Applications of Frobenius maps

许多交换代数定理可以证明表明:(1)如果定理失败,可以找到一个反例的环p主要特征(例如,一个环包含整数环模素数p),和(2)没有这样的反例存在于p.Step特点(2)往往比在更容易证明以上特点存在的零,因为弗罗贝尼乌斯函数f (r),提出了r的甲状旁腺素的力量。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函数的存在性的形式化方法是紧闭理论，该理论在大约20年前首次提出，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棘手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。该理论的实质是在特征为p的交换环的环中取一个理想，并产生另一个具有有用性质的更大理想的运算。即使在看似简单的示例中，这种操作也很难掌握，本项目的目的之一是生成一种算法来计算紧密闭包操作中涉及的关键组件，即参数-测试-理想和测试-理想。本项目采用的方法是通过对偶来研究这些测试理想，对偶将它们与某些大而复杂的物体的某些子物体联系起来，即环的剩余场的注入壳。这种方法在解决该问题的一个相对简单的实例方面非常成功，该项目将尝试推广这些结果。环残馀场的注入壳的研究，使我们对代数集的一组被广泛研究的数值不变量，即它们的跳跃系数，有了新的认识。这就证明了这些由一个条件定义的曲面的不变量是有理数的离散集合。这个项目将尝试将这个结果推广到其他表面，并将尝试产生一个计算这些数字的算法。

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.