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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
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?)

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