This monograph is the result of the
conference on higher local fields held in Muenster, August 29 to September
5, 1999. The aim is to provide an introduction to higher local fields
(more generally complete discrete valuation fields with arbitrary residue
field) and render the main ideas of this theory (Part I), as well as to
discuss several applications and connections to other areas (Part II).
The volume grew as an extended version of talks given at the conference.
The two parts are separated by a paper of K. Kato, an IHES preprint from
1980 which has never been published.
An n-dimensional local field is a complete discrete
valuation field whose residue field is an (n-1)-dimensional local field;
0-dimensional local fields are just perfect (e.g. finite) fields of
positive characteristic. Given an arithmetic scheme, there is a higher
local field associated to a flag of subschemes on it. One of central
results on higher local fields, class field theory, describes abelian
extensions of an n-dimensional local field via (all in the case of finite
0-dimensional residue field; some in the case of infinite 0-dimensional
residue field) closed subgroups of the n-th Milnor K-group of F.
We hope that the volume will be a useful introduction and guide to the
subject. The contributions to this volume were received over the period
November 1999 to August 2000 and the electronic publication date is 10
Ivan Fesenko and Masato