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Value groups, residue fields, and bad places of rational function fields


Author: Franz-Viktor Kuhlmann
Journal: Trans. Amer. Math. Soc. 356 (2004), 4559-4600
MSC (2000): Primary 12J10; Secondary 12J15, 16W60
DOI: https://doi.org/10.1090/S0002-9947-04-03463-4
Published electronically: May 28, 2004
MathSciNet review: 2067134
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Abstract: We classify all possible extensions of a valuation from a ground field $K$ to a rational function field in one or several variables over $K$. We determine which value groups and residue fields can appear, and we show how to construct extensions having these value groups and residue fields. In particular, we give several constructions of extensions whose corresponding value group and residue field extensions are not finitely generated. In the case of a rational function field $K(x)$ in one variable, we consider the relative algebraic closure of $K$ in the henselization of $K(x)$ with respect to the given extension, and we show that this can be any countably generated separable-algebraic extension of $K$. In the ``tame case'', we show how to determine this relative algebraic closure. Finally, we apply our methods to power series fields and the $p$-adics.


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  • [AP] Alexandru, V. - Popescu, N.$\,$: Sur une classe de prolongements à $K(X)$ d'une valuation sur une corps $K$, Rev. Roumaine Math. Pures Appl. 33 (1988), 393-400 MR 89h:12015
  • [APZ1] Alexandru, V. - Popescu, N. - Zaharescu, A.$\,$: A theorem of characterization of residual transcendental extensions of a valuation, J. of Math. Kyoto Univ. 28 (1988), 579-592 MR 90c:12011
  • [APZ2] Alexandru, V. - Popescu, N. - Zaharescu, A.$\,$: Minimal pairs of definition of a residual transcendental extension of a valuation, J. of Math. Kyoto Univ. 30 (1990), 207-225 MR 92b:12014
  • [APZ3] Alexandru, V. - Popescu, N. - Zaharescu, A.$\,$: All valuations on $K(X)$, J. of Math. Kyoto Univ. 30 (1990), 281-296 MR 92c:12011
  • [B] Bourbaki, N.$\,$: Commutative algebra, Paris (1972)
  • [C] Cutkosky, D.$\,$: Local factorization and monomialization of morphisms, Astérisque 260 (1999) MR 2001c:14027
  • [CP] Cutkosky, D. - Piltant, O.$\,$: Ramification of valuations, to appear in: Advances in Mathematics
  • [E] Endler, O.$\,$: Valuation theory, Springer, Berlin (1972) MR 50:9847
  • [GMP1] Green, B. - Matignon, M. - Pop, F.$\,$: On valued function fields I, manuscripta math. 65 (1989), 357-376 MR 91g:12010
  • [GMP2] Green, B. - Matignon, M. - Pop, F.$\,$: On valued function fields II, III, J. reine angew. Math. 412 (1990), 128-149; 432 (1992), 117-133
  • [KA] Kaplansky, I.$\,$: Maximal fields with valuations I, Duke Math. Journ. 9 (1942), 303-321 MR 3:264d
  • [KH1] Khanduja, S. K.$\,$: Value groups and simple transcendental extensions, Mathematika 38 (1991), 381-385 MR 93c:12016
  • [KH2] Khanduja, S. K.$\,$: Prolongations of valuations to simple transcendental extensions with given residue field and value group, Mathematika 38 (1991), 386-390 MR 93c:12017
  • [KH3] Khanduja, S. K.$\,$: On valuations of $K(X)$, Proc. Edinburgh Math. Soc. 35 (1992), 419-426 MR 94a:12008
  • [KH4] Khanduja, S. K.$\,$: A uniqueness problem in simple transcendental extensions of valued fields, Proc. Edinburgh Math. Soc. 37 (1993), 13-23 MR 95b:12010
  • [KH5] Khanduja, S. K.$\,$: On value groups and residue fields of some valued function fields, Proc. Edinburgh Math. Soc. 37 (1994), 445-454 MR 95h:12006
  • [KH6] Khanduja, S. K.$\,$: A note on residually transcendental prolongations with uniqueness property, J. Math. Kyoto Univ. 36 (1996), 553-556 MR 97j:12007
