Extending the Plus Closure for Mixed Characteristic Rings

扩展混合特征环的 Plus 闭合

• 批准号: 0100731
• 负责人: Raymond Heitmann
• 金额: $ 9万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-01 至 2005-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0100731&HistoricalAwards=false
• 关键词: Extending; Plus; Closure; Mixed; Characteristic

In 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 four variants of an extended plus closure. As the name suggests, these closures 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 the closures were demonstrated, In this project, additional properties will be demonstrated, While the benefits of this project will probably not be restricted to these, the objectives are the properties which shall allow the extended plus closure to fill the role of tight closure. Assuming the program is successful, the following will be demonstrated. Ideals in regular local rings will be shown to be closed. The colon-capturing property will be proved. The persistence property will be proved - an element in the closure of an ideal will remain in the closure upon taking homomorphic images. It should also be shown that an element which is not in the closure of an ideal of a local ring will also not be in the closure when the ideal is extended to the completion. A successful project will have major ramifications for the homological understanding of mixed characteristic rings. Among other things, this will imply the truth of the Direct Summand Conjecture. One of the most fundamental subjects in algebra is the understanding of ideals and modules in local rings. For those local rings which contain a field, 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 also led to the discovery of new properties. Understanding of local rings which do not contain a field has always lagged behind. The principal investigator has proposed several closely related and highly promising candidates to play the role of tight closure in the alternate setting. In this project, the investigator will attempt to determine to what extent these new closures fill the void.

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，这个闭包也很好地推广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四个变体。 顾名思义，这些闭包是基于理想的正闭包，即在原环的某个整数扩张中的理想扩张中的元素集合。 在早期的工作中，证明了封闭的一些特性，在本项目中，将证明其他特性，虽然本项目的好处可能不限于这些，但目标是允许扩展加封闭以填补紧密封闭的作用的特性。假设程序成功，将演示以下内容。 正则局部环中的理想将被证明是闭的。 结肠捕获性质将被证明。 持久性财产将被证明-一个元素在封闭的理想将保持在封闭后采取同态图像。 还应该证明，当理想扩张到完备时，不在局部环的理想的闭包中的元素也将不在闭包中。 一个成功的项目将有重大分歧的同调理解的混合特征环。 除此之外，这将意味着直接和数猜想的真理。代数中最基本的课题之一是理解局部环中的理想和模。 对于那些包含一个场的局部环，紧闭包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和简化表示的方法。 作为一种天然的副产品，它也导致了新特性的发现。 对不含场的局部环的理解一直比较落后。 主要研究者已经提出了几个密切相关的和非常有前途的候选人，以发挥在替代设置紧密关闭的作用。 在本项目中，研究者将试图确定这些新的密封件在多大程度上填补了空白。