On ramification theory in the imperfect residue field case

Abstract

We consider the class of complete discretely valued fields such that the residue field is of prime characteristic p and the cardinality of a $p$-base is 1. This class includes two-dimensional local and local-global fields. A new definition of ramification filtration for such fields is given. It appears that a Hasse-Herbrand type functions can be defined with all the usual properties. Therefore, a theory of upper ramification groups, as well as the ramification theory of infinite extensions, can be developed. Next, we consider an equal characteristic two-dimensional local field $K$. We introduce some filtration on the second K-group of a given field. This filtration is other than the filtration induced by the valuation. We prove that the reciprocity map of two-dimensional local class field theory identifies this filtration with the ramification filtration.Comment: This is a corrected and extended version of my 1998 Nottingham preprint; many details are added. AmSTeX, 28 pages. To appear in Proceedings of the conference "Ramification theory of arithmetic schemes" (Luminy, 1999