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A generalisation of the Abhyankar Jung theorem to associated graded rings of valuations

Published online by Cambridge University Press:  10 December 2015

STEVEN DALE CUTKOSKY
STEVEN DALE CUTKOSKY*
Affiliation:
Department of Mathematics, University of Missouri, Columbia, MO 65211, U.S.A. email: [email protected]
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Abstract

Suppose that R → S is an extension of local domains and ν* is a valuation dominating S. We consider the natural extension of associated graded rings along the valuation grν*(R) → grν*(S). We give examples showing that in general, this extension does not share good properties of the extension R → S, but after enough blow ups above the valuations, good properties of the extension R → S are reflected in the extension of associated graded rings. Stable properties of this extension (after blowing up) are much better in characteristic zero than in positive characteristic. Our main result is a generalisation of the Abhyankar–Jung theorem which holds for extensions of associated graded rings along the valuation, after enough blowing up.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 160 , Issue 2 , March 2016 , pp. 233 - 255
Copyright
Copyright © Cambridge Philosophical Society 2015 

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References

REFERENCES

[1]Abhyankar, S.On the ramification of algebraic functions. Amer. J. Math. 77 (1955), 575–592.CrossRefGoogle Scholar
[2]Abhyankar, S.On the valuations centered in a local domain. Amer. J. Math. 78 (1956), 321–348.CrossRefGoogle Scholar
[3]Abhyankar, S.Local uniformisation of algebraic surfaces over ground fields of characteristic p ≠ 0. Anns. of Math. 63 (1956), 491–526.CrossRefGoogle Scholar
[4]Abhyankar, S.Ramification Theoretic Methods in Algebraic Geometry (Princeton University Press, 1959).CrossRefGoogle Scholar
[5]Abhyankar, S.Resolution of Singularities of Embedded Algebraic Surfaces, second edition (Springer Verlag, New York, Berlin, Heidelberg, 1998).CrossRefGoogle Scholar
[6]Bruns, W. and Gubeladze, J.Polytopes, Rings and K-theory (Springer Verlag, 2009).Google Scholar
[7]Cutkosky, S. D.Local factorisation and monomialisation of morphisms, Astérisque 260 (1999).Google Scholar
[8]Cutkosky, S. D. Ramification of valuations and local rings in positive characteristic. To appear in Comm. Algebra.Google Scholar
[9]Cutkosky, S. D. and Ghezzi, L.Completions of valuation rings. Contemp. Math. 386 (2005), 13–34.CrossRefGoogle Scholar
[10]Cutkosky, S. D. and Piltant, O.Ramification of valuations, Adv. Math. 183 (2004), 1–79.CrossRefGoogle Scholar
[11]Cutkosky, S. D. and Vinh, P. A.Valuation semigroups of two dimensional local rings. Proc. London Math. Soc. 108 (2014), 350–384.CrossRefGoogle Scholar
[12]Ghezzi, L., Tài Hà, H. and Kashcheyeva, O.Toroidalisation of generating sequences in dimension two function fields. J. Algebra 301 (2006). 838–866.CrossRefGoogle Scholar
[13]Ghezzi, L. and Kashcheyeva, O.Toroidalisation of generating sequences in dimension two function fields of positive characteristic. J. Pure Appl. Algebra 209 (2007), 631–649.CrossRefGoogle Scholar
[14]Herrera Govantes, F. J., Olalla Acosta, F. J., Spivakovsky, M. and Teissier, G.Extending a valuation centered in a local domain to its formal completion. Proc. London Math. Soc. 105 (2012), 571–621.CrossRefGoogle Scholar
[15]Grothendieck, A. and Dieudonné, A.Eléments de géométrie algébrique IV, vol. 2. Publ. Math. Inst. Hautes Études Sci. 24 (1965).Google Scholar
[16]Grothendieck, A. and Murre, J.The Tame Fundamental Group of a Formal Neighbourhood of a Divisor with Normal Crossings on a Scheme. Lecture Notes in Math. 208 (Springer Verlag, Berlin – New York, 1971).CrossRefGoogle Scholar
[17]Heinzer, W. and Sally, J.Extensions of valuations to the completion of a local domain. J. Pure Appl. Algebra 71 (1991), 175–185.CrossRefGoogle Scholar
[18]Jung, H. E. W.Darstellung der Funktionen eines Algebraischen Körpers zweier unabhängigier Varänderlichen x, y in der Umbegung einer Stelle x = a, y = b. J. Reine Angew. Math. 133 (1908), 289–314.CrossRefGoogle Scholar
[19]Kuhlmann, F. V.Valuation theoretic and model theoretic aspects of local uniformisation, in Resolution of Singularities - A Research Textbook in Tribute to Oscar Zariski, Hauser, H., Lipman, J., Oort, F., Quiros, A. (es.). Progr. Math. 181 (Birkhäuser, 2000), 4559–4600.Google Scholar
[20]Kuhlmann, F.-V.A classification of Artin Schreier defect extensions and a characterisation of defectless fields. Illinois J. Math. 54 (2010), 397–448.CrossRefGoogle Scholar
[21]Luengo, I.A new proof of the Jung-Abhyankar theorem. J. Algebra 85 (1983), 399–409.CrossRefGoogle Scholar
[22]Parusínski, A. and Rond, G.The Abhyankar-Jung theorem. J. Algebra 365 (2012), 29–41.CrossRefGoogle Scholar
[23]Spivakovsky, M.Valuations in function fields of surfaces. Amer. J. Math. 112 (1990), 107–156.CrossRefGoogle Scholar
[24]Teissier, B.Valuations, deformations and toric geometry, in Valuation theory and its applications II, Kuhlmann, F. V., Kuhlmann, S. and Marshall, M., editors. Fields Institute Communications 33 (Amer. Math. Soc. Providence, RI, 2003), 361–459.Google Scholar
[25]Teissier, B.Overweight deformations of affine toric varieties and local uniformisation. In Valuation Theory in Interaction. Proceedings of the Second International Conference on Valuation Theory (Segovia–El Escorial, 2011). Edited by Campillo, A., Kuhlmann, F-V. and Teissier, B.European Math. Soc. Publishing House, Congress Reports Series (Sept. 2014), 474–565.CrossRefGoogle Scholar
[26]Villamayor, O. E.U. Equimultiplicity, algebraic elemination and blowing up. Adv. Math. 262 (2014), 313–369.CrossRefGoogle Scholar
[27]Zariski, O. Le problème des modules pour les branches planes. Course given at the Centre de Mathématiques de l'École Polytechnique, Paris, 1973, with an appendix by Bernard Teissier, Second Edition (Hermann, Paris, 1983).Google Scholar
[28]Zariski, O. and Samuel, P.Commutative Algebra Volume II (Van Nostrand, 1960).CrossRefGoogle Scholar