Elementary Matrix-computational Proof of Quillen-Suslin Theorem for Ore Extensions

Article type: Research Article

Authors: Fajardo, William^{a; *; †} | Lezama, Oswaldo^{b; ‡}

Affiliations: [a] Universidad Nacional de Colombia, Department of Mathematics, Bogot, Colombia. [email protected] | [b] Universidad Nacional de Colombia, Department of Mathematics, Bogot, Colombia. [email protected]

Correspondence: [*] Address for correspondence: Universidad Nacional de Colombia, Department of Mathematics, Bogot, Colombia

Note: [†] The first author was supported by Institucin Universitaria Politcnico Grancolombiano

Note: [‡] The second author was supported by the project New trends of non-commutative algebra and skew PBW extensions, HERMES CODE 26872.

Abstract: In this short note we present an elementary matrix-constructive algorithmic proof of the Quillen-Suslin theorem for Ore extensions A := K[x; σ, δ], where K is a division ring, σ : K → K is a division ring automorphism and σ : K → K is a σ-derivation of K. It asserts that every finitely generated projective A-module is free. We construct a symbolic algorithm that computes the basis of a given finitely generated projective A-module. The algorithm is implemented in a computational package. Its efficiency is illustrated by four representative examples.

Keywords: Symbolic algorithms, Maple, non-commutative computational algebra, projective modules, Ore extensions

DOI: 10.3233/FI-2019-1754

Journal: Fundamenta Informaticae, vol. 164, no. 1, pp. 41-59, 2019