  • [KH7] Khanduja, S. K.$\,$: On residually transcendental valued function fields of conics, Glasgow Math. J. 38 (1996), 137-145 MR 98a:12007
  • [KH8] Khanduja, S. K.$\,$: On extensions of valuations with prescribed value groups and residue fields, J. Indian Math. Soc. 62 (1996), 57-60 MR 98c:12013
  • [KH9] Khanduja, S. K.$\,$: Valued function fields with given genus and residue field, J. Indian Math. Soc. 63 (1997), 115-121 MR 99c:12008
  • [KH10] Khanduja, S. K.$\,$: An independence theorem in simple transcendental extensions of valued fields, J. Indian Math. Soc. 63 (1997), 243-248 MR 99g:12008
  • [KH11] Khanduja, S. K.$\,$: Tame fields and tame extensions, J. Alg. 201 (1998), 647-655 MR 99a:12010
  • [KH12] Khanduja, S. K.$\,$: The minimum property for Krasner's constant, in: Valuation Theory and its Applications, Proceedings of the Valuation Theory Conference Saskatoon 1999, Volume I, eds. F.-V. Kuhlmann, S. Kuhlmann and M. Marshall, Fields Institute Communications 32, Amer. Math. Soc. (2002) MR 2003c:12001
  • [KHG1] Khanduja, S. K. - Garg, U.$\,$: On extensions of valuations to simple transcendental extensions, Proc. Edinburgh Math. Soc. 32 (1989), 147-156 MR 90e:12019
  • [KHG2] Khanduja, S. K. - Garg, U.$\,$: Rank 2 valuations of $K(x)$, Mathematika 37 (1990), 97-105 MR 91j:12016
  • [KHG3] Khanduja, S. K. - Garg, U.$\,$: On the rank of extensions of valuations, Colloq. Math. 59 (1990), 25-29 MR 92c:12008
  • [KHG4] Khanduja, S. K. - Garg, U.$\,$: On residually generic prolongations to a simple transcendental extension, J. Indian Math. Soc. 57 (1990), 1-8 MR 93g:12010
  • [KHG5] Khanduja, S. K. - Garg, U.$\,$: Residue fields of valued function fields of conics, Proc. Edinburgh Math. Soc. 36 (1993), 469-478 MR 94i:12002
  • [KHG6] Khanduja, S. K. - Garg, U.$\,$: Prolongations of a Krull valuation to a simple transcendental extension, J. Indian Math. Soc. 59 (1993), 13-21 MR 94j:12012
  • [KHPR] Khanduja, S. K. - Popescu, N. - Roggenkamp, K. W.$\,$: On minimal pairs and residually transcendental extensions of valuations, to appear in: Mathematika
  • [KHS] Khanduja, S. K. - Saha, J.$\,$: A uniqueness problem in valued function fields of conics, Bull. London Math. Soc. 28 (1996), 455-462 MR 98a:12008
  • [KKU1] Knaf, H. - Kuhlmann, F.-V.$\,$: Abhyankar places admit local uniformization in any characteristic, submitted, available at: http://math.usask.ca/fvk/Valth.html
  • [KKU2] Knaf, H. - Kuhlmann, F.-V.$\,$: Every place admits local uniformization in a finite extension of the function field, preprint available at: http://math.usask.ca/fvk/Valth.html
  • [KU1] Kuhlmann, F.-V.$\,$: Henselian function fields and tame fields, preprint (extended version of Ph.D. thesis), Heidelberg (1990)
  • [KU2] Kuhlmann, F.-V.$\,$: Valuation theory of fields, abelian groups and modules, to appear in the ``Algebra, Logic and Applications'' series (formerly Gordon and Breach, eds. A. Macintyre and R. Göbel). Preliminary versions of several chapters available at: http://math.usask.ca/ $\,\tilde{ }\,$fvk/Fvkbook.htm
  • [KU3] Kuhlmann, F.-V.$\,$: Valuation theoretic and model theoretic aspects of local uniformization, in: Resolution of Singularities - A Research Textbook in Tribute to Oscar Zariski. Herwig Hauser, Joseph Lipman, Frans Oort, Adolfo Quiros (eds.), Progress in Mathematics Vol. 181, Birkhäuser Verlag Basel (2000), 381-456 MR 2001c:14001
  • [KU4] Kuhlmann, F.-V.$\,$: The model theory of tame valued fields, in preparation
  • [KU5] Kuhlmann, F.-V.$\,$: Algebraic independence of elements in completions and maximal immediate extensions of valued fields, in preparation
  • [KUKMZ] Kuhlmann, F.-V. - Kuhlmann, S. - Marshall, M. - Zekavat, M.$\,$: Embedding ordered fields in formal power series fields, J. Pure Appl. Algebra 169 (2002), 71-90 MR 2002m:12003
