opening up vast new areas for research.