Topological examples of projective modules

Swan, Richard G.

doi:10.1090/S0002-9947-1977-0448350-9

Topological examples of projective modules
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by Richard G. Swan

Trans. Amer. Math. Soc. 230 (1977), 201-234

DOI: https://doi.org/10.1090/S0002-9947-1977-0448350-9

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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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