Integral Closures of Ideals and Coefficient Ideals of Monomial Ideals
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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