2

VICTOR ABRASHKIN

i-dimensional constants with 1

~

i n. The techniques developed earlier by the

author

[Abl-3]

to study the classical ramification filtration can be adjusted to

obtain similar results for higher dimensional local fields. In particular, the paper

[Ab5]

contains an explicit description of the ramification filtration of the maxi-

mal p-extension of 2-dimensional local field of characteristic p with Galois group

of nilpotence class 2 (p;:?;: 3). Following the strategy from

[Ab4]

one can use this

description to prove a local analogue of the Grothendieck Conjecture for higher

dimensional local fields. This result is stated in n.6 below. It justifies that the

proposed ramification theory is sufficiently nice because it carries practically all

information about the original local field. Complete proofs of announced Theorems

1 and 2 are given in the papers

[Ab6, 7]

in the case of local fields of dimension 2.

It would be interesting to compare our theory with recent approach to ramification

theory from

[A-S]

as well as with earlier approaches to such a theory by K.Kato,

O.Hyodo, etc., which were basically related to the study of arithmetical proper-

ties of abelian extensions of higher dimensional local fields. One can find a brief

exposition of related results together with necessary references in the book

[HLF].

1

1.

n-dimensional local fields

By definition L is a local n-dimensional field if either

n

= 0 and L is a finite

field, or

n

;:?;: 1 and L is the quotient field of a complete discrete valuation ring

0l1)

with residue field

L,

which is a local field of dimension n- 1. With the obvious

notation there is the following sequence of epimorphic maps and embeddings of

valuation rings and residue fields

L

:=

L(o)

:: Oll)

-----+

L

:=

£(1)

::

oi\)1)

-----+ ••. -----+

L(n-1)

=

L(n)'

(1)

where

£(n)

is a finite field. For 0

~

i

~ n,

denote by

0~)

the preimage of

£(i)

in L with respect to the composition of corresponding morphisms from (1). The

kernel of the natural projection from

0~)

to

£(i)

will be denoted by

m~).

Notice

that

0l0)

=Land

m~)

= 0. The ring

Oln)

will be denoted also by

th

and will be

called the valuation ring of

L.

A subfield

E

of

L

is closed if it is either finite or it is the fraction field of a closed

non-discrete (with respect to the corresponding valuation topology) subring of

0l1)

and the corresponding residue field Eisa closed subfield of the (n -I)-dimensional

local field

L.

Then

E

is provided with a unique induced structure of local field

of dimension

~

n. On the other hand, if M is a finite extension of L, then M is

provided uniquely with a structure of an n-dimensional local field such that L is a

closed n-dimensional subfield of M.

In this paper we are going to consider only local fields

L,

which satisfy one of

the following two basic assumptions:

a) the finite characteristic case, i.e. char

£( 0)

= char

£(n);

in this case the field L

is always standard, that is

L

~

k((tn)) ... ((h)),

where

k

=

£(n)

(for any field

F,

F((t)) is a field offormal Laurent series with coefficients in

F);

b) the mixed characteristic case, i.e. char £(0) = 0 but char

£(l)

= char

£(n)

=

p 0; in this case L is a finite extension of some standard field K {{

tn}} ... { {

t2}},

where [K:

Qp]

oo (ifF::

Qp

then

F{

{t}} =

F®zv

UmZ/pMZ((t))).

1

The author is very grateful to the referee and I.Fesenko for pointing out several inaccuracies

in the original version of this paper