A Computational Approach to the Quillen-Suslin Theorem, Buchsbaum-Eisenbud Matrices, and Generic Hilbert-Burch Matrices

<p> We investigate certain classes of modules of low projective dimension over polynomial rings whose free resolutions have known special structure. We begin with projective modules and investigate a computational approach to a famous theorem of Quillen-Suslin, which states that every finitely generated projective module over S = R[x_1,...,x_n], with R a principal ideal domain, is free. We describe a package for the computer algebra system <italic>Macaulay2</italic> which we have developed to compute free generating sets for projective modules. We give special attention to the algorithms when R is the ring of integers and provide a constructive proof of a result of Suslin-Vaserstein, for which we could not find a constructive proof in the literature. The approach to this proof involves the ideas of strong Groebner bases for ideals of polynomials with integral coefficients and ``leading coefficient ideals.'' </p> <p>Moving to projective dimension two, we fix the ring B = k[x,y], with k a field, and consider homogeneous height two perfect ideals I = (g_1,g_2,g_3) generated by homogeneous forms g_i with deg g_i = d_i. Motivated by work of Cox-Kustin-Polini-Ulrich, we study the problem of constructing a local inverse to a particular morphism Phi which sends a 3 x 2 matrix of homogeneous forms of B to a triple of its signed 2 x 2 minors. For each possibility for the graded Betti numbers of B/I we describe an open cover of the parameter space of coefficients of the generators, and on each open set we describe the precise relationship between the coefficients of the forms g_i and the coefficients of the forms appearing in a presentation matrix P such that Phi(P) = (g_1,g_2,g_3). Furthermore, in the case where the degrees of each of the columns of P are the same, we generalize results of [CKPU] on the existence of universal projective resolutions for algebras B/I whose graded Betti numbers satisfy certain conditions. </p> <p>In projective dimension three, we fix the ring B = k[x,y,z], with k a field, and consider homogeneous grade three Gorenstein ideals I in B. The Buchsbaum-Eisenbud structure theorem for grade three Gorenstein algebras such as B/I implies that there exists an alternating presentation matrix psi. In a recent paper, Fisher describes how to produce such an alternating presentation matrix in the case when I is generated in a single degree. We extend this result and provide an algorithm to compute an alternating presentation matrix when I has generators in any degrees.</p>

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Campus Access Dissertation

Discipline thesis:degree_discipline

Mathematics

Year

2012

Author dc:creator

• Barwick, Jonathan Brett

Contributors dc:contributor

• Andrew R Kustin

• Mathematics
• Physical Sciences and Mathematics
• buchsbaum-eisenbud
• hilbert-burch
• leading coefficient ideal
• quillen-suslin

Statement dc:rights

• © 2012, Jonathan Brett Barwick

Repository record dc:identifier

https://scholarcommons.sc.edu/etd/1584

OAI identifier oai:identifier

oai:scholarcommons.sc.edu:etd-2585