4
VICTOR ABRASHKIN
of "0-dimensional constants" Le o. In the mixed characteristic case L contains
Qp,
therefore, Le 1 is the algebraic closure of
Qp
in L, and we take its maximal
unramified over
Qp
subfield as Leo.
If
E is a finite extension of L then E is provided with a unique induced F-
structure such that for any 1
~
i
~
n, Ee i is the algebraic closure of Lei in
E. Inversely, any given F-structure onE induces the F-structure of L given by its
subfields Lei
:=
LnEei· In the both cases above we shall call F-structures of E and
L
compatible. Throughout all this paper all local fields are assumed to be provided
with some
F
-structure. When considering any algebraic extension of n-dimensional
local fields we always assume that the corresponding F -structures are compatible.
Notice also that for 1
~
i
~
n, the subfields
L~~]_
1
:=
(Lei
n 0}_1))
mod
m}_1l
give
the induced F-structure of the first residue field
L(1)
of
L.
So, while giving an
F-structure on L we provide automatically all residue fields of L with uniquely
determined induced F-structures.
Suppose
L
is a standard field. Then either
L
=
k((tn)) ...
((t1)), where
k
is
a finite field, or
L = K { {
tn}} ... { { t2} }, where
K
is a !-dimensional local field
with uniformizing element t1. In the both cases t1, ... , tn form a system of local
parameters in L. Associate to it the F-structure of L such that for
1
~
j
~
n, the
subfield Lej consists of elements l given in terms of corresponding formal series
(2)
by the condition
O:i
1
...
in
=
0 if at least one of the indices iH1, ... , in is not zero.
In other words, for 1
~
j
~
n, the subfield Lej consists of elements presented as
formal series in variables t1, ... , tj. This F-structure of (a standard field) L will be
called standard. The following proposition is very well-known application of Epp's
result on eliminating wild ramification.
PROPOSITION
1. Let L be an n-dimensional local field with F -structure. Then
there is a finite separable extension E' of Le,n-1 such that the induced F-structure
on E
:=
LE' is standard.
PROOF.
Apply induction on n.
If n = 1
then there is nothing to prove.
Let n
1
and let
Lalg
be an algebraic closure of
L.
By Epp's Theorem [Epp,
KZ] there is a finite separable extension M1 of Le 1 in
Lalg
such that if M
=
LM1,
then any uniformizing element t1 of M1 appears also as a uniformizing element of
M
(with respect to its first valuation). Let M'
=
M(1)
be the first residue field of M.
Consider its induced F- structure {M~i 11 ~
i
~ n- 1}, where M~i
=
(Me,i+1)(1)
is the first residue field of Me,i+1· By induction there is a finite separable extension
E'
of
M~,n- 2
in
L~~
such that the induced F- structure
{Eei
I
1
~
i
~
n- 1}
of
E
:=
E'
M' is standard, i.e. it is associated to some system of local parameters
f1, ... , tn-1 of
E.
Let E be one of unramified extensions of Min
La1g
with the (first) residue field
E.
For 1
~ i ~
n, denote by Eei the maximal unramified extension of Mei in E.
It is easy to see that {Eei
I
1
~
i
~
n} is F-structure on E, this F-structure is
associated to a collection of local parameters t1, ... ,
tn
such that for
i = 2, ... ,
n,
ti
E
OE(
1)
and ti mod
mE(
1)
=
fi-1· The proposition is proved.
c• c'
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