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Transactions of the American Mathematical Society

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

 
 

 

Algebraic fiber bundles


Author: Steven E. Landsburg
Journal: Trans. Amer. Math. Soc. 266 (1981), 259-273
MSC: Primary 14F05; Secondary 13C10, 16A50, 55R25
DOI: https://doi.org/10.1090/S0002-9947-1981-0613795-1
MathSciNet review: 613795
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Abstract: When $ X$ is a finite simplicial complex and $ G$ is any of a certain class of groups, a classification of $ G$-principal bundles over $ X$ in terms of projective modules over a ring $ R(G,X)$ is given. This generalizes Swan's classification of vector bundles and uses the results of Mulvey. Often, $ R$ can be taken to be noetherian; in this case $ {\text{Spec}}(R)$ is usually reducible with "cohomologically trivial" irreducible components. Information is derived concerning the nature of projective modules over such rings, and some results are obtained indicating how such information reflects information about $ X$.


References [Enhancements On Off] (What's this?)

  • [1] A. Dold, Lectures on algebraic topology, Springer-Verlag, Berlin, 1972. MR 0415602 (54:3685)
  • [2] E. G. Evans, Jr., Projective modules as fiber bundles, Proc. Amer. Math. Soc. 27 (1971), 623-626. MR 0269642 (42:4537)
  • [3] R. Fossum, Vector bundles over spheres are algebraic, Invent. Math. 8 (1969), 222-225. MR 0250298 (40:3537)
  • [4] R. Gurjar, Thesis, University of Chicago, 1979.
  • [5] K. Lønsted, An algebraization of vector bundles on compact manifolds, J. Pure Appl. Algebra 2 (1972), 193-207. MR 0337964 (49:2733)
  • [6] -, Vector bundles over finite CW complexes are algebraic, Proc. Amer. Math. Soc. 105 (1962), 264-277.
  • [7] J. Milnor, Introduction to algebraic $ K$-theory, Princeton Univ. Press, Princeton, N. J., 1971. MR 0349811 (50:2304)
  • [8] C. Mulvey, A generalization of Swan's Theorem, Math. Z. 151 (1976), 57-70. MR 0429900 (55:2909)
  • [9] R. S. Palais, Lusternik-Schnirelman theory on Banach manifolds, Topology 5 (1966), 115-132. MR 0259955 (41:4584)
  • [10] J.-P. Serre, Faisceaux algébriques cohérents, Ann. of Math. (2) 6 (1955), 197-278. MR 0068874 (16:953c)
  • [11] R. G. Swan, Vector bundles and projective modules, Trans. Amer. Math. Soc. 105 (1962), 264-277. MR 0143225 (26:785)
  • [12] -, Topological examples of projective modules, Trans. Amer. Math. Soc. 230 (1977), 201-234. MR 0448350 (56:6657)
  • [13] R. G. Swan and J. Towber, A class of projective modules which are nearly free, J. Algebra 36 (1975), 427-434. MR 0376682 (51:12857)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1981-0613795-1
Keywords: Projective module, $ G$-principal bundle, vector bundle, affine scheme
Article copyright: © Copyright 1981 American Mathematical Society

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