On ramification theory in the imperfect residue field case

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.