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

This article is cited in 8 scientific papers (total in 8 papers)

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
Received: 25.05.2003

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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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    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  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    2. Morrow M., “Integration on Valuation Fields Over Local Fields”, Tokyo J. Math., 33:1 (2010), 235–281  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    3. Morrow M., “An Explicit Approach to Residues on and Dualizing Sheaves of Arithmetic Surfaces”, N. Y. J. Math., 16 (2010), 575–627  mathscinet  zmath  isi
    4. Stefan Wewers, “Fiercely ramified cyclic extensions of p-adic fields with imperfect residue field”, manuscripta math, 2013  crossref  mathscinet  isi  scopus  scopus
    5. St. Petersburg Math. J., 26:5 (2015), 695–740  mathnet  crossref  mathscinet  isi  elib  elib
    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  mathscinet  zmath  isi
    7. I. B. Zhukov, G. K. Pak, “Approximational approach to ramification theory”, St. Petersburg Math. J., 27:6 (2016), 967–976  mathnet  crossref  mathscinet  isi  elib
    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  crossref  mathscinet  isi  scopus
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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