Extending 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扩展到等特征为零的局部环。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，这些闭包与等特性中的紧闭包一致，它们基于理想的正闭包，即在原环的某个积分扩展中处于理想扩展中的元素集。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冒号捕获特性意味着理想的不规则环是闭合的，而且冒号捕获特性实际上在三维空间中是成立的。因此，直接和猜想是三维空间的定理。当前项目的一个关键目标是将这些结果扩展到维度四及以上。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。它还将确定三个特定的兴趣环是否属于科恩-麦考利。三者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。对于等特征完全局部域和维数不等于3的所有完全局部域，答案是已知的。代数中最基本的课题之一是理解局部环的理想和模的概念。对于那些包含字段的本地化，紧密闭包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，它促进了新特性的发现。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哪一个是最好的候选人，以及新的闭包在多大程度上填补了空白。

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.