Homological Methods and Ideal Closures in Commutative Algebra

huneke, Craig l .摘要：本课题是交换代数领域，特别是Noetherian环的同调理论。用于研究这一领域的杠杆是发展对最大Cohen-Macaulay模和模的扩展的理解。此外，该项目还研究了交换代数的几个核心领域，包括理想的紧闭包、理想的积分闭包、理想的核心、演化、符号幂和理性奇点。这一提议的重点是几个开放的猜想，特别是关于有限可数的Cohen-Macaulay型环，Auslander关于可拓模消失的猜想，以及Schreyer的问题等。所使用的方法中，最大cohen - macaulay模的研究不仅是通过无限分辨率的研究，而且采用了从紧闭和约简到特征p的技术。这些方法一部分是经典方法，一部分是作者正在开发的方法。交换代数起源于19世纪对多变量多项式方程及其解的研究。多项式方程和几何之间的关系至少可以追溯到笛卡尔和平面坐标的思想。交换代数通过形成一个称为环的代数对象来研究这种多项式或幂级数方程的解，该代数对象由“一般”解组成。这些泛型解的代数性质可以让我们深入了解方程的几何和代数性质。该领域的一项重要技术是通过对所有大素数进行系数模化来研究这类方程。一个特别的例子是近15年来紧闭理论的爆炸性发展。交换代数结合了许多其他领域的技术，包括组合学、拓扑学和分析学。

DMS-0244405Huneke, Craig L.Abstract:This project is in the field of commutative algebra, especially in thehomological theory of Noetherian rings. The lever being used to study thisarea is in developing the understanding of maximal Cohen-Macaulay modulesand extensions of modules. In addition, the project also studies several areas central to commutative algebra, including the tight closure ofideals, integral closures of ideals, the core of an ideal, evolutions,symbolic powers, and rational singularities. The focus of this proposalis on several open conjectures, especially regarding rings of finite andcountable Cohen-Macaulay type, conjectures of Auslander on the vanishingof extension modules, and questions of Schreyer, among others. Amongthe methods being used is the study of maximal Cohen-Macaulaymodules not only through study of infinite resolutions, but with techniquescoming from tight closure and reduction to characteristic p. The methods arein part classical methods as well as those being developed by the proposer.Commutative algebra arose from the 19th century study of polynomial equations inmany variables, and their solutions. The relationship between polynomial equations andgeometry goes back at least to Descartes and the idea of coordinatizing the plane.Commutative algebra studies the solutions of such polynomial or power series equationsby forming an algebraic object, called a ring, which consists of the 'generic' solutions.The algebraic properties of these generic solutions then give insight into thegeometric and algebraic nature of the equations. An important technique in thisfield has been to study such equations by reducing the coefficients modulo prime numbersfor all large primes. A particular example of this has been the explosive developmentof the theory of tight closure over the last fifteen years. Commutative algebra combinestechniques from a number of other areas including combinatorics, topology, and analysis.