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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Topological examples of projective modules


Author: Richard G. Swan
Journal: Trans. Amer. Math. Soc. 230 (1977), 201-234
MSC: Primary 55F25; Secondary 13J99, 16A80, 14F05
DOI: https://doi.org/10.1090/S0002-9947-1977-0448350-9
MathSciNet review: 0448350
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Abstract: A new and more elementary proof is given for LØnsted's theorem that vector bundles over a finite complex can be represented by projective modules over a noetherian ring. The rings obtained are considerably smaller than those of LØnsted. In certain cases, methods associated with Hilbert's 17th problem can be used to give a purely algebraic description of the rings. In particular, one obtains a purely algebraic characterization of the homotopy groups of the classical Lie groups. Several examples are given of rings such that all projective modules of low rank are free. If $ m \equiv 2 \bmod 4$, there is a noetherian ring of dimension m with nontrivial projective modules of rank m such that all projective modules of $ {\text{rank}} \ne m$ are free.


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DOI: https://doi.org/10.1090/S0002-9947-1977-0448350-9
Keywords: Projective modules, vector bundles
Article copyright: © Copyright 1977 American Mathematical Society