Integral Closure and Related Operations on Monomial Ideals

Abstract: The motivation for this thesis starts with the theory of Hilbert coefficients. It is a well known fact that given an ideal I the integral closure ? can be defined as the largest ideal with the same multiplicity as I. For monomial ideals there is an alternative definition. We give a review of this material and discuss the lattice of integrally closed monomial ideals.Ideals in two-dimensional regular local rings have the special property that the product of integrally closed ideals is again integrally closed. The study of this subject has a long tradition. Our characterization of integrally closed monomial ideals, presented in the first half of the thesis, is useful when studying other properties of and operations on ideals.The concept of reduction is tightly connected with the integral closure, since given two ideals I ? J we know that I is a reduction of J if and only if J ? ?. It is well known that minimal reductions exist in local rings and in polynomial rings. Moreover, the number of generators of a minimal reduction is interestingly connected to the dimension of the fibre cone F(I) = R/m ? I/mI ? ??? of an ideal I. In general, minimal reductions are not easy to find. We present a process of determing a minimal reduction in a two-dimensional power series ring and in some cases in the two-dimensional polynomial ring over any field k. The method can then be applied to some classes of ideals in integral domains and monomial subrings.The last section of the thesis concerns associated Ratliff-Rush ideals, an operation defined as ? = Ul?1 (Il+1:Il) where I is a regular ideal. An equivalent definition is that ? is the unique largest ideal containing I and with the same Hilbert polynomial. The notion was introduced almost thirty years ago, but the subject was not studied until the beginning of the nineties. The Ratliff-Rush operation behaves quite irrationally with respect to other ideal operations. We prove some results on numerical semigroups, that we use in our description of Ratliff-Rush ideals of certain classes of monomial ideals. Moreover, we establish new classes of Ratliff-Rush ideals, that is ideals such that I = ? and answer some questions from one of the early papers on this subject.