The Full Extended Plus Closure

在等特征p的局部环的研究中，紧闭被证明是非常有用的。这种闭包也可以很好地推广到等特征零的局部环。不幸的是，这种闭包不能自然地扩展到混合特征环。该项目旨在填补这一空白。在早期的工作中，首席研究员定义了扩展+闭包的几种变体。顾名思义，这些闭包与等特征p中的紧闭包一致，它们是基于理想的正闭包，即在原环的某个积分扩展中处于理想扩展中的元素集合。在前面的工作中，展示了这些闭包的许多属性。最值得注意的是，首席研究员已经证明，冒号捕获特性意味着正则环中的理想是封闭的，而且冒号捕获特性实际上在三维空间中也是成立的。因此，直接和猜想是三维空间中的一个定理。当前项目的主要目标是将这些结果扩展到维度4及以上。一个完全成功的程序将确定这些扩展+闭包中的一个-或其近亲-满足Huneke对紧密闭包的混合特征模拟所提出的所有要求。除了前面提到的那些属性之外，最值得注意的是持久性属性，即当同态应用于环时，理想闭包中的元素仍留在闭包中。另一个目标是模拟紧闭的相应理论的测试元素理论。代数中最基本的课题之一是理解局部环中的“理想”和“模”的概念。对于那些包含场的局部环，紧闭的概念已经演变成一种统一的表示方式，并且简化了这些对象的许多已知属性。作为一种自然的副产品，它导致了新特性的发现。对不包含场的局部环的理解总是落后的。首席研究员提出了几个密切相关的和有希望的候选人在替代环境中扮演紧密关闭的角色。这些候选人已经带来了一个重要的新结果。在这个项目中，研究者将继续努力确定新的闭包在多大程度上填补了空白。

DMS-0355486Raymond C. HeitmannIn the study of local rings of equicharacteristic p, the tight closure has proved very useful. This closure also extends nicely to local rings of equicharacteristic zero. Unfortunately this closure does not naturally extend to mixed characteristic rings. This project is designed to fill this void. In earlier work, the principal investigator defined several variants of an extended plus closure. As the name suggests, these closures, which coincide with tight closure in equicharacteristic p, are based upon the plus closure of an ideal, the set of elements which are in the extension of the ideal in some integral extension of the original ring. In the earlier work, a number of properties of these closures were demonstrated. Most notably, the principal investigator has demonstrated that the colon-capturing property implies that ideals in regular rings are closed and also that the colon-capturing property does in fact hold in dimension three. Hence the Direct Summand Conjecture is a theorem in dimension three. The primary objective of the current project is to extend these results to dimension four and above. A completely successful program would establish that one of these extended plus closures - or a close relative - satisfies all of the requirements suggested by Huneke for a mixed characteristic analog of tight closure. In addition to those properties already mentioned, the most notable is the persistence property, the property that elements in the closure of an ideal remain in the closure when a homomorphism is applied to the ring. An additional objective is a theory of test elements that mimics the corresponding theory for tight closure. One of the most fundamental subjects in algebra is the understanding of the concepts of "ideals" and "modules" in local rings. For those local rings that contain a field, the notion of tight closure has evolved as a way to give a unified presentation - and a simplified one - for many of the known properties of these objects. As a natural byproduct, it has led to the discovery of new properties. Understanding of local rings that do not contain a field has always lagged behind. The principal investigator has proposed several closely related and promising candidates to play the role of tight closure in the alternate setting. These candidates have already led to one significant new result. In this project, the investigator will continue his efforts to determine to what extent the new closures fill the void.