Homotopy classification of module bundles via Grassmannians

Given a Waelbroeck ring R, we prove that the Grassmannian of a projective finitely generated R -module is a topological manifold modeled on a topological abelian group of R -linear maps. Fibre bundles of fibre type a module as above, over a compact base space B, admitting R -valued partitions of unity, are classified by the homotopy classes of continuous maps on B with values in the respective Grassmannian. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)