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On Serre's problem on projective modules


Author: Moshe Roitman
Journal: Proc. Amer. Math. Soc. 50 (1975), 45-52
MSC: Primary 13C10; Secondary 14F05
DOI: https://doi.org/10.1090/S0002-9939-1975-0387266-7
MathSciNet review: 0387266
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Abstract: The main purpose of this paper is to prove the following result concerning Serre's problem: Any projective module of rank $ n$ over $ k[{X_1}, \cdots ,{X_n}]$ (where $ k$ is an infinite field) is free. We give also simple proofs (based on Serre's theorem that $ {K_0}(k[{X_1}, \cdots ,{X_n}]) = Z)$ to the following particular case of Bass' theorem: any projective module of rank $ > n$ over $ k[{X_1}, \cdots ,{X_n}]$ ($ k$ any field) is free, and to Seshadri's theorem: finitely generated projective modules over $ k[X,Y]$ are free.


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

  • [1] Aron Simis, When are projective modules free? Queen's Papers in Pure and Appl. Math., no. 21, Queen's University, Kingston, Ont., 1969. MR 41 #260. MR 0255599 (41:260)
  • [2] O. Zariski and P. Samuel, Commutative algebra. Vol. 1, University Series in Higher Math., Van Nostrand, Princeton, N. J., 1958. MR 19, 833. MR 0090581 (19:833e)
  • [3] H. Bass, $ K$-theory and stable algebra, Inst. Hautes Études Sci. Publ. Math. No. 22 (1964), 5-60. MR 30 #4805. MR 0174604 (30:4805)
  • [4] B. L. van der Waerden, Modern algebra. Vol. I, Springer, Berlin, 1930; English transl., Ungar, New York, 1949. MR 10, 587.

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DOI: https://doi.org/10.1090/S0002-9939-1975-0387266-7
Article copyright: © Copyright 1975 American Mathematical Society

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