Extending Plus Closure

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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.