Problems in Commutative Algebra

该奖项支持交换代数的研究。研究人员将与学生和合作者一起研究四个相关领域的问题：紧闭包理论，F-有限Cohen-Macaulay型环，积分闭合理想理论及其在演化中的应用，以及无限自由分解的研究。紧闭包理论的核心是归结为正特征。首席研究员将使用紧闭闭包来研究两个主要问题。第一个是关于正则局部环中符号幂的行为的最新工作，第二个是关于紧闭包是否与局部化交换的问题。为了研究有限(或F-有限)Cohen-Macaulay型环，这一建议将研究置于更广泛的背景下，即理解固定Cohen-Macaulay域的纯不可分扩张分解为不可分解Cohen-Macaulay模的直和。理想的积分闭包是交换代数中的一个基本目标。这一建议集中在与复数上的演化的存在性有关的问题上。关于无限自由分解和到Tors的零点的研究是本文最后一部分的重点，特别是在零维Gorenstein环上。交换代数研究代数方程，如多项式方程和几何之间的关系。这个想法可以追溯到笛卡尔和使平面协调的想法，并被证明是一个强大的工具。各种各样的问题都可以放在解方程组的背景下来处理。例如，线性代数研究线性(一次)方程组。交换代数通过形成一个由“一般”解组成的代数对象来研究高阶多项式或幂级数方程的解。通过这些一般解的代数性质，我们可以深入了解方程的几何和代数性质。

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.