Book Review
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MathSciNet review:
3077137
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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$99.00$
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
- 1.
- H. Bass, K-theory and stable algebra, Publ. Math. I.H.E.S. 22 (1964), 5-60. MR 0174604
- 2.
- H. Bass, Big projective modules are free, Ill. J. Math. 7 (1963), 24-31. MR 0143789
- 3.
- H. Bass, Algebraic K-Theory, W. A. Benjamin, New York, 1968. MR 0249491
- 4.
- H. Bass, Some problems in `classical' algebraic K-theory, Algebraic K-Theory II, Lecture Notes in Math. 342, 1-70, Springer-Verlag, New York, 1973. MR 0409606
- 5.
- D. Eisenbud and G. Evans, Generating modules efficiently: Theorems from algebraic K-theory, J. Alg. 27 (1973), 278-305. MR 0327742
- 6.
- H. Grauert, Analytische Faserungen über holomorph-vollständigen Räumen, Math. Ann. 135 (1958), 263-273. MR 0098199
- 7.
- J. Gubeladze, The Anderson conjecture and a maximal class of monoids over which projective modules are free, Mat. Sbornik 135 (1988), 14169-185 (Math. USSR-Sbornik 63 (1989), 165-180). MR 0937805
- 8.
- G. Horrocks, Projective modules over an extension of a local ring, Proc. London Math. Soc. 14 (1964), 714-718. MR 0169878
- 9.
- T. Y. Lam, Serre's Conjecture, Lect. Notes in Math. 635, Springer-Verlag, Berlin, 1978. MR 0485842
- 10.
- H. Lindel, On the Bass-Quillen conjecture concerning projective modules over polynomial rings, Invent. Math. 65 (1981), 319-323. MR 0641133
- 11.
- M. P. Murthy, Projective
-modules, J. London Math. Soc. 41 (1966), 453-456. MR 0200289
- 12.
- M. P. Murthy and J. Towber, Algebraic vector byndles over
are trivial, Invent. Math. 24 (1974), 173-189. MR 0422276
- 13.
- Parimala, Failure of a quadratic analogue of Serre's conjecture, Amer. J. Math. 100 (1978), 913-924. MR 0517136
- 14.
- D. Popescu, Polynomial rings and their projective modules, Nagoya Math. J. 113 (1989), 121-128. MR 0986438
- 15.
- D. Quillen, Projective modules over polynomial rings, Invent. Math. 36 (1976), 167-171. MR 0427303
- 16.
- R. A. Rao, The Bass-Quillen conjecture in dimension 3 but characteristic
via a question of A. Suslin, Invent. Math. 93 (1988), 609-618. MR 0952284
- 17.
- M. Roitman, On Serre's problem on projective modules, Proc. Amer. Math. Soc. 50 (1975), 45-52. MR 0387266
- 18.
- C. S. Seshadri, Triviality of vector bundles over the affine space
, Proc. Nat'l. Acad. Sci. USA 44 (1958), 456-458. MR 0102527
- 19.
- J-P. Serre, Faisceaux algébrique coherént, Ann. of Math. 61 (1955), 191-278. MR 0068874
- 20.
- J-P. Serre, Modules projectifs et espaces fibrés à fibre vectorielle, Sem. Dubreil-Pisot no. 23, Paris, 1957/1958. 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.
- 23.
- A. A. Suslin, Projective modules over a polynomial ring are free, Dokl. Akad Nauk SSSR 229 (1976), 1063-1066 (Soviet Math. Doklady 17 (1976), 1160-1164). MR 0469905
- 24.
- A. A. Suslin, Stably free modules, Math. Sbornik (N.S.) 102 (1977), 537-550 (Math. USSR Sbornik 31 (1977), 479-491). MR 0441949
- 25.
- A. A. Suslin, On the structure of the special linear group over polynomial rings, Izv. Akad. Nauk SSSR Ser. Math. 41 (1977), 235-252 (Math. USSR-Izvestia 11 (1977), 221-238). MR 0472792
- 26.
- A. A. Suslin and L. N. Vaserstein, Serre's problem on projective modules over polynomial rings and algebraic K-theory, Izv. Akad. Nauk SSSR Ser. Math. 40 (1976), 993-1054 (Math. USSR-Izvestia 10 (1976), 937-1001). MR 0447245
- 27.
- R. G. Swan and J. Towber, A class of projective modules which are nearly free, J. Alg. 36 (1975), 427-434. MR 0376682
- 28.
- L. N. Vaserstein, On the stabilization of the general linear group over a ring, Mat. Sbornik 79(1969), 405-424 (Math USSR-Sbornik 8 (1969), 383-400). MR 0267009
- 29.
- T. Vorst, The Serre problem for discrete Hodge algebras, Math. Zeit. 184 (1983), 425-433. MR 0716288
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.