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 Mat. Sb., 2003, Volume 194, Number 12, Pages 3–30 (Mi msb785)

On ramification theory in the imperfect residue field case

I. B. Zhukov

Saint-Petersburg State University

Abstract: This paper is devoted to the ramification theory of complete discrete valuation fields such that the residue field has prime characteristic $p$ and the cardinality of a $p$-base is 1. This class contains two-dimensional local and local-global fields. A new definition of ramification filtration for such fields is given. It turns out that Hasse–Herbrand type functions can be defined with all the usual properties. Thanks to this, a theory of upper ramification groups and the ramification theory of infinite extensions can be developed.
The case of two-dimensional local fields of equal characteristic is studied in detail. A filtration on the second $K$-group of the field in question is introduced that is different from the one induced by the standard filtration on the multiplicative group. The reciprocity map of two-dimensional local class field theory is proved to identify this filtration with the ramification filtration.

DOI: https://doi.org/10.4213/sm785

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English version:
Sbornik: Mathematics, 2003, 194:12, 1747–1774

Bibliographic databases:

UDC: 512.62
MSC: Primary 12F05; Secondary 11S15, 19F05

Citation: I. B. Zhukov, “On ramification theory in the imperfect residue field case”, Mat. Sb., 194:12 (2003), 3–30; Sb. Math., 194:12 (2003), 1747–1774

Citation in format AMSBIB
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This publication is cited in the following articles:
1. Kedlaya K., “Swan conductors for $p$-adic differential modules. I. A local construction”, Algebra Number Theory, 1:3 (2007), 269–300
2. Morrow M., “Integration on Valuation Fields Over Local Fields”, Tokyo J. Math., 33:1 (2010), 235–281
3. Morrow M., “An Explicit Approach to Residues on and Dualizing Sheaves of Arithmetic Surfaces”, N. Y. J. Math., 16 (2010), 575–627
4. Stefan Wewers, “Fiercely ramified cyclic extensions of p-adic fields with imperfect residue field”, manuscripta math, 2013
5. St. Petersburg Math. J., 26:5 (2015), 695–740
6. Xiao L., Zhukov I., “Ramification of Higher Local Fields, Approaches and Questions”, Valuation Theory in Interaction, EMS Ser. Congr. Rep., eds. Campillo A., Kuhlmann F., Teissier B., Eur. Math. Soc., 2014, 600–656
7. I. B. Zhukov, G. K. Pak, “Approximational approach to ramification theory”, St. Petersburg Math. J., 27:6 (2016), 967–976
8. Vostokov S.V. Afanas'eva S.S. Bondarko M.V. Volkov V.V. Demchenko O.V. Ikonnikova E.V. Zhukov I.B. Nekrasov I.I. Pital P.N., “Explicit Constructions and the Arithmetic of Local Number Fields”, Vestnik St. Petersburg Univ. Math., 50:3 (2017), 242–264
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