Tight closure, Frobenius maps and Frobenius splittings

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，我最近研究的目的之一就是产生一个算法来计算紧闭包操作中的一个关键组成部分，即参数测试理想和测试理想。在过去的几年里，我开发了一种新的方法来研究这些测试理想通过一个二元性，涉及到某些子对象的某些大型和复杂的对象，即内射外壳的剩余领域的环。这种方法在探索其他问题方面也非常成功。

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.