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
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.
","abstract_html":"<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>","abstract_has_math":false,"creators":["Barwick, Jonathan Brett"],"institution":null,"degree_name":"Ph.D.","degree_level":"Campus Access Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Andrew R Kustin"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2012,"date_issued":"2012-01-01T08:00:00Z","date_published":"2012-01-01T08:00:00Z","updated_at":"2026-07-24T04:38:31Z","subjects":["Mathematics","Physical Sciences and Mathematics","buchsbaum-eisenbud","hilbert-burch","leading coefficient ideal","quillen-suslin"],"languages":[],"rights":["© 2012, Jonathan Brett Barwick"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://scholarcommons.sc.edu/etd/1584","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Andrew R Kustin"]},{"key":"dc:creator","label":"Author","values":["Barwick, Jonathan Brett"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Campus Access Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics","Physical Sciences and Mathematics","buchsbaum-eisenbud","hilbert-burch","leading coefficient ideal","quillen-suslin"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["© 2012, Jonathan Brett Barwick"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://scholarcommons.sc.edu/etd/1584"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":[" 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
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.
"]},{"key":"dc:title","label":"Title","values":["A Computational Approach to the Quillen-Suslin Theorem, Buchsbaum-Eisenbud Matrices, and Generic Hilbert-Burch Matrices"]}]}],"canonical_facts":{"dc:contributor":["Andrew R Kustin"],"dc:creator":["Barwick, Jonathan Brett"],"dc:description.abstract":[" 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
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.
"],"dc:identifier":["https://scholarcommons.sc.edu/etd/1584"],"dc:rights":["© 2012, Jonathan Brett Barwick"],"dc:subject":["Mathematics","Physical Sciences and Mathematics","buchsbaum-eisenbud","hilbert-burch","leading coefficient ideal","quillen-suslin"],"dc:title":["A Computational Approach to the Quillen-Suslin Theorem, Buchsbaum-Eisenbud Matrices, and Generic Hilbert-Burch Matrices"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Campus Access Dissertation"],"thesis:degree_name":["Ph.D."]},"updated_at":"2026-07-24T04:38:31Z"}