Prime characteristic methods in commutative algebra

交换代数中的质数特征方法

基本信息

批准号：
EP/I031405/1
负责人：
Mordechai Katzman
金额：
$ 7.28万
依托单位：
University of Sheffield
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2011
资助国家：
英国
起止时间：
2011 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FI031405%2F1
关键词：
Prime characteristic methods commutative algebra

项目摘要

Many theorems in Commutative Algebra can be proved by showing that:(1) if the theorem fails, one can find a counter-example in a ring of prime characteristic p (i.e., a ring which contains the ring of integers modulo a prime number p), and(2) no such counter-example exists in characteristic p.Step (2) above is often much easier to prove than in characteristic zero because of the existence of the Frobenius function f(r) which raises r to the pth power. This functon is an endomorphism of the rings, i.e., it has the property that f(r+s)=f(r)+f(s), and surprisingly, gives a good handle on many problems in characteristic p.A formal method to exploit the existence of these Frobenius function is the theory of Tight Closure which was first developed about 20 years ago to tackle old problems in the field. Since its inception it has been very successful in giving short and elegant solutions to hard old questions. Tight Closure also found surprising applications in other fields, especially in Algebraic Geometry.The essence of this theory is an operation which takes an ideal in a ring of commutative ring of characteristic p and produces another larger ideal with useful properties. This operation is very difficult to grasp, even in seemingly simple examples, and one of the aims of my recent research has been to produce an algorithm to compute a crucial component involved in the tight closure operation, namely parameter-test-ideals and test-ideals. During the last few years I developed a new way to study these test-ideals via a duality which relates them to certain sub-objects of certain large and complicated objects, namely injective hulls of the residue field of the ring. This approach has been very successful in exploring other problems as well.I propose to expand my research of commutative rings of prime characteristic by continuing my collaboration with fellow researchers in my field who work in the US. These include Prof. Gennady Lyubeznik (University of Minnesota),Prof. Karl Schwede (Penn State University) and participants of the special programme in Commutative Algebra organized by the Mathematical Sciences Research Institute in California, which I would like to attend.

可以证明：（1）如果理论失败，可以证明许多定理的定理，可以在主要特征p的环中找到反例（即，包含整数ring ring on of the Integers a Modulo a prime d p），并且（2）在特征范围内（2）的特征（2）在特征上（2）的特征（2）在特征上（2）的特征（2） Frobenius函数F（R）将R提高到PTH功率。该功能子是环的内态性，即，它具有f（r+s）= f（r）+f（r）+f（s）的属性，令人惊讶的是，它很好地解决了许多特征性的问题。自成立以来，它在为艰难的旧问题提供简短而优雅的解决方案方面非常成功。紧密闭合还发现了其他领域的惊人应用，尤其是在代数几何形状中。该理论的本质是一种在特征P的通勤环中采用理想的操作，并在具有有用属性的情况下产生了另一个更大的理想。即使在看似简单的示例中，这项操作也很难掌握，我最近的研究的目的之一是产生一种算法来计算与紧密闭合操作有关的关键组件，即参数测试 - 理想和测试 - 理想。在过去的几年中，我开发了一种新的方法来通过二元性研究这些测试思想，这将它们与某些大型且复杂物体的某些亚物体相关联，即环形残基域的外观壳体。这种方法在探索其他问题方面也非常成功。我建议通过继续与在美国工作的研究人员的合作来扩大我对主要特征的交换戒指的研究。其中包括Gennady Lyubeznik教授（明尼苏达大学），教授。 Karl Schwede（宾夕法尼亚州立大学）和由加利福尼亚州数学科学研究所组织的交换代数特别计划的参与者，我想参加。