Studies in Commutative Algebra and Algebraic Geometry

本文提出了正特征线中关于紧闭包的几个中心问题，如紧闭包是否与局部化互换，以及理想的紧闭包是否与理想的收缩扩张一致等.另一个主要目的是探讨不含域环中的几个闭包概念，如整数环上的紧生成代数，希望将紧闭包理论推广到这类环上，从而解决许多悬而未决的问题。 一个新的方法来证明混合特征的大Cohen-Macaulay模的存在性的问题将被追求，以及几条线的研究，旨在解决长期存在的和重要的问题，是否正则环是其模有限扩张的直和。 整数和实数或复数都是例子，但也有各种各样的环，包括有限环。 一个环可以在任何环中引入可变元素，形成一个更大的环。 环理论可以用来研究多变量的大型方程组的行为。 从研究真实的或复系数方程到研究有限环中系数的相关系统有过渡的方法。 这些方法产生了关于原始方程组解的定性信息：有时人们可以确定是否存在解，如果存在，解空间的维数是多少。 建议中的大部分问题都可以看作是解方程的问题。 该计划中的项目将提供关于许多类型方程组的解的性质的基本信息。

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.