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Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

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Book Review

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MathSciNet review: 3077137
Full text of review: PDF   This review is available free of charge.
Book Information:

Author: T. Y. Lam
Title: Serre's problem on projective modules
Additional book information: Springer Monographs in Mathematics, Springer, Berlin, Heidelberg, New York, 2006, xxi + 401 pp., ISBN 978-3-540-23317-6, US$99.00$

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

  • H. Bass, $K$-theory and stable algebra, Inst. Hautes Études Sci. Publ. Math. 22 (1964), 5–60. MR 174604
  • Hyman Bass, Big projective modules are free, Illinois J. Math. 7 (1963), 24–31. MR 143789
  • Hyman Bass, Algebraic $K$-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0249491
  • Hyman Bass, Some problems in “classical” algebraic $K$-theory, Algebraic $K$-theory, II: “Classical” algebraic $K$-theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Lecture Notes in Math., Vol. 342, Springer, Berlin, 1973, pp. 3–73. MR 0409606
  • David Eisenbud and E. Graham Evans Jr., Generating modules efficiently: theorems from algebraic $K$-theory, J. Algebra 27 (1973), 278–305. MR 327742, DOI 10.1016/0021-8693(73)90106-3
  • Hans Grauert, Analytische Faserungen über holomorph-vollständigen Räumen, Math. Ann. 135 (1958), 263–273 (German). MR 98199, DOI 10.1007/BF01351803
  • I. Dzh. Gubeladze, The Anderson conjecture and a maximal class of monoids over which projective modules are free, Mat. Sb. (N.S.) 135(177) (1988), no. 2, 169–185, 271 (Russian); English transl., Math. USSR-Sb. 63 (1989), no. 1, 165–180. MR 937805, DOI 10.1070/SM1989v063n01ABEH003266
  • G. Horrocks, Projective modules over an extension of a local ring, Proc. London Math. Soc. (3) 14 (1964), 714–718. MR 169878, DOI 10.1112/plms/s3-14.4.714
  • T. Y. Lam, Serre’s conjecture, Lecture Notes in Mathematics, Vol. 635, Springer-Verlag, Berlin-New York, 1978. MR 0485842
  • Hartmut Lindel, On the Bass-Quillen conjecture concerning projective modules over polynomial rings, Invent. Math. 65 (1981/82), no. 2, 319–323. MR 641133, DOI 10.1007/BF01389017
  • M. Pavaman Murthy, Projective $A[X]$-modules, J. London Math. Soc. 41 (1966), 453–456. MR 200289, DOI 10.1112/jlms/s1-41.1.453
  • M. Pavaman Murthy and Jacob Towber, Algebraic vector bundles over $A^{3}$ are trivial, Invent. Math. 24 (1974), 173–189. MR 422276, DOI 10.1007/BF01390050
  • S. Parimala, Failure of a quadratic analogue of Serre’s conjecture, Amer. J. Math. 100 (1978), no. 5, 913–924. MR 517136, DOI 10.2307/2373953
  • Dorin Popescu, Polynomial rings and their projective modules, Nagoya Math. J. 113 (1989), 121–128. MR 986438, DOI 10.1017/S0027763000001288
  • Daniel Quillen, Projective modules over polynomial rings, Invent. Math. 36 (1976), 167–171. MR 427303, DOI 10.1007/BF01390008
  • Ravi A. Rao, The Bass-Quillen conjecture in dimension three but characteristic $\not =2,3$ via a question of A. Suslin, Invent. Math. 93 (1988), no. 3, 609–618. MR 952284, DOI 10.1007/BF01410201
  • Moshe Roitman, On Serre’s problem on projective modules, Proc. Amer. Math. Soc. 50 (1975), 45–52. MR 387266, DOI 10.1090/S0002-9939-1975-0387266-7
  • C. S. Seshadri, Triviality of vector bundles over the affine space $K^{2}$, Proc. Nat. Acad. Sci. U.S.A. 44 (1958), 456–458. MR 102527, DOI 10.1073/pnas.44.5.456
  • Jean-Pierre Serre, Faisceaux algébriques cohérents, Ann. of Math. (2) 61 (1955), 197–278 (French). MR 68874, DOI 10.2307/1969915
  • J.-P. Serre, Modules projectifs et espaces fibrés à fibre vectorielle, Séminaire P. Dubreil, M.-L. Dubreil-Jacotin et C. Pisot, 1957/58, Fasc. 2, Exposé 23, Secrétariat mathématique, Paris, 1958, pp. 18 (French). MR 0177011
  • 21.
    J-P. Serre, Sur les modules projectifs, Sem. Dubreil-Pisot no. 2, Paris, 1960/1961.
    22.
    N. E. Steenrod, The Topology of Fibre Bundles, Princeton, 1951.
  • A. A. Suslin, Projective modules over polynomial rings are free, Dokl. Akad. Nauk SSSR 229 (1976), no. 5, 1063–1066 (Russian). MR 0469905
  • A. A. Suslin, Stably free modules, Mat. Sb. (N.S.) 102(144) (1977), no. 4, 537–550, 632 (Russian). MR 0441949
  • A. A. Suslin, The structure of the special linear group over rings of polynomials, Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), no. 2, 235–252, 477 (Russian). MR 0472792
  • L. N. Vaseršteĭn and A. A. Suslin, Serre’s problem on projective modules over polynomial rings, and algebraic $K$-theory, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), no. 5, 993–1054, 1199 (Russian). MR 0447245
  • Richard G. Swan and Jacob Towber, A class of projective modules which are nearly free, J. Algebra 36 (1975), no. 3, 427–434. MR 376682, DOI 10.1016/0021-8693(75)90143-X
  • L. N. Vaseršteĭn, On the stabilization of the general linear group over a ring, Math. USSR-Sb. 8 (1969), 383–400. MR 0267009
  • Ton Vorst, The Serre problem for discrete Hodge algebras, Math. Z. 184 (1983), no. 3, 425–433. MR 716288, DOI 10.1007/BF01163515

  • Review Information:

    Reviewer: Richard G. Swan
    Affiliation: University of Chicago
    Email: swan@math.uchicago.edu
    Journal: Bull. Amer. Math. Soc. 45 (2008), 451-457
    DOI: https://doi.org/10.1090/S0273-0979-08-01171-3
    Published electronically: January 7, 2008
    Review copyright: © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.