Residual Intersections and Their Generators

The goal of this dissertation is to broaden the classes of ideals for which the generators of residual intersections are known. This is split into two main parts.

The first part is Chapter 5, where we prove that, for an ideal I in a local Cohen-Macaulay ring R, under suitable technical assumptions, we are able to express s-residual intersections, for s ≥ μ(I) − 2, in terms of (μ(I) − 2)-residual intersections. This result implies that s- residual intersections can be expressed in terms of links, if μ(I) ≤ ht(I) + 3 and some other hypotheses are satisfied. In Chapter 5, we prove our result using two different methods and two different sets of technical assumptions on the depth conditions satisfied by the ideal I. For Section 5.2 and Section 5.3 we use the properties of Fitting ideals and methods developed in [33] to prove our main result. In these sections, we require I to satisfy the Gs condition and be weakly (s − 2)-residually S2. In Section 5.4, we prove analogous results to those in Section 5.2 and Section 5.3 using disguised residual intersections, a notion developed by Bouca and Hassansadeh in [5].

The second part is Chapter 6 where we prove that the n-residual intersections of ideals generated by maximal minors of a 2 × n generic matrix for n ≥ 4 are sums of links. To prove this, we require a series of technical results. We begin by proving the main theorem for this chapter in a special case, using the results of Section 6.1 to compute the generators of the relevant links in a our special case, and then using these generators to compute the Gro ̈bner Basis for the sum of links in Section 6.2. The computation of the Gro ̈bner basis, as well as an application of graph theoretic results about binomial edge ideals [17], allow us to show that our main theorem holds in this special case. Lastly, we conclude our proof in Section 6.3, where we show that n-residual intersections of ideals generated by maximal minors of 2 × n generic matrices commute with specialization maps, and use this to show that the generic n-residual intersections of ideals generated by maximal minors of a 2 × n generic matrix for n ≥ 4 are sums of links. This allows us to prove the main theorem of Chapter 6.