Studies in Commutative Algebra and Algebraic Geometry

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Several central open questions concerning tight closure in positivecharacteristic, such as whether tight closure commutes withlocalization, and whether the tight closure of an ideal of a domaincoincides, in good cases, with the contracted expansion of the idealfrom the absolute integral closure of the domain are proposed for study.Another main thrust is to explore several notions of closurein rings that do not contain a field, such as finitely generated algebrasover the integers, with the hope of extending tight closure theory to suchrings, thereby solving many open questions. A new approach to theproblem of proving existence of big Cohen-Macaulay modules inmixed characteristic will be pursued, as well as several lines ofresearch aimed at solving the long standing and important questionof whether regular rings are direct summands of their module-finiteextensions.Commutative rings are abstract systems in which one can performaddition, subtraction and multiplication. The integers andreal or complex numbers are examples, but there are vastlydifferent sorts of rings as well, including finite rings. Onecan introduce variable elements into any ring, forming a largerring. Ring theory can be used to study the behavior of large systemsof equations in many variables. There are methods of transitionfrom the study of equations with real or complex coefficients to thestudy of related systems with coefficients in a finite ring. Thesemethods produce qualtitative information about the solutions ofthe original systems of equations: sometimes one can determine whetherthere exist solutions and, if so, what the dimension of the solutionspace is. Most of the problems in the proposal can be viewed asproblems about solving equations. The projects in the proposalwill provide fundamental information about the nature of thesolutions for many sorts of systems of equations.