Homological Methods and Ideal Closures in Commutative Algebra

DMS-0244405 Huneke，克雷格L.摘要：本项目是在交换代数领域，特别是在诺特环的同调理论方面。杠杆被用来研究这一领域是在发展最大的科恩-麦考利模和扩展模块的理解。 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。这个proposalis的重点是几个开放的命题，特别是关于环的有限和可数的科恩-麦考利型，命题Auslander的vanishingof扩展模块，和问题的Schreyer，等等。 其中正在使用的方法是研究最大的科恩-Macaulay模不仅通过研究无限的决议，但与techniquescoming从紧封闭和减少到特征p的方法是部分经典的方法，以及那些正在开发的proposer.Commutative代数产生于19世纪研究多项式方程在许多变量，和他们的解决方案。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化的思想。交换代数通过形成一个代数对象来研究这种多项式或幂级数方程的解，称为环，它由“通用”解组成。这些通用解的代数性质然后让我们深入了解方程的几何和代数性质。在这一领域的一个重要技术一直是研究这样的方程通过减少系数模素数为所有大素数。一个特别的例子是在过去的15年里紧封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。

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.