Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-09T16:02:08.186Z Has data issue: false hasContentIssue false
  • English
  • Français

Chern Classes of Projective Modules

Published online by Cambridge University Press:  22 January 2016

Hideki Ozeki
Hideki Ozeki*
Affiliation:
Brown University and Nagoya University
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

[1] Cartan, H. and Eilenberg, S., Homological algebra, Princeton, 1955.Google Scholar
[2] Chern, S. S., Topics in differential geometry.Google Scholar
[3] Chevalley, C. and Eilenberg, S., Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc, vol. 63 (1948) pp. 85–124.CrossRefGoogle Scholar
[4] Frölicher, A. and Nijenhuis, A., Theory of vector-valued differential forms. I, Proc. Kon. Ned. Acad. Wet. Amsterdam, vol. 59 (1956), pp. 338–359.Google Scholar
[5] Grauert, H., Analytische Faserungen über holomorphvollständigen Räumen, Math. Ann., vol. 135 (1958), pp. 203–273.Google Scholar
[6] Grothendieck, A., La théorie des classes de Chern, Bull. Soc. Math. France, vol. 86 (1958), pp. 137–154.Google Scholar
[7] Hirzebruch, F., Neue topologische Methoden in der alg. Geometrie, Springer, 1956.Google Scholar
[8] Kobayashi, S. and Nomizu, K., Differential geometry, to appear.Google Scholar
[9] Milnor, J., Lectures on characteristic classes, Princeton, 1957.Google Scholar
[10] Milnor, J., Lectures on differential topology, Princeton, 1958.Google Scholar
[11] Nomizu, K., Invariant affine connehtions on homogeneous spaces, Amer. J. Math., vol. 76 (1954), pp. 33–65.Google Scholar
[12] Northcott, D. G., An introduction to homological algebra, Cambridge, 1960.Google Scholar
[13] Palais, R. S., The cohomology of Lie rings, Proc. Symp. Pure Math., vol. Ill, 1961.Google Scholar
[14] Rham, G. de, Variétés differentiates, Paris, 1954.Google Scholar
[15] Steenrod, N., The topology of fibre bundles, Princeton, 1951.Google Scholar
[16] Weil, A., Sur la théorèmede de Rham, Comm. Math. Helv., vol. 26, pp. 119–145.Google Scholar