Projective modules over Laurent polynomial rings

Swan, Richard G.

doi:10.1090/S0002-9947-1978-0469906-4

Projective modules over Laurent polynomial rings

Author: Richard G. Swan
Journal: Trans. Amer. Math. Soc. 237 (1978), 111-120
MSC: Primary 13C10; Secondary 13F20, 14F05
DOI: https://doi.org/10.1090/S0002-9947-1978-0469906-4
MathSciNet review: 0469906
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Abstract: Quillen's solution of Serre's problem is extended to Laurent polynomial rings. An example is given of a $A[T,{T^{ - 1}}]$ -module P which is not extended even though A is regular and ${P_\mathfrak{m}}$ is extended for all maximal ideals $\mathfrak{m}$ of A.

[1] H. Bass, Algebraic K-theory, Benjamin, New York, 1968. MR 40 #2736. MR 0249491 (40:2736)
[2] -, Some problems in ``classical'' algebraic K-theory, Algebraic K-Theory, II (Proc. Conf., Seattle Res. Center, Battelle Mem. Inst., 1972), Lecture Notes in Math., vol. 342, Springer-Verlag, Berlin and New York, 1973, pp. 3-73. MR 48 #3656b. MR 0409606 (53:13358)
[3] -, Libération des modules projectifs sur certains anneaux de polynômes, Séminaire Bourbaki 1973/1974, Exposé no. 448, Lecture Notes in Math., vol. 431, Springer-Verlag, Berlin and New York, 1975. pp. 228-254. MR 51 # 18. MR 0472826 (57:12516)
[4] A. Borel and J.-P. Serre, Groupes de Lie et puissances réduites de Steenrod, Amer. J. Math. 75 (1953), 409-448. MR 15, 338. MR 0058213 (15:338b)
[5] J.-L. Loday, Applications algébriques du tore dans la sphère et de ${S^p} \times {S^q}$ dans ${S^{p + q}}$ , Algebraic K-Theory, II (Proc. Conf., Seattle Res. Center, Battelle Memorial Inst., 1972), Lecture Notes in Math., vol. 342, Springer-Verlag, Berlin and New York, 1973, pp. 79-91. MR 51 #4276. MR 0368034 (51:4276)
[6] M. P. Murthy, Projective -modules, J. London Math. Soc. 41 (1966), 453-456. MR 34 #188. MR 0200289 (34:188)
[7] D. Quillen, Projective modules over polynomial rings. Invent. Math. 36 (1976), 167-171. MR 0427303 (55:337)
[8] N. Steenrod, The topology of fibre bundles, Princeton Univ. Press, Princeton, N. J., 1951. MR 12,522. MR 0039258 (12:522b)
[9] R. G. Swan, Vector bundles and projective modules, Trans. Amer. Math. Soc. 105 (1962), 264-277. MR 26 #785. MR 0143225 (26:785)
[10] -, Serre's problem, Conf. on Commutative Algebra-1975 (A. Geramita, Editor), Queen's Papers in Pure and Appl. Math., no. 42, Queen's Univ., Kingston, Ontario, Canada. MR 0384769 (52:5642)
[11] 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)