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Transactions of the American Mathematical Society

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Elimination of ramification I: The generalized stability theorem

Author: Franz-Viktor Kuhlmann
Journal: Trans. Amer. Math. Soc. 362 (2010), 5697-5727
MSC (2010): Primary 12J10, 13A18; Secondary 12L12, 14B05
Published electronically: June 11, 2010
MathSciNet review: 2661493
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Abstract: We prove a general version of the ``Stability Theorem'': if $ K$ is a valued field such that the ramification theoretical defect is trivial for all of its finite extensions, and if $ F\vert K$ is a finitely generated (transcendental) extension of valued fields for which equality holds in the Abhyankar inequality, then the defect is also trivial for all finite extensions of $ F$. This theorem is applied to eliminate ramification in such valued function fields. It has applications to local uniformization and to the model theory of valued fields in positive characteristic.

References [Enhancements On Off] (What's this?)

  • [B] Bourbaki, N.$ $: Elements of mathematics. Commutative algebra. Translated from the French. Hermann, Paris; Addison-Wesley Publ. Co., Reading, MA (1972). MR 0360549 (50:12997)
  • [B-G-R] Bosch, S. - Güntzer, U. - Remmert, R.$ $: Non-Archimedean Analysis, Springer-Verlag, Berlin (1984). MR 746961 (86b:32031)
  • [En] Endler, O.$ $: Valuation theory, Springer, Berlin (1972). MR 0357379 (50:9847)
  • [Ep] Epp, Helmut H. P.$ $: Eliminating Wild Ramification, Inventiones Math. 19 (1973), 235-249. MR 0321929 (48:294)
  • [G] Gruson, L.$ $: Fibrés vectoriels sur un polydisque ultramétrique, Ann. Sci. Ec. Super., IV. Ser. 177 (1968), 45-89. MR 0229654 (37:5228)
  • [G-M-P] Green, B. - Matignon, M. - Pop, F.$ $: On valued function fields I, Manuscripta Math. 65 (1989), 357-376. MR 1015661 (91g:12010)
  • [G-R] Grauert, H. - Remmert, R.$ $: Über die methode der diskret bewerteten ringe in der nicht archimedischen Analysis, Invent. Math. 2 (1966), 87-133. MR 0206039 (34:5864)
  • [H] Huppert, B.$ $: Endliche Gruppen I, Springer, Berlin (1967). MR 0224703 (37:302)
  • [K-K1] Knaf, H. - Kuhlmann, F.-V.$ $: Abhyankar places admit local uniformization in any characteristic, Ann. Scient. Ec. Norm. Sup. 38 (2005), 833-846. MR 2216832 (2007k:13006)
  • [K-K2] Knaf, H. - Kuhlmann, F.-V.$ $: Every place admits local uniformization in a finite extension of the function field, Adv. Math. 221 (2009), no. 2, 428-453. MR 2508927
  • [K1] Kuhlmann, F.-V.: Henselian function fields and tame fields, extended version of Ph.D. thesis, Heidelberg (1990).
  • [K2] Kuhlmann, F.-V.: On local uniformization in arbitrary characteristic, The Fields Institute Preprint Series, Toronto (1997).
  • [K3] Kuhlmann, F.-V.$ $: Valuation theoretic and model theoretic aspects of local uniformization, in: Resolution of Singularities -- A Research Textbook in Tribute to Oscar Zariski. H. Hauser, J. Lipman, F. Oort, A. Quiros (eds.), Progress in Mathematics Vol. 181, Birkhäuser-Verlag, Basel (2000), 381-456. MR 1748629 (2001c:14001)
  • [K4] Kuhlmann, F.-V.$ $: A correction to Epp's paper ``Elimination of wild ramification'', Inventiones Math. 153 (2003), 679-681. MR 2000472 (2004g:13017)
  • [K5] Kuhlmann, F.-V.$ $: Additive Polynomials and Their Role in the Model Theory of Valued Fields, Logic in Tehran, 160-203, Lect. Notes Log., 26, Assoc. Symbol. Logic, La Jolla, CA, 2006. MR 2262319 (2007k:03095)
  • [K6] Kuhlmann, F.-V.$ $: A classification of Artin Schreier defect extensions and a characterization of defectless fields, submitted.
  • [K7] Kuhlmann, F.-V.$ $: The model theory of tame valued fields, in preparation.
  • [K8] Kuhlmann, F.-V.$ $: Elimination of Ramification II: Henselian Rationality, in preparation.
  • [K-P] Kuhlmann, F.-V. - Prestel, A.$ $: On places of algebraic function fields, J. Reine Angew. Math. 353 (1984), 181-195. MR 765832 (86d:12014)
  • [K-P-R] Kuhlmann, F.-V. - Pank, M. - Roquette, P.$ $: Immediate and purely wild extensions of valued fields, Manuscripta Math. 55 (1986), 39-67. MR 828410 (87d:12012)
  • [L] Lang, S.$ $: Algebra, Springer, New York, 3rd. ed. (2002). MR 1878556 (2003e:00003)
  • [O] Ohm, J.$ $: The Henselian defect for valued function fields, Proc. Amer. Math. Soc. 107 (1989), no. 2, 299-308. MR 975654 (90h:12012)
  • [R] Ribenboim, P.$ $: Théorie des valuations, Les Presses de l'Université de Montréal, Montréal, 2nd ed. (1968). MR 0249425 (40:2670)
  • [W] Warner, S.$ $: Topological fields, Mathematics studies 157, North-Holland, Amsterdam (1989). MR 1002951 (90i:12012)
  • [Z-S] Zariski, O. - Samuel, P.$ $: Commutative Algebra, Vol. II, The University Series in Higher Mathematics, D. Von Nostrand Co., Inc., Princeton, NJ-Toronto-London-New York. MR 0120249 (22:11006)

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Additional Information

Franz-Viktor Kuhlmann
Affiliation: Department of Mathematics and Statistics, University of Saskatchewan, 106 Wiggins Road, Saskatoon, Saskatchewan, Canada S7N 5E6

Received by editor(s): May 27, 2008
Published electronically: June 11, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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