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Invitation to higher local elds
Conference in M unster, August–September 1999
Editors: I. Fesenko and M. Kurihara
ISSN 1464-8997 (on line) 1464-8989 (printed)
iii
Geometry & Topology Monographs
Volume 3: Invitation to higher local elds
Pages iii–xi: Introduction and contents
Introduction
This volume is a result of the conference on higher local elds in Munster, August 29–
September 5, 1999, which was supported by SFB 478 “Geometrische Strukturen in
der Mathematik”. The conference was organized by I. Fesenko and F. Lorenz. We
gratefully acknowledge great hospitality and tremendous efforts of Falko Lorenz which
made the conference vibrant.
Class eld theory as developed in the rst half of this century is a fruitful generaliza-
tion and extension of Gauss reciprocity law; it describes abelian extensions of number
elds in terms of objects associated to these elds. Since its construction, one of the
important themes of number theory was its generalizations to other classes of elds or
to non-abelian extensions.
In modern number theory one encounters very naturally schemes of nite type over
. A very interesting direction of generalization of class eld theory is to develop a
theory for higher dimensional elds—nitely generated elds over their prime subelds
(or schemes of nite type over Z in the geometric language). Work in this subject,
higher (dimensional) class eld theory, was initiated by A.N. Parshin and K. Kato
independently about twenty ve years ago. For an introduction into several global
aspects of the theory see W. Raskind’s review on abelian class eld theory of arithmetic
schemes.
One of the rst ideas in higher class eld theory is to work with theMilnor K -groups
instead of the multiplicative group in the classical theory. It is one of the principles of
class eld theory for number elds to construct the reciprocity map by some blending of
class eld theories for local elds. Somewhat similarly, higher dimensional class eld
theory is obtained as a blending of higher dimensional local class eld theories, which
treat abelian extensions of higher local elds . In this way, the higher local elds were
introduced in mathematics.
A precise denition of higher local elds will be given in section 1 of Part I; here
we give an example. A complete discrete valuation eld K whose residue eld is
isomorphic to a usual local eld with nite residue eld is called a two-dimensional
local eld. For example, elds F p (( T ))(( S )), Q p (( S )) and
=
+ X
: ai Q p inf vp ( ai )
lim
( ai )=+
Published 10 December 2000: c Geometry & Topology Publications
iv
Invitation to higher local elds
Gal( K ab K )
is close to an isomorphism (it induces an isomorphism between the group K N LK L
and Gal( LK ) for a nite abelian extension LK , and it is injective with everywhere
dense image). For two-dimensional local elds K as above, instead of the multiplicative
group K , the Milnor K -group K 2 ( K ) (cf. Some Conventions and section 2 of Part I)
plays an important role. For these elds there is a reciprocity map
2 ( K ) Gal( K ab K )
which is approximately an isomorphism (it induces an isomorphism between the group
2 ( K ) N LK K 2 ( L ) and Gal( LK ) for a nite abelian extension LK , and it has
everywhere dense image; but it is not injective: the quotient of K 2 ( K ) by the kernel of
the reciprocity map can be described in terms of topological generators, see section 6
Part I).
Similar statements hold in the general case of an n -dimensional local eld where
one works with the Milnor Kn -groups and their quotients (sections 5,10,11 of Part I);
and even class eld theory of more general classes of complete discrete valuation elds
can be reasonably developed (sections 13,16 of Part I).
Since K 1 ( K )= K , higher local class eld theory contains the classical local class
eld theory as its one-dimensional version.
The aim of this book is to provide an introduction to higher local elds and render
the main ideas of this theory. The book grew as an extended version of talks given at the
conference in Munster. Its expository style aims to introduce the reader into the subject
and explain main ideas, methods and constructions (sometimes omitting details). The
contributors applied essential efforts to explain the most important features of their
subjects.
Hilbert’s words in Zahlbericht that precious treasures are hidden in the theory of
abelian extensions are still up-to-date. We hope that this volume, as the rst collection
of main strands of higher local eld theory, will be useful as an introduction and guide
on the subject.
The rst part presents the theory of higher local elds, very often in the more
general setting of complete discrete valuation elds.
Section 1, written by I. Zhukov, introduces higher local elds and topologies on their
additive and multiplicative groups. Subsection 1.1 contains all basic denitions and is
referred to in many other sections of the volume. The topologies are dened in such a
Geometry & Topology Monographs, Volume 3 (2000) – Invitation to higher local elds
( v p is the p -adic valuation map) are two-dimensional local elds. Whereas the rst
two elds above can be viewed as generalizations of functional local elds, the latter
eld comes in sight as an arithmetical generalization of Q p .
In the classical local case, where K is a complete discrete valuation eld with nite
residue eld, the Galois group Gal( K ab K ) of the maximal abelian extension of K is
approximated by the multiplicative group K ; and the reciprocity map
Invitation to higher local elds
v
m ( F ) where F is either the rational function eld in
one variable F = k ( t ) or the formal power series F = k (( t )).
Appendix to Section 2, written by M. Kurihara and I. Fesenko, contains some
basic denitions and properties of differential forms and Kato’s cohomology groups
in characteristic p and a sketch of the proof of Bloch–Kato–Gabber’s theorem which
describes the differential symbol from the Milnor K -group Kn ( F ) p of a eld F of
positive characteristic p to the differential module
F .
Section 4, written by J. Nakamura, presents main steps of the proof of Bloch–Kato’s
theorem which states that the norm residue homomorphism
Ω
( K ) m H q ( K Z m ( q ))
is an isomorphism for a henselian discrete valuation eld K of characteristic 0 with
residue eld of positive characteristic. This theorem and its proof allows one to simplify
Kato’s original approach to higher local class eld theory.
Section 5, written by M. Kurihara, is a presentation of main ingredients of Kato’s
higher local class eld theory.
Section 6, written by I. Fesenko, is concerned with certain topologies on the Milnor
-groups of higher local elds K which are related to the topology on the multiplicative
group; their properties are discussed and the structure of the quotient of the Milnor
-groups modulo the intersection of all neighbourhoods of zero is described. The latter
quotient is called a topological Milnor K -group; it was rst introduced by Parshin.
Section 7, written by I. Fesenko, describes Parshin’s higher local class eld theory
in characteristic p , which is relatively easy in comparison with the cohomological
approach.
Section 8, written by S. Vostokov, is a review of known approaches to explicit
formulas for the (wild) Hilbert symbol not only in the one-dimensional case but in
the higher dimensional case as well. One of them, Vostokov’s explicit formula, is of
importance for the study of topological Milnor K -groups in section 6 and the existence
theorem in section 10.
Section 9, written by M. Kurihara, introduces his exponential homomorphism for
a complete discrete valuation eld of characteristic zero, which relates differential
forms and the Milnor K -groups of the eld, thus helping one to get an additional
information on the structure of the latter. An application to explicit formulas is discussed
in subsection 9.2.
Section 10, written by I. Fesenko, presents his explicit method to construct higher
local class eld theory by using topological K -groups and a generalization of Neukirch–
Geometry & Topology Monographs, Volume 3 (2000) – Invitation to higher local elds
n ( F ), and computation of H n +1
way that the topology of the residue eld is taken into account; the price one pays is
that multiplication is not continuous in general, however it is sequentially continuous
which allows one to expand elements into convergent power series or products.
Section 2, written by O. Izhboldin, is a short review of the Milnor K -groups and
Galois cohomology groups. It discusses p -torsion and cotorsion of the groups K n ( F )
and Kn ( F )= K n ( F ) l 1 lK n ( F ), an analogue of Satz 90 for the groups K n ( F ) and

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