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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Vector bundles and projective modules


Author: Leonid N. Vaserstein
Journal: Trans. Amer. Math. Soc. 294 (1986), 749-755
MSC: Primary 18F25; Secondary 13C10, 14F05, 19A13, 19A15
MathSciNet review: 825734
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Abstract: Serre and Swan showed that the category of vector bundles over a compact space $ X$ is equivalent to the category of finitely generated projective modules over the ring of continuous functions on $ X$. In this paper, titled after the famous paper by Swan, this result is extended to an arbitrary topological space $ X$. Also the well-known homotopy classification of the vector bundles over compact $ X$ up to isomorphism is extended to arbitrary $ X$. It is shown that the $ {K_0}$-functor and the Witt group of the ring of continuous functions on $ X$ coincide, and they are homotopy-type invariants of $ X$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1986-0825734-3
PII: S 0002-9947(1986)0825734-3
Keywords: Projective modules, vector bundles, Witt group
Article copyright: © Copyright 1986 American Mathematical Society