Extending the Plus Closure for Mixed Characteristic Rings

在等特征p局部环的研究中，紧闭包被证明是非常有用的，这种紧闭包也可以很好地推广到等特征0局部环上。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，首席研究员定义了四种扩展+闭包的变体。顾名思义，这些闭包是基于理想的正闭包，即在原环的某个积分扩展中处于理想扩展中的元素集合。在早期的工作中，展示了闭包的一些特性，在这个项目中，将展示更多的特性，虽然这个项目的好处可能并不局限于这些，但目标是允许延长+闭包来填补紧闭包的角色。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合的。将证明冒号捕获特性。将证明持久性——在取同态图像时，理想闭包中的元素将留在闭包中。还应表明，当理想扩展到补全时，不在局部环理想闭包中的元素也不在闭包中。一个成功的项目将对混合特征环的同调理解产生重大影响。除其他事项外，这将意味着直接求和猜想的真实性。代数中最基本的课题之一是局部环的理想和模的理解。对于那些包含场的局部环，紧密闭包已经演变成一种统一的表达方式——并且是一种简化的表达方式——对于这些物体的许多已知属性。作为一种天然的副产品，它还导致了新特性的发现。对于不包含场的局部环的认识一直比较落后。首席研究员提出了几个密切相关的，非常有前途的候选人，以发挥紧密关闭在交替设置的作用。在这个项目中，研究者将试图确定这些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 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.