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
Previous Page Next Page