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Failure of cancellation for direct sums of line bundles


Author: Richard G. Swan
Journal: Trans. Amer. Math. Soc. 336 (1993), 581-605
MSC: Primary 55N15; Secondary 13D15, 19A13
DOI: https://doi.org/10.1090/S0002-9947-1993-1091709-9
MathSciNet review: 1091709
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Abstract: In answer to a question of Murthy and Wiegand, examples are given of finitely generated projective modules $ L$ of rank $ 1$ over a commutative ring $ R$ such that $ L \oplus {L^{ - 1}}$ is stably free but not free.

Examples are also given of projective modules for which the determinant map det: $ \operatorname{Aut}(P) \to {R^{\ast} }$ is not onto. Some related topological questions are also discussed.


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

  • [AT] E. Artin and J. T. Tate, A note on finite ring extensions, J. Math. Soc. Japan 3 (1951), 74-77. MR 0044509 (13:427c)
  • [B] H. Bass, Algebraic $ K$-theory, Benjamin, New York, 1968. MR 0249491 (40:2736)
  • [Br] R. Brown, Elements of modern topology, McGraw-Hill, Maidenhead, 1968. MR 0227979 (37:3563)
  • [H] M. Hajja, A note on monomial automorphisms, J. Algebra 85 (1983), 243-250. MR 723077 (85k:12001)
  • [Hu] S. T. Hu, Homotopy theory, Academic Press, New York, 1959. MR 0106454 (21:5186)
  • [HFB] D. Husemoller, Fibre bundles, Springer-Verlag, New York 1975. MR 0370578 (51:6805)
  • [MS] J. W. Milnor and J. D. Stasheff, Characteristic classes, Ann. of Math. Stud., no. 76, Princeton Univ. Press, Princeton, N.J., 1974. MR 0440554 (55:13428)
  • [O] M. Ojanguren, Unités représentées par des formes quadratiques ou par des normes réduites, Lecture Notes in Math., vol. 967, Springer-Verlag, Berlin and New York, 1982, pp. 291-299. MR 689397 (84e:10031)
  • [R] M. Raynaud, Anneaux locaux Henséliens, Lecture Notes in Math., vol. 169, Springer-Verlag, Berlin and New York, 1970. MR 0277519 (43:3252)
  • [SCL] J-P. Serre, Corps locaux, Actualités Sci. Indust, no. 1296, Hermann, Paris, 1966. MR 0354618 (50:7096)
  • [SFB] N. E. Steenrod, The topology of fibre bundles, Princeton Univ. Press, Princeton, N.J., 1952.
  • [SGA] A. Grothendieck, Revêtements étales et groupe fondamental, SGA 1, Lecture Notes in Math., vol. 224, Springer-Verlag, Berlin and New York, 1971. MR 2017446 (2004g:14017)
  • [SM] R. G. Swan, Vector bundles, projective modules, and the $ K$-theory of spheres, Proc. of the John Moore Conference, Algebraic Topology and Algebraic $ K$-Theory (W. Browder, ed.), Ann. of Math. Stud., no. 113, Princeton Univ. Press, Princeton, N.J., 1987, pp. 432-522. MR 921488 (89f:18009)
  • [SN] -, Noether's problem in Galois theory, Emmy Noether in Bryn Mawr (B. Srinivasan and J. Sally, eds.), Springer-Verlag, New York, 1983, pp. 21-40. MR 713788 (84g:01001)
  • [ST] 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)
  • [SV] R. G. Swan, Vector bundles and projective modules, Trans. Amer. Math. Soc. 105 (1962), 264-277. MR 0143225 (26:785)
  • [SX] -, Topological examples of projective modules, Trans. Amer. Math. Soc. 230 (1977), 201-234. MR 0448350 (56:6657)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1993-1091709-9
Article copyright: © Copyright 1993 American Mathematical Society

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