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

Kuhlmann, Franz-Viktor

doi:10.1090/S0002-9947-04-03463-4

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 to a rational function field in one or several variables over . 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 in one variable, we consider the relative algebraic closure of in the henselization of with respect to the given extension, and we show that this can be any countably generated separable-algebraic extension of . In the ``tame case'', we show how to determine this relative algebraic closure. Finally, we apply our methods to power series fields and the -adics.

[AP] Victor Alexandru and Nicolae Popescu, Sur une classe de prolongements à 𝐾(𝑋) d’une valuation sur un corps 𝐾, Rev. Roumaine Math. Pures Appl. 33 (1988), no. 5, 393–400 (French). MR 950136
[APZ1] Victor Alexandru, Nicolae Popescu, and Alexandru Zaharescu, A theorem of characterization of residual transcendental extensions of a valuation, J. Math. Kyoto Univ. 28 (1988), no. 4, 579–592. MR 981094, https://doi.org/10.1215/kjm/1250520346
[APZ2] V. Alexandru, N. Popescu, and Al. Zaharescu, Minimal pairs of definition of a residual transcendental extension of a valuation, J. Math. Kyoto Univ. 30 (1990), no. 2, 207–225. MR 1068787, https://doi.org/10.1215/kjm/1250520067
[APZ3] V. Alexandru, N. Popescu, and A. Zaharescu, All valuations on 𝐾(𝑋), J. Math. Kyoto Univ. 30 (1990), no. 2, 281–296. MR 1068792, https://doi.org/10.1215/kjm/1250520072
[B] Bourbaki, N. $\,$ : Commutative algebra, Paris (1972)
[C] Steven Dale Cutkosky, Local monomialization and factorization of morphisms, Astérisque 260 (1999), vi+143 (English, with English and French summaries). MR 1734239
[CP] Cutkosky, D. - Piltant, O. $\,$ : Ramification of valuations, to appear in: Advances in Mathematics
[E] Otto Endler, Valuation theory, Springer-Verlag, New York-Heidelberg, 1972. To the memory of Wolfgang Krull (26 August 1899–12 April 1971); Universitext. MR 0357379
[GMP1] B. Green, M. Matignon, and F. Pop, On valued function fields. I, Manuscripta Math. 65 (1989), no. 3, 357–376. MR 1015661, https://doi.org/10.1007/BF01303043
[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] Sudesh K. Khanduja, Value groups and simple transcendental extensions, Mathematika 38 (1991), no. 2, 381–385 (1992). MR 1147837, https://doi.org/10.1112/S0025579300006720
[KH2] Sudesh K. Khanduja, Prolongations of valuations to simple transcendental extensions with given residue field and value group, Mathematika 38 (1991), no. 2, 386–390 (1992). MR 1147838, https://doi.org/10.1112/S0025579300006732
[KH3] Sudesh K. Khanduja, On valuations of 𝐾(𝑥), Proc. Edinburgh Math. Soc. (2) 35 (1992), no. 3, 419–426. MR 1187004, https://doi.org/10.1017/S0013091500005708
[KH4] Sudesh K. Khanduja, A uniqueness problem in simple transcendental extensions of valued fields, Proc. Edinburgh Math. Soc. (2) 37 (1994), no. 1, 13–23. MR 1258026, https://doi.org/10.1017/S0013091500018642
[KH5] Sudesh K. Khanduja, On value groups and residue fields of some valued function fields, Proc. Edinburgh Math. Soc. (2) 37 (1994), no. 3, 445–454. MR 1297313, https://doi.org/10.1017/S0013091500018897
[KH6] Sudesh K. Khanduja, A note on residually transcendental prolongations with uniqueness property, J. Math. Kyoto Univ. 36 (1996), no. 3, 553–556. MR 1417826, https://doi.org/10.1215/kjm/1250518509
[KH7] Sudesh K. Khanduja, On residually transcendental valued function fields of conics, Glasgow Math. J. 38 (1996), no. 2, 137–145. MR 1397168, https://doi.org/10.1017/S0017089500031360
[KH8] Sudesh K. Khanduja, On extensions of valuations with prescribed value groups and residue fields, J. Indian Math. Soc. (N.S.) 62 (1996), no. 1-4, 57–60. MR 1458473
[KH9] Sudesh K. Khanduja, Valued function fields with given genus and residue field, J. Indian Math. Soc. (N.S.) 63 (1997), no. 1-4, 115–121. MR 1618074
[KH10] Sudesh K. Khanduja, An independence theorem in simple transcendental extensions of valued fields, J. Indian Math. Soc. (N.S.) 63 (1997), no. 1-4, 243–248. MR 1618022
[KH11] Sudesh K. Khanduja, Tame fields and tame extensions, J. Algebra 201 (1998), no. 2, 647–655. MR 1612351, https://doi.org/10.1006/jabr.1997.7298
[KH12] Franz-Viktor Kuhlmann, Salma Kuhlmann, and Murray Marshall (eds.), Valuation theory and its applications. Vol. I, Fields Institute Communications, vol. 32, American Mathematical Society, Providence, RI, 2002. MR 1928358
[KHG1] Sudesh K. Khanduja and Usha Garg, On extensions of valuations to simple transcendental extensions, Proc. Edinburgh Math. Soc. (2) 32 (1989), no. 1, 147–156. MR 982002, https://doi.org/10.1017/S0013091500007008
[KHG2] Sudesh K. Khanduja and Usha Garg, Rank 2 valuations of 𝐾(𝑥), Mathematika 37 (1990), no. 1, 97–105. MR 1067891, https://doi.org/10.1112/S0025579300012833
