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

Publication Year:
2012
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Abstract Views 52
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Repository URL:
https://scholarcommons.sc.edu/etd/1584
Author(s):
Barwick, Jonathan Brett
Tags:
buchsbaum-eisenbud; hilbert-burch; leading coefficient ideal; quillen-suslin; Mathematics; Physical Sciences and Mathematics
interview description
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 Macaulay2 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.''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.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.