The Full Extended Plus Closure

完全扩展的 Plus 闭合

• 批准号: 0355486
• 负责人: Raymond Heitmann
• 金额: --
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-06-01 至 2008-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0355486&HistoricalAwards=false
• 关键词: Full; Extended; Plus; Closure

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.

DMS-0355486 Raymond C. Heitmann在等特征p的局部环的研究中，紧闭包被证明是非常有用的. 这个闭包也很好地扩展到等特征零的局部环。 不幸的是，这种封闭性并不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，主要研究者定义了扩展加闭包的几个变体。 顾名思义，这些闭包与等特征p中的紧闭包相一致，是基于理想的正闭包，理想是在原环的某个积分扩展中的理想扩展中的元素集合。 在早期的工作中，证明了这些闭包的一些性质。 最值得注意的是，首席研究员已经证明了结肠捕获性质意味着正则环中的理想是封闭的，并且结肠捕获性质实际上在三维中也成立。 因此，直接和数猜想是一个三维定理。 目前项目的主要目标是将这些成果扩展到第四个层面及以上。 一个完全成功的程序将确定这些扩展的正闭包之一-或一个近亲-满足Huneke提出的紧闭包的混合特征模拟的所有要求。 除了已经提到的那些性质之外，最值得注意的是持久性性质，即当对环应用同态时，理想闭包中的元素仍保留在闭包中的性质。 一个额外的目标是一个理论的测试元素，模仿相应的理论紧封闭。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。 对于那些包含一个域的局部环，紧闭包的概念已经发展成为一种对这些对象的许多已知性质给出统一表示的方法。 作为一种天然副产品，它导致了新特性的发现。 对不包含场的局部环的理解一直落后。 主要研究者提出了几个密切相关的和有前途的候选人，以发挥作用，在交替设置紧密关闭。 这些候选人已经导致了一个重要的新结果。 在这个项目中，调查人员将继续努力确定新的封闭物在多大程度上填补了空白。