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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On Serre's problem on projective modules


Author: Moshe Roitman
Journal: Proc. Amer. Math. Soc. 50 (1975), 45-52
MSC: Primary 13C10; Secondary 14F05
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 Applied Mathematics, No. 21, Queen’s University, Kingston, Ont., 1969. MR 0255599 (41 #260)
  • [2] Oscar Zariski and Pierre Samuel, Commutative algebra, Volume I, The University Series in Higher Mathematics, D. Van Nostrand Company, Inc., Princeton, New Jersey, 1958. With the cooperation of I. S. Cohen. MR 0090581 (19,833e)
  • [3] H. Bass, 𝐾-theory and stable algebra, Inst. Hautes Études Sci. Publ. Math. 22 (1964), 5–60. 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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0387266-7
PII: S 0002-9939(1975)0387266-7
Article copyright: © Copyright 1975 American Mathematical Society