Coefficient ideals of the Hilbert polynomial and integral closures of parameter ideals

Kishor Mohan Shah, Purdue University

Abstract

We make contributions to two mainstream topics in commutative algebra: Hilbert Polynomials and Integral Closures of Ideals. We also initiate the algebraic study of the fiber cone of an ideal. Coefficient ideals of the Hilbert polynomial. Let R be a local ring with maximal ideal m. Let I be a m-primary ideal. In this thesis, we set down the discovery of certain unique largest ideals which contain I and are contained in the integral closure of I. These ideals form a chain and arise from the Hilbert coefficients of the Hilbert polynomial of I. We dub these ideals as the Coefficient Ideals of the Hilbert Polynomial. We obtain surprising structure theorems for each coefficient ideal. We further study the associated primes of $gr\sb{I}R$ via the coefficient ideals. Integral closures of parameter ideals. Let R be a quasi-unmixed local ring. Let $x\sb1,\dots,x\sb{d}$ be a system of parameters for R or a part of a system of parameters for R. We shall call the powers of the ideal generated by $x\sb1,\dots,x\sb{d}$ as the parameter ideals of $x\sb1,\dots,x\sb{d}$. Suppose that a homogeneous polynomial in $x\sb1,\dots,x\sb{d}$ lies in the integral closure of some parameter ideal. In this thesis, we precisely point out the ideals in which the coefficients of the homogeneous polynomial lie by studying prime characteristics and applying (as well as rigorously explaining) the standard Peskine-Szpiro-Hochster technique of reduction to prime characteristics. The fiber cone of an ideal. Let R be a local ring with maximal ideal m containing a fixed ideal I. The fiber cone of an ideal I is the ring F(I) = $\oplus I\sp{n}/I\sp{n}m$. We study the dimension of F(I), Cohen-Macaulayness of F(I), and the Hilbert polynomial of F(I).

Degree

Ph.D.

Advisors

Huneke, Purdue University.

Subject Area

Mathematics

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