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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Vector bundles and projective modules
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by Leonid N. Vaserstein PDF
Trans. Amer. Math. Soc. 294 (1986), 749-755 Request permission

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
  • © Copyright 1986 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 294 (1986), 749-755
  • MSC: Primary 18F25; Secondary 13C10, 14F05, 19A13, 19A15
  • DOI: https://doi.org/10.1090/S0002-9947-1986-0825734-3
  • MathSciNet review: 825734