Extending Plus Closure

在等特征p的局部环的研究中，紧闭包已被证明是非常有用的。这个闭包也很好地扩展到局部等特征零环。不幸的是，这种封闭性并不自然地扩展到混合特征环。这个项目就是为了填补这个空白。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，这些封闭，这符合紧封闭在equicharteristicp，是基于加封闭的理想，一套元素，这是在扩展的理想在一些整体扩展的原始环。在早期的工作中，这些封闭的一些性质被证明。最值得注意的是，首席研究员已经证明了结肠捕获属性意味着理想不规则环是封闭的，并且结肠捕获属性实际上在三维中成立。因此，直接和数猜想是三维空间中的一个定理。当前项目的一个关键目标是将这些结果扩展到第四维度及以上。一个完全成功的程序将确定这些扩展的正闭包之一--或者一个近亲--满足Huneke对紧闭包的混合特征模拟提出的所有要求。它还将确定三个特定的兴趣环是否是Cohen-Macaulay。其中最引人注目的是三维完全混合特征局部域的绝对积分闭包。对于等特征完备局部整环和维数不等于3的完备局部整环，答案是已知的.代数学中最基本的课题之一是理解局部环中的理想和模的概念.对于那些包含域的局部环，紧闭包的概念已经发展成为一种方法，可以对这些对象的许多已知性质给出统一的表示--而且是一种简化的表示。作为一种自然的副产品，它导致了新特性的发现。对不含场的局部环的认识一直比较滞后。主要研究者已经提出了几个密切相关的和有希望的候选人在交替设置中扮演紧密闭合的角色。这些候选人已经导致了一个重要的新结果。在这个项目中，研究人员将继续努力，以确定哪一个是最好的候选人，并在多大程度上新的封闭填补空白。

In the study of local rings of equicharacteristic p, the tight closurehas proved very useful. This closure also extends nicely to local ringsof equicharacteristic zero. Unfortunately this closure does notnaturally extend to mixed characteristic rings. This project is designedto fill this void. In earlier work, the principal investigator definedseveral variants of an extended plus closure. As the name suggests,these closures, which coincide with tight closure in equicharacteristicp, are based upon the plus closure of an ideal, the set of elementswhich are in the extension of the ideal in some integral extension ofthe original ring. In the earlier work, a number of properties of theseclosures were demonstrated. Most notably, the principal investigator hasdemonstrated that the colon-capturing property implies that ideals inregular rings are closed and also that the colon-capturing property doesin fact hold in dimension three. Hence the Direct Summand Conjecture isa theorem in dimension three. A key objective of the current project isto extend these results to dimension four and above. A completelysuccessful program would establish that one of these extended plusclosures - or a close relative - satisfies all of the requirementssuggested by Huneke for a mixed characteristic analog of tight closure.It would also determine whether or not three particular rings ofinterest are Cohen-Macaulay. The most compelling of the three is theabsolute integral closure of a complete mixed characteristic localdomain of dimension three. For equicharacteristic complete local domainsand all complete local domains of dimension not equal to three, theanswer is already known.One of the most fundamental subjects in algebra is the understanding ofthe concepts of ideals and modules in local rings. For those localrings which contain a field, the notion of tight closure has evolved asa way to give a unified presentation - and a simplified one - for manyof the known properties of these objects. As a natural byproduct, it hasled to the discovery of new properties. Understanding of local ringswhich do not contain a field has always lagged behind. The principalinvestigator has proposed several closely related and promisingcandidates to play the role of tight closure in the alternate setting.These candidates have already led to one significant new result. In thisproject, the investigator will continue his efforts to determine whichis the best candidate and to what extent the new closures fill the void.