Problems in Commutative Algebra

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This award supports research in commutative algebra. The investigator together with students and collaborators will study problems in four connected areas: the theory of tight closure, rings of F-finite Cohen-Macaulay type, the theory of integrally closed ideals with applications to evolutions, and the study of infinite free resolutions. At its core, tight closure theory rests on reduction to positive characteristic. The principal investigator will study two main problems using tight closure. The first deals with recent work concerning the behavior of symbolic powers in regular local rings, and the second concerns the question of whether tight closure commutes with localization. To study rings of finite (or F-finite) Cohen-Macaulay type, this proposal places the study in the broader context of understanding the decomposition of purely inseparable extensions of a fixed Cohen-Macaulay domain into direct sums of indecomposable Cohen-Macaulay modules. Integral closures of ideals are a basic object in commutative algebra. This proposal concentrates on questions pertaining to the existence of evolutions over the complex numbers. The study of infinite free resolutions and vanishing to Tors is the focus of the last part of the proposal, especially over Gorenstein rings of dimension zero.Commutative algebra studies the relationship between algebraic equations, such as polynomial equations, and geometry. This idea goes back to Descartes and the idea of coordinatizing the plane, and has proved to be a powerful tool. A wide range of problems can be put into the context of solving systems of equations. For example, linear algebra studies systems of linear (degree one) equations. Commutative algebra studies the solutions of polynomial or power series equations of higher order by forming an algebraic object consisting of the 'generic' solutions. The algebraic properties of these generic solutions then give insight into the geometric and algebraic nature of the equations.