Homological Methods and Ideal Closures in Commutative Algebra

DMS-0244405Huneke，Craig L.摘要：这个项目是在交换代数领域，特别是在Notherian环的同调理论方面。用来研究这一领域的杠杆是发展对极大Cohen-Macaulay模和模的扩张的理解。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、符号幂和有理奇点。本文的重点是几个公开猜想，特别是关于有限可数Cohen-Macaulay型环，Auslander关于扩张模的零点的猜想，以及Schreyer的问题等。在这些方法中，极大Cohen-Macaulay模的研究不仅是通过无限分辨率的研究，而且是利用从紧闭包和约化到特征p的技巧。这些方法部分是经典的方法，也是由作者发展的方法。交换代数起源于19世纪对多元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坐标平面的思想。交换代数研究这种多项式或幂等式的解，通过形成一个代数对象，称为环，它由‘一般’解组成。然后，这些一般解的代数性质使我们深入了解方程的几何和代数性质。这一领域的一项重要技术是通过降低所有大素数的模素数系数来研究这类方程。一个特别的例子是，在过去的15年里，紧密封闭理论得到了爆炸性的发展。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.