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

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Barwick, Jonathan Brett
buchsbaum-eisenbud; hilbert-burch; leading coefficient ideal; quillen-suslin; Mathematics; Physical Sciences and Mathematics
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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.