Topics in Commutative Algebra

University Of Kansas Center For Research Inc, Lawrence KS

Investigators

Abstract

This project is in the field of commutative algebra, especially Noetherian algebras and more especially polynomial rings over fields. The research in this proposal is directed at understanding the asymptotics of equations and their reduction modulo large prime numbers. Such asymptotics are captured through local cohomology, Hilbert-Kunz multiplicities, symbolic powers, and tight closure. The methods proposed are in part classical methods as well as those being developed by the proposer. The project also studies homological algebra, especially in terms of the graded Betti numbers and in terms of rings of finite Cohen-Macaulay type. A new asymptotic length function for local cohomology is proposed with applications to the homological conjectures, especially the monomial conjecture. Additional problems are proposed on the theory of liaison, especially regarding comparing the local and homogeneous versions, as well as understanding Gorenstein liaison classes. 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 solutions. 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 twenty years. Commutative algebra combines techniques from a number of other areas including combinatorics, topology, and analysis.