The Henselian defect for valued function fields

Ohm, Jack

doi:10.1090/S0002-9939-1989-0975654-7

The Henselian defect for valued function fields
HTML articles powered by AMS MathViewer

by Jack Ohm

Proc. Amer. Math. Soc. 107 (1989), 299-308

DOI: https://doi.org/10.1090/S0002-9939-1989-0975654-7

PDF | Request permission

Abstract:

The notion of defect for finite algebraic extensions of valued fields is classical and due to Ostrowski. Recently Matignon has generalized Ostrowski’s definition to rk 1 (residually transcendental) valued function fields and used it to prove a very sharp version of the genus reduction inequality for $1 - \dim$ function fields. The further generalization of the notion of defect to valued function fields of arbitrary rk is treated here.

References

James Ax, A metamathematical approach to some problems in number theory, 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969) Amer. Math. Soc., Providence, R.I., 1971, pp. 161–190. MR 0316419
S. Bosch, U. Güntzer, and R. Remmert, Non-Archimedean analysis, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 261, Springer-Verlag, Berlin, 1984. A systematic approach to rigid analytic geometry. MR 746961, DOI 10.1007/978-3-642-52229-1
N. Bourbaki, Éléments de mathématique. Fasc. XXX. Algèbre commutative. Chapitre 5: Entiers. Chapitre 6: Valuations, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1308, Hermann, Paris, 1964 (French). MR 0194450
Otto Endler, Valuation theory, Universitext, Springer-Verlag, New York-Heidelberg, 1972. To the memory of Wolfgang Krull (26 August 1899–12 April 1971). MR 0357379
Hans Grauert and Reinhold Remmert, Über die Methode der diskret bewerteten Ringe in der nicht-archimedischen Analysis, Invent. Math. 2 (1966), 87–133 (German). MR 206039, DOI 10.1007/BF01404548

Remarks on good reduction in valued function fields

Laurent Gruson, Fibrés vectoriels sur un polydisque ultramétrique, Ann. Sci. École Norm. Sup. (4) 1 (1968), 45–89 (French). MR 229654

Ordinary defect, Matignon’s defect and other notions of a defect for finite extensions of valued fields and a special class of valued algebraic function fields

Serge Lang, Introduction to algebraic geometry, Interscience Publishers, Inc., New York-London, 1958. MR 0100591
Michel Matignon, Genre et genre résiduel des corps de fonctions valués, Manuscripta Math. 58 (1987), no. 1-2, 179–214 (French, with English summary). MR 884992, DOI 10.1007/BF01169090

Genre et genre résiduel des corps de fonctions valués (pour des valuations de rk arbitraire)

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, DOI 10.1090/S0002-9939-1988-0962804-0
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, DOI 10.1215/kjm/1250520076
Jack Ohm, Simple transcendental extensions of valued fields, J. Math. Kyoto Univ. 22 (1982/83), no. 2, 201–221. MR 666971, DOI 10.1215/kjm/1250521810
Jack Ohm, Simple transcendental extensions of valued fields. II. A fundamental inequality, J. Math. Kyoto Univ. 25 (1985), no. 3, 583–596. MR 807499, DOI 10.1215/kjm/1250521073
Alexander Ostrowski, Untersuchungen zur arthmetischen Theorie der Körper, Math. Z. 39 (1935), no. 1, 269–320 (German). MR 1545505, DOI 10.1007/BF01201361
Peter Roquette, On the prolongation of valuations, Trans. Amer. Math. Soc. 88 (1958), 42–56. MR 100589, DOI 10.1090/S0002-9947-1958-0100589-6
André Weil, Foundations of algebraic geometry, American Mathematical Society, Providence, R.I., 1962. MR 0144898
Oscar Zariski and Pierre Samuel, Commutative algebra, Volume I, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, New Jersey, 1958. With the cooperation of I. S. Cohen. MR 0090581
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