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).