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 December 2000.

Ivan Fesenko and Masato Kurihara

Geometry & Topology Monographs 3 (2000)

DOI: 10.2140/gtm.2000.3