  • [KUPR] Kuhlmann, F.-V. - Pank, M. - Roquette, P.$\,$: Immediate and purely wild extensions of valued fields, manuscripta mathematica 55 (1986), 39-67 MR 87d:12012
  • [KW] Knebusch, M. - Wright, M.$\,$: Bewertungen mit reeller Henselisierung, J. reine angew. Math. 286/287 (1976), 314-321 MR 54:7440
  • [L] Lam, T. Y.$\,$: The theory of ordered fields, in: Ring Theory and Algebra III (ed. B. McDonald), Lecture Notes in Pure and Applied Math. 55, Dekker, New York (1980), 1-152 MR 82e:12033
  • [M] MacLane, S.$\,$: A construction for absolute values in polynomial rings, Trans. Amer. Math. Soc. 40 (1936), 363-395
  • [MO1] Matignon, M. - Ohm, J.$\,$: A structure theorem for simple transcendental extensions of valued fields, Proc. Amer. Math. Soc. 104 (1988), 392-402 MR 90h:12011
  • [MO2] Matignon, M. - Ohm, J.$\,$: Simple transcendental extensions of valued fields III: The uniqueness property, J. of Math. Kyoto Univ. 30 (1990), 347-366 MR 91j:12017
  • [MS] MacLane, S. - Schilling, O.F.G.$\,$: Zero-dimensional branches of rank 1 on algebraic varieties, Annals of Math. 40 (1939), 507-520 MR 1:26c
  • [MOSW1] Mosteig, E. - Sweedler, M.$\,$: Well-ordered valuations on characteristic zero rational function fields of transcendence degree two, manuscript (2001), available at: http://math.usask.ca/fvk/Valth.html
  • [MOSW2] Mosteig, E. - Sweedler, M.$\,$: Valuations and Filtrations, J. Symbolic Comput. 34 (2002), 399-435 MR 2003j:12008
  • [N] Nagata, M.$\,$: A theorem on valuation rings and its applications, Nagoya Math. J. 29 (1967), 85-91 MR 34:7503
  • [O1] Ohm, J.$\,$: Simple transcendental extensions of valued fields, J. of Math. Kyoto Univ. 22 (1982), 201-221 MR 84a:13005
  • [O2] Ohm, J.$\,$: The Ruled Residue Theorem for simple transcendental extensions of valued fields, Proc. Amer. Math. Soc. 89 (1983), 16-18 MR 85b:12010
  • [O3] Ohm, J.$\,$: Simple transcendental extensions of valued fields II: A fundamental inequality, J. of Math. Kyoto Univ. 25 (1985), 583-596 MR 87c:12013
  • [PL] Polzin, M.$\,$: Prolongement de la valeur absolue de Gauss et problème de Skolem, Bull. Soc. Math. France 116 (1988), 103-132 MR 90e:12020
  • [PP] Popescu, L. - Popescu, N.$\,$: On the residual transcendental extensions of a valuation. Key polynomials and augmented valuation, Tsukuba J. Math. 1 (1991), 57-78 MR 92h:12008
  • [PR] Prestel, A.$\,$: Lectures on Formally Real Fields, Lecture Notes in Math. 1093, Springer, Berlin-Heidelberg-New York-Tokyo (1984) MR 86h:12013
  • [R] Ribenboim, P.$\,$: Théorie des valuations, Les Presses de l'Université de Montréal, Montréal, 1st ed. (1964), 2nd ed. (1968) MR 40:2670
  • [S] Spivakovsky, M.$\,$: Valuations on function fields of surfaces, Amer. J. Math. 112 (1990), 107-156 MR 91c:14037
  • [SW] Sweedler, M.$\,$: Ideal Bases and Valuation Rings, manuscript (1986), available at: http://myweb.lmu.edu/faculty/emosteig/
  • [V] Vaquié, M.$\,$: Extension d'une valuation, preprint (2002), available at: http://math.usask.ca/fvk/Valth.html
  • [ZS] Zariski, O. - Samuel, P.$\,$: Commutative Algebra, Vol. II, Springer, New York-Heidelberg-Berlin (1960) MR 22:11006

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Additional Information

Franz-Viktor Kuhlmann
Affiliation: Mathematical Sciences Group, University of Saskatchewan, 106 Wiggins Road, Saskatoon, Saskatchewan, Canada S7N 5E6
Email: fvk@math.usask.ca

DOI: https://doi.org/10.1090/S0002-9947-04-03463-4
Received by editor(s): July 12, 2002
Received by editor(s) in revised form: July 15, 2003
Published electronically: May 28, 2004
Article copyright: © Copyright 2004 American Mathematical Society

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