10 VICTOR ABRASHKIN
As usually, introduce the upper numbering of the ramification filtration of r
E/ L
by setting
r r('PE/L(jl)
E/L,j
=
E/L
for all
j
E
J(n). By the above Prop.6 the ramification filtration in upper number-
ing behaves well in
the projective system of all finite Galois extensions E
I
L with
compatible F-structures and we can introduce for all
j
E
J(n), the ramification
subgroups
of the absolute Galois group r
L.
Notice that if His an open subgroup of
rL
and
E
=
L{;!P,
then the decreasing
sequence of subgroups
rL
"J
r~)H
"J
r~· 0 )H
"J ... "J
r~n)H
"J
H
corresponds to the tower of algebraic extensions
L
c
LEco
c
LEc1
c · · · c
LEc,n-1
C
E.
In particular, if eE/L = (ell ... ' en), then e1 =
(r~)
H :
r~·O)
H), ... ' en =
(r~n)
H : H), i.e. the ramification filtration contains all information about the
vector eE/L·
Similarly to the classical !-dimensional case the composition property from
Prop. 6b) allows to extend the definition of Herbrand's function 'PE/L to the case
of all not necessarily normal finite extensions E
I
L of n-dimensional local fields
with induced
F
-structures. Equivalently, the Her brand function can be introduced
directly
(cf.
e.g. [De] for !-dimensional case): it will be sufficient to replace in
all the above constructions the group fE/L by an appropriate subset JE/L of£-
isomorphic embeddings of
E
into
L0

Then the Herbrand function is a piece-wise
linear function on J(n) and its "edge points" correspond to the jumps of the corre-
sponding filtration {IE/L,j}jEJ(n)· The above definitions and formal computations
with Herbrand's functions imply the following proposition.
PROPOSITION 7. Suppose E is a finite extension of an n-dimensionallocal field
L
and let '!fJE/L be the inverse Herbrand function. Then
a) for any
j
E
J(n), r~E/L(j))
=
r~)
n
rE;
b)
if
j
E
Ji with
1
~
i
~
n, then
. - t (
(j)
(0;) (0;))
-1 .
'1/JE;L(J)
=
eE/L,~i
lo;
r L r E : r E
d]
(6)
(if eE/L =(ell ... , en) then
eE/L,~i
:=(ell··., ei)).
PROOF. We can assume that
j
E
Jn.
Let E
1
be any finite Galois extension of L containing E. Then for any
j
E
Jn,
'!fJEdE ('1/JE;L(j))
=
'!fJEdL(j) and, therefore,
r
(.PE/L(jl) r r r r(j) r
El/E = EI/E,'r/JE1 /L(j) = El/L,'r/JE
1
JL(j)
n
EI/E = EI/L
n
EI/E·
Taking the projective limit on E
1
we obtain the property a).
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