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On 2-dimensional local rings with Artin's approximation property

Published online by Cambridge University Press:  24 October 2008

M. L. Brown
M. L. Brown
Affiliation:
University College, Cardiff
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Abstract

Extending results of Popescu and Brown, the main result of this paper is that excellent henselian R1 and S1 2-dimensional local rings, at least in characteristic zero, have the approximation property of M. Artin.

Most of the paper consists of an extension of Néron's desingularization to rings which are R1 and S1; such a theorem was previously known for factorial domains. The main theorem is then deduced from this desingularization theorem using a theorem of Elkik.

Because of cohomological obstructions, the desingularization theorem is proved only for quasi-projective varieties. In the previously known case for factorial domains, these obstructions are always zero and the desingularization can be obtained by blowing up subschemes. The more general desingularization of this paper is obtained by blowing up locally free sheaves instead, the obstructions being zero for this case.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 94 , Issue 1 , July 1983 , pp. 35 - 52
Copyright
Copyright © Cambridge Philosophical Society 1983

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References

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