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Chern Classes of Projective Modules

Published online by Cambridge University Press:  22 January 2016

Hideki Ozeki
Hideki Ozeki*
Affiliation:
Brown University and Nagoya University
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In topology, one can define in several ways the Chern class of a vector bundle over a certain topological space (Chern [2], Hirzebruch [7], Milnor [9], Steenrod [15]). In algebraic geometry, Grothendieck has defined the Chern class of a vector bundle over a non-singular variety. Furthermore, in the case of differentiable vector bundles, one knows that the set of differentiable cross-sections to a bundle forms a finitely generated projective module over the ring of differentiable functions on the base manifold. This gives a one to one correspondence between the set of vector bundles and the set of f.g.-projective modules (Milnor [10]). Applying Grauert’s theorems (Grauert [5]), one can prove that the same statement holds for holomorphic vector bundles over a Stein manifold.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 23 , December 1963 , pp. 121 - 152
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1963

References

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