Applications of Frobenius maps

弗罗贝尼乌斯图的应用

基本信息

批准号：
EP/G060967/1
负责人：
Mordechai Katzman
金额：
$ 1.73万
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：
2009
资助国家：
英国
起止时间：
2009 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FG060967%2F1
关键词：
Applications Frobenius maps

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 this project is to produce an algorithm to compute a crucial component involved in the tight closure operation, namely parameter-test-ideals and test-ideals. The approach taken by this project is to study this 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 tackling a relatively simple instance of this problem and the project will attempt the generalize those results.The study of injective hulls of the residue field of the ring yielded new insights into a certain widely studied set numerical invariants of algebraic sets, namely their jumping coefficients. This resulted in a proof that these invariants for surfaces defined by one condition form a discrete set of rational numbers. This project will attempt to generalize this result for other surfaces and it will try to produce an algorithm for computing these numbers.

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，该操作也很难掌握，该项目的目的之一是生成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。该项目采用的方法是通过二元性研究这种测试理想，将它们与某些大型且复杂物体的某些亚物体相关联，即环形残基场的注射式壳体。这种方法在解决此问题的相对简单实例方面非常成功，该项目将尝试概括这些结果。对戒指残留场的注射式船体的研究产生了对某些经过广泛研究的代数组合的数值不变性的新见解，即它们的跳跃系数。这导致了一个证明，这些不变的表面由一个条件定义的表面形成了一个离散的有理数集。该项目将尝试将此结果推广到其他表面，并尝试生成用于计算这些数字的算法。