RAMIFICATION THEORY FOR HIGHER DIMENSIONAL LOCAL FIELDS

7

REMARK. Notice that if

M{

is any finite extension of M 1 and E1

=EM{

and

L1 =

LM{

then we still have the monogeneity property Opj1

=

Oz

1

[B]. This follows

easily from Remark in n.2.

Let

VE

be the valuation of rank

n

on

E

from Prop.3. Use the natural identifi-

cation

fElL= fElL

to set for any

g

E

fElL•

ZEIL(g)

=

VE(gB- B)- VE(B).

Then

'iEI L(g)

E

Jn

U { oo} does not depend on the above special choices of the aux-

iliary field

M'

and the generator

B

(but it definitely depends on the corresponding

F-structure on

L).

For any j E

Jn,

set

rE1L,j

=

{g

E

rEIL

1

zEidg)

~

J}.

This is a decreasing filtration of

rEI L

by its normal subgroups, which is parametrized

by elements of

Jn.

Define the auxiliary Herbrand function

PEIL: Jn----- Jn

by the

relation

if3EIL(j)

=

ei;~L

1j

lfEIL,jl dj.

On

The value of this integral coincides with that of the corresponding integral sum

taken for the partition

On

~

]1

· · ·

j

8

~

j

where

al~

breaking points j1, ... ,

j

8

are the indices of jumps of the ramification filtration

{f

E

1 L,j} between

On

and j.

This implies for any j E

Jn,

that

PEIL(j)

=

ei;~L

L

min{'iE1L(g),j}.

gEf'E/L

Suppose a subfield

F

of

E

is normal over

L.

With the above notation we have

a

to~er

of nor_:nal extensions

E ::) F

:=

F M' ::)

L.

C_?nsider the natural projection

1f:

rEIL----- rFIL

:=

rFIL"

Then Ker'lf

=

rEIF

=

rEIF·

Clear~,

for

an~ 8

E

r~F,

it holds

ZEI£(8)

=

ZEIF(8)

and, therefore, one has

for all

j, fEIF,j

=

fEIF

n

fEIL,j·

Notice that the extension of penultimate residue fields

jf(n-

1)/

~n-

1

) is totally

ramified, so

h, ... , tn-

1

,

N E

1

_pB

is a system of local parameters of

F,

we still have

the

monogenei~

property

O_p

=

Oz[NEIFB],

and we can introduce for all

j

E

Jn,

the subgroups

fFIL,j·

PROPOSITION 4. For any

j

E

Jn, 7r(fEIL,j)

=

fFIL,PE;F(j)·

PROOF. We follow arguments from the proof of corresponding 1-dimensional

property from

[AN],

Ch.l.

Clearly, we have wE

1r(rEIL,j)

{=

j ~ d(w)

:=

max{'iEILh)

l1rb)

=

w} and

wE

rFIL,cpE/F(j)

{=

PEIF(j)

~

'iFIL(w).

So, it is sufficient to prove that

(/!EI F(d(w)) ='iF I L(w).

Suppose

'Yo

E

rEIL

is such that

7r(/'o)

=

w

and

'iEid'Yo)

=

d(w).

For any

8

E

rEIF•

we have

'iEid'Yo8)

=

min{'iEIL(8),

d(w)}.

(4)