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)
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