Algebraic vector bundles on the 2-sphere
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 \(m = \pm n\).
scientific article; zbMATH DE number 599735
scientific article