On Serre’s problem on projective modules
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- by Moshe Roitman PDF
- Proc. Amer. Math. Soc. 50 (1975), 45-52 Request permission
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
- 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
- Oscar Zariski and Pierre Samuel, Commutative algebra, Volume I, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, New Jersey, 1958. With the cooperation of I. S. Cohen. MR 0090581
- H. Bass, $K$-theory and stable algebra, Inst. Hautes Études Sci. Publ. Math. 22 (1964), 5–60. MR 174604 B. L. van der Waerden, Modern algebra. Vol. I, Springer, Berlin, 1930; English transl., Ungar, New York, 1949. MR 10, 587.
Additional Information
- © Copyright 1975 American Mathematical Society
- 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