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.
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