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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.

交换代数中的许多定理可以通过证明：（1）如果定理失败，可以在素特征p的环中找到反例（即，一个环，它包含整数模一个素数p的环），以及（2）在特征p中不存在这样的反例。上面的步骤（2）通常比在特征零中更容易证明，因为存在将r提升到p次幂的Frobenius函数f（r）。这个函数是环的自同态，即，它具有f（r+s）=f（r）+f（s）的性质，令人惊讶的是，它很好地处理了特征p中的许多问题。利用这些Frobenius函数的存在性的一种形式化方法是紧闭包理论，该理论最初是在大约20年前发展起来的，用于解决该领域的老问题。自成立以来，它一直非常成功地为困难的老问题提供简短而优雅的解决方案。紧闭包在其他领域也有惊人的应用，特别是在代数几何中。紧闭包理论的本质是一种运算，它取特征为p的交换环的环中的一个理想，并产生另一个具有有用性质的更大的理想。即使在看似简单的例子中，这个操作也很难掌握，这个项目的目标之一是产生一个算法来计算紧闭包操作中涉及的关键组件，即参数测试理想和测试理想。该项目所采取的方法是研究这个测试理想通过一个对偶关系，将它们与某些大而复杂的对象的某些子对象，即环的剩余域的内射壳。这种方法已经非常成功地解决了这个问题的一个相对简单的例子，该项目将尝试推广这些结果。环的剩余域的内射壳的研究产生了新的见解，某些广泛研究的代数集的数值不变量，即其跳跃系数。这导致了一个证明，这些不变量的表面定义的一个条件形成一个离散的一组有理数。这个项目将尝试把这个结果推广到其他表面，并尝试产生一个计算这些数字的算法。