[KHG3] Sudesh K. Khanduja and Usha Garg, On rank of extensions of valuations, Colloq. Math. 59 (1990), no. 1, 25–29. MR 1078288, https://doi.org/10.4064/cm-59-1-25-29
[KHG4] Sudesh K. Khanduja and Usha Garg, On residually generic prolongations of a valuation to a simple transcendental extension, J. Indian Math. Soc. (N.S.) 57 (1991), no. 1-4, 101–108. MR 1161327
[KHG5] Sudesh K. Khanduja and Usha Garg, Residue fields of valued function fields of conics, Proc. Edinburgh Math. Soc. (2) 36 (1993), no. 3, 469–478. MR 1242758, https://doi.org/10.1017/S0013091500018551
[KHG6] Sudesh K. Khanduja and Usha Garg, Prolongations of a Krull valuation to a simple transcendental extension, J. Indian Math. Soc. (N.S.) 59 (1993), no. 1-4, 13–21. MR 1248942
[KHPR] Khanduja, S. K. - Popescu, N. - Roggenkamp, K. W. $\,$ : On minimal pairs and residually transcendental extensions of valuations, to appear in: Mathematika
[KHS] Sudesh K. Khanduja and Jayanti Saha, A uniqueness problem in valued function fields of conics, Bull. London Math. Soc. 28 (1996), no. 5, 455–462. MR 1396143, https://doi.org/10.1112/blms/28.5.455
[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] Franz-Viktor Kuhlmann, Valuation theoretic and model theoretic aspects of local uniformization, Resolution of singularities (Obergurgl, 1997) Progr. Math., vol. 181, Birkhäuser, Basel, 2000, pp. 381–456. MR 1748629
[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] F.-V. Kuhlmann, S. Kuhlmann, M. Marshall, and M. Zekavat, Embedding ordered fields in formal power series fields, J. Pure Appl. Algebra 169 (2002), no. 1, 71–90. MR 1890186, https://doi.org/10.1016/S0022-4049(01)00064-0
[KUPR] Franz-Viktor Kuhlmann, Matthias Pank, and Peter Roquette, Immediate and purely wild extensions of valued fields, Manuscripta Math. 55 (1986), no. 1, 39–67. MR 828410, https://doi.org/10.1007/BF01168612
[KW] Manfred Knebusch and Michael J. Wright, Bewertungen mit reeller Henselisierung, J. Reine Angew. Math. 286(287) (1976), 314–321 (German). MR 419419
[L] T. Y. Lam, The theory of ordered fields, Ring theory and algebra, III (Proc. Third Conf., Univ. Oklahoma, Norman, Okla., 1979) Lecture Notes in Pure and Appl. Math., vol. 55, Dekker, New York, 1980, pp. 1–152. MR 584611
[M] MacLane, S. $\,$ : A construction for absolute values in polynomial rings, Trans. Amer. Math. Soc. 40 (1936), 363-395
[MO1] Michel Matignon and Jack Ohm, A structure theorem for simple transcendental extensions of valued fields, Proc. Amer. Math. Soc. 104 (1988), no. 2, 392–402. MR 962804, https://doi.org/10.1090/S0002-9939-1988-0962804-0
[MO2] Michel Matignon and Jack Ohm, Simple transcendental extensions of valued fields. III. The uniqueness property, J. Math. Kyoto Univ. 30 (1990), no. 2, 347–365. MR 1068796, https://doi.org/10.1215/kjm/1250520076
[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] Edward Mosteig and Moss Sweedler, Valuations and filtrations, J. Symbolic Comput. 34 (2002), no. 5, 399–435. MR 1937467, https://doi.org/10.1006/jsco.2002.0565
[N] Masayoshi Nagata, A theorem on valuation rings and its applications, Nagoya Math. J. 29 (1967), 85–91. MR 207688
[O1] Jack Ohm, Simple transcendental extensions of valued fields, J. Math. Kyoto Univ. 22 (1982/83), no. 2, 201–221. MR 666971, https://doi.org/10.1215/kjm/1250521810
[O2] Jack Ohm, The ruled residue theorem for simple transcendental extensions of valued fields, Proc. Amer. Math. Soc. 89 (1983), no. 1, 16–18. MR 706500, https://doi.org/10.1090/S0002-9939-1983-0706500-9
[O3] Jack Ohm, Simple transcendental extensions of valued fields. II. A fundamental inequality, J. Math. Kyoto Univ. 25 (1985), no. 3, 583–596. MR 807499, https://doi.org/10.1215/kjm/1250521073
[PL] Marc Polzin, Prolongement de la valeur absolue de Gauss et problème de Skolem, Bull. Soc. Math. France 116 (1988), no. 1, 103–132 (French, with English summary). MR 946280
[PP] Liliana Popescu and Nicolae Popescu, On the residual transcendental extensions of a valuation. Key polynomials and augmented valuation, Tsukuba J. Math. 15 (1991), no. 1, 57–78. MR 1118582, https://doi.org/10.21099/tkbjm/1496161567
[PR] Alexander Prestel, Lectures on formally real fields, Lecture Notes in Mathematics, vol. 1093, Springer-Verlag, Berlin, 1984. MR 769847
[R] Paulo Ribenboim, Théorie des valuations, Deuxième édition multigraphiée. Séminaire de Mathématiques Supérieures, No. 9 (Été, vol. 1964, Les Presses de l’Université de Montréal, Montreal, Que., 1968 (French). MR 0249425
[S] Mark Spivakovsky, Valuations in function fields of surfaces, Amer. J. Math. 112 (1990), no. 1, 107–156. MR 1037606, https://doi.org/10.2307/2374856
[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] Oscar Zariski and Pierre Samuel, Commutative algebra. Vol. II, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N. J.-Toronto-London-New York, 1960. MR 0120249