RAMIFICATION THEORY FOR HIGHER DIMENSIONAL LOCAL FIELDS

9

PROOF. The both functions are piecewise linear functions taking the same

value

On

in

On.

Notice that Prop.4 gives for any

j

E

Jn,

the following natural exact

sequence of ramification subgroups

Therefore,

lfEIL,jl

=

lfEIF,jllfFIL,PE;F(1)1·

This relation implies the equality of

derivatives of the both sides of (5) in all j except a finite number of edge points

coming from jumps of the corresponding ramification filtrations. The proposition

is proved.

5. Ramification filtration of

r

L

As earlier, let L be an n-dimensionallocal field inside

L0

provided with induced

F

-structure {

Lc

i

I 1

~

i

~

n}. Denote by Lsep the separable closure of

L

in

L0

and set

rL

= Gal(Lsep/L). Consider the set of indices

J(n)

=

JoUJ1· · · UJn,

where as earlier

Ji

=

{j

E

Qi

I

j

~

oi

E

Qi} for all 1

~

i

~

n

and

Jo

= {c},

where cis just a symbol. The set

J(n)

is provided with the ordering coming from

lexicographical orderings inside each of its component J

8

,

1

~

s

~

n, and by setting

that for 0

~

s

1

s

~

n,

every element of J81 is less than any element of J

8

•

We are

going to define the ramification filtration

{f(1llJEJ(n)

of the absolute Galois group

r£.

Consider a finite Galois extension

E / L

with the induced

F

-structure

{Eci

I 1

~

i

~

n}. Then for all 1

~

i

~

n, EcdLci

is a finite Galois exten-

sion of i-dimensional local fields provided with induced compatible F-structures.

Besides, for all 1

~

i

~

n,

we have the natural exact sequences

Let

VE

be the valuation of rank

n

on

E

from Prop.3. Then

VEIEc,

=

VEe,

is

also the valuation of rank i on

Ec

i from Prop.3 whilst Qi being identified with

Qi EB

On-i

C

Qn.

Let

j

E

J(n).

If

j

= c

E

Jo

we

~t

fEIL,c

=

fElL·

Suppose that

j

E

Ji

with 1

~

i

~

n.

Consider the subgroup

rEc;ILc;,j

of

rEc;/Lci

from n.4 and denote

by

rEIL,j

its preimage with respect to the composition of all projections

7r8

with

s

= i

+

1, ... ,

n.

It is easy to see that

{rEIL,j

}jEJ(n)

is a decreasing filtration by

normal subgroups of

r

E

1

L.

This completes the definition of ramification filtration

of the group

fElL

in lower numbering.

Define the Her brand function

cp

E

1

L : J ( n)

---

J ( n)

as follows. For c

E

J0

,

set

'PEIL(c)

=c. For 1

~

i

~nand

j

E

Ji

C

J(n),

set

'PEIL(j)

=

fEc;/Lc,(j).

Clearly,

'PEIL

is a bijection of

J(n)

such that

'PEIL(Ji)

=

Ji

for all 0

~

i

~

n.

Prop.4 implies obviously the following property.

PROPOSITION 6.

Let E

:J

F

:J

L be a tower of finite Galois extensions provided

with compatible

F

-structures and let

7r

be a natural epimorphism from rEI L to

r

F I L.

Then for any j

E

J(n),

a) n(fEIL,j)

=

fFIL,pE;F(j)i

b)

'PEIL(j)

=

'PFIL ('PEIF(J)).