Algebraic vector bundles on the 2-sphere
type: Item
instance of: scholarly article
title: Algebraic vector bundles on the 2-sphere
zbMATH Open document ID: 0819.13002
author: Richard G. Swan
published in: Rocky Mountain Journal of Mathematics
publication date: 1995-08-21T00:00:00Z
review text: Let \(A\) be \(\mathbb{R} [x,y,z]/(x^ 2 + y^ 2 + z^ 2 - 1)\) and let \(P_ 2 (A)\) be the set of isomorphism classes of finitely generated rank-2 \(A\)- modules. \textit{J. Barge} and \textit{M. Ojanguren} [Comment. Math. Helv. 62, 616-629 (1987; Zbl 0647.14011)] showed that, if \({\mathfrak p}\) is the invertible ideal \((x + iy,z - 1)\) of \(\mathbb{C} \otimes_ \mathbb{R} A\), considered as an \(A\)-module, then \(P_ 2 (A) = \{A^ 2, {\mathfrak p}, {\mathfrak p}^ 2, \dots\}\), the elements being distinct. The paper gives an algebraic proof of the following result: \({\mathfrak p}^ m \approx {\mathfrak p}^ n\) as \(A\)-modules if and only if \(m = \pm n\).
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 13E15
zbMATH DE Number: 599735
zbMATH Keywords: vector bundles
zbMATH Keywords: 2-sphere
reviewed by: Paolo Valabrega
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181072502
OpenAlex ID: W1974701919
cites work: Fibrés algébriques sur une surface réelle. (Algebraic fibers on a real surface)
cites work: Q5562621
cites work: Q4053793
cites work: Semialgebraic topology over a real closed field. II: Basic theory of semialgebraic spaces
cites work: Generating modules efficiently: Theorems from algebraic K-theory
cites work: Determinants in Projective Modules
cites work: Euler classes of inner product modules
cites work: An Algebraic Proof of the Borsuk-Ulam Theorem for Polynomial Mappings
cites work: Algebraic vector bundles over the 2-sphere
cites work: Vector bundles over affine surfaces
cites work: Symplectic bundles over affine surfaces
cites work: A Simple Proof of Gabber's Theorem on Projective Modules over a Localized Local Ring
cites work: Q3490638
cites work: Quaternionic bundles on algebraic spheres
cites work: Topological Examples of Projective Modules
DOI: 10.1216/RMJM/1181072502
description: scientific article; zbMATH DE number 599735