Contemporary Mathematics

Volume 300, 2002

Ramification Theory for Higher Dimensional Local Fields

Victor Abrashkin

To 60th birthday of A.N.Parshin.

ABSTRACT.

The paper contains a construction of ramification theory for higher

dimensional local fields K provided with additional structure given by an in-

creasing sequence of their "subfields of i-dimensional constants", where 0

~

i

~

nand n is the dimension of K. It is also announced that a local analogue

of the Grothendieck Conjecture still holds: all automorphisms of the abso-

lute Galois group of K, which are compatible with ramification filtration and

satisfy some natural topological conditions appear as conjugations via some

automorphisms of the algebraic closure of K.

0. Introduction

This paper deals with the formalism of ramification theory of higher dimen-

sional local fields. It comes from I.Zhukov's approach

[Zh], [Ab5]

to such a theory

in the case of 2-dimensional local fields K, which is based on the introduction of

the additional structure on K given by its closed 1-dimensional local subfield Kc

of dimension 1 - "the subfield of 1-dimensional constants". Then the filtration

of

rK

=

Gal(Ksep/K) by its ramification subgroups appears in the form of de-

creasing filtration of rK by normal subgroups {r~l}}EJ( 2 ). Here J(2)

=

J

1

II

J

2

,

where J

1 =

{j

E

Q

I

j

~

0},

h

=

{j

E

Q2

I

j

~

(0, 0)} (with respect to the

lexicographical ordering on

Q2

),

and by definition each element of J2 is greater

than every element of J1. For

j

E

J1, the groups r~l appear as the preimages of

the classical ramification subgroups of r

Kc

=

Gal(Kc,sep/ Kc) with respect to the

natural projection 7r from r

K

to r

Kc.

The "2-dimensional part" of ramification

filtration of rK appears as a decreasing filtration {r~lhEJ2 of rK

=

Ker7r and its

definition can be given in terms of semistable reduction of the arithmetical scheme

SpecOK--+ SpecOKc attached to the field extension K

::J

Kc (here

OK

and

OKc

are corresponding valuation rings).

The above interpretation of Zhukov's approach admits a direct generalization

to the case of local fields of arbitrary dimension n, which are supposed to be pro-

vided with an additional F-structure given by increasing sequence of subfields of

2000 Mathematics Subject Classification. Primary 11815, 11820, Secondary 11870.

Key words and phrases. local fields, ramification, anabelian conjecture.

©

2002 American Mathematical Society

1

http://dx.doi.org/10.1090/conm/300/05140