Extending the Plus Closure for Mixed Characteristic Rings

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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.