"Integral Closures of Ideals and Coefficient Ideals of Monomial Ideals" by Lindsey Hill
 

Integral Closures of Ideals and Coefficient Ideals of Monomial Ideals

Lindsey Hill, Purdue University

Abstract

The integral closure I of an ideal I in a ring R consists of all elements x ∈ R that are integral over I. If R is an algebra over an infinite field k, one can define general elements of belonging to a Zariski-open subset of kn.We prove that for any ideal I of height at least 2 in a local, equidimensional excellent algebra over a field of characteristic zero, the integral closure specializes with respect to a general element of I. That is, we show that In a Noetherian local ring (R, m) of dimension d, one has a sequence of ideals approximating the integral closure of I for Ian m-primary ideal. The ideals.are the coefficient ideals of I. The ith coefficient ideal I{i} of I is the largest ideal containing I and integral over I for which the first i + 1 Hilbert coefficients of I and I{i} coincide. With a goal of understanding how coefficient ideals behave under specialization by general elements, we turn to the case of monomial ideals in polynomial rings over a field. A consequence of the specialization of the integral closure is that the ith coefficient ideal specializes when the ith coefficient ideal coincides with the integral closure. To this end, we give a formula for first coefficient ideals of m-primary monomial ideals generated in one degree in 2 variables in order to describe when I{1} = I. In the 2-dimensional case, we characterize the behavior of all coefficient ideals with respect to specialization by general elements.In the d-dimensional case for d ≥ 3, we give a characterization of when I{1} = I for mprimary monomial ideals generated in one degree. In the final chapter, we give an application to the core, by characterizing when core(I) = adj(I d) for such ideals. Much of this dissertation is based on joint work with Rachel Lynn.

Degree

Ph.D.

Advisors

Ulrich, Purdue University.

Subject Area

Mathematics

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