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)).
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