Homological Methods and Ideal Closures in Commutative Algebra

University Of Kansas Center For Research Inc, Lawrence KS

Investigators

Abstract

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