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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

REFERENCES

(1)Altman, A. and Kleiman, K.Introduction to Grothendieck Duality Theory. Lecture Notes in Mathematics, no. 146 (Springer-Verlag, 1970).CrossRefGoogle Scholar
(2)Artin, M.Algebraic approximation of structures over complete local rings. Inst. Hautes Études Sci. Publ. Math. 36 (1969), 23–58.CrossRefGoogle Scholar
(3)Borelli, M.Divisorial varieties. Pacific J. Math. 13 (1963), 375–388.CrossRefGoogle Scholar
(4)Brown, M. L. Artin's approximation property. Thesis, Cambridge University (1979).Google Scholar
(5)Brown, M. L. A class of 2-dimensional local rings with Artin's approximation property. (Preprint.)Google Scholar
(6)Fossum, R. M.The divisor class group of a Krull domain (Springer-Verlag, 1973).CrossRefGoogle Scholar
(7)Grothendieck, A. and Dieudonné, J.Éléments de géométrie algébrique. Inst. Hautes Études Sci. Publ. Math. (1960–1967).CrossRefGoogle Scholar
(8)Hartshorne, R.Algebraic geometry (Springer-Verlag, 1977).CrossRefGoogle Scholar
(9)Kaplansky, I.Commutative rings (Allyn & Bacon Inc., Boston, 1970).Google Scholar
(10)Kurke, K., Mostowski, T., Pfister, G., Popescu, D., Roczen, M.Die Approximations - eigenschaft lokaler Ringe. Lecture Notes in Mathematics, no. 634 (Springer-Verlag, 1978).CrossRefGoogle Scholar
(11)Matsumura, H.Commutative algebra, 2nd ed. (Benjamin/Cummings, Reading, Mass., 1980).Google Scholar
(12)Néron, A.Modèles minimaux des variétés abéliennes sur les corps locaux et globaux. Inst. Hautes Études Sci. Publ. Math. 21 (1964), 1–128.CrossRefGoogle Scholar
(13)Popescu, D.A remark on two-dimensional local rings with the property of approximation. Math. Zeit. 173, 235–240 (1980).CrossRefGoogle Scholar
(14)Serre, J.-P.Algèbre locale: multiplicités. Lecture Notes in Mathematics, no. 11 (Springer-Verlag, 1965).Google Scholar
(15)Zariski, O. and Samuel, P.Commutative algebra (Van Nostrand, Princeton, 1958).Google Scholar