Specialization and Complexity of Integral Closure of Ideals

This dissertation is based on joint work with Lindsey Hill. There are two main parts, which are linked by the common theme of the integral closure of the Rees algebra.

In the first part of this dissertation, comprised of Chapter 3 and Chapter 4, we study the integral closure of the Rees algebra directly. In Chapter 3 we identify a bound for the multiplicity of the Rees algebra R[It] of a homogeneous ideal I generated in the same degree, and combine this result with theorems of Ulrich and Vasconcelos to obtain upper bounds on the number of generators of the integral closure of the Rees algebra as a module over R[It]. We also find various other upper bounds for this number, and compare them in the case of a monomial ideal generated in the same degree. In Chapter 4, inspired by the large depth assumption on the integral closure of R[It] in the results of Chapter 3, we obtain a lower bound for the depth of the associated graded ring and the Rees algebra of the integral closure filtration in terms of the dimension of the Cohen-Macaulay local ring R and the equimultiple ideal I. We finish the first part of this dissertation with a characterization of when the integral closure of R[It] is Cohen-Macaulay for height 2 ideals.

In the second part of this dissertation, Chapter 5, we use the integral closure of the Rees algebra as a tool to discuss specialization of the integral closure of an ideal I. We prove that for ideals of height at least two in a large class of rings, the integral closure of I is compatible with specialization modulo general elements of I. This result is analogous to a result of Itoh and an extension by Hong and Ulrich which show that for ideals of height at least two in a large class of rings, the integral closure of I is compatible with specialization modulo generic elements of I. We then discuss specialization modulo a general element of the maximal ideal, rather than modulo a general element of the ideal I itself. In general it is not the case that the operations of integral closure and specialization modulo a general element of the maximal ideal are compatible, even under the assumptions of our main theorem. We prove that the two operations are compatible for local excellent algebras over fields of characteristic zero whenever R/I is reduced with depth at least 2, and conclude with a class of ideals for which the two operations appear to be compatible based on computations in Macaulay2.