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 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.
