RAMIFICATION THEORY FOR HIGHER DIMENSIONAL LOCAL FIELDS 11
In order to prove
b)
notice that
(f~lf~n)
:
f~nl)
=
(f~)
:
f~)
n
f~;l) (f~)
n
f~nl
:
f~l
n
f~,nl)
-
1
,
where the first factor equals
If~~
1
L
I
=
If
E,
1 L,'lf
E,
1
L(j)
I and the second factor equals
(
f('lfE;dJ)). f('lfE;dJ))
n
f(On))- 1 -If .
~-1
E · E
Et - E,jE,'IfE1
;L(J) ·
So, the derivative of the right-hand side in (6) equals
eE,; L
If
Et/
L,'lfE,
1
dJl
l-
1
e:E~
1 E
If
Et/
E,'lfE,
1
dJl I
=
'P'e,jL
('l/JE,;L(j))-1'P'e,jE
('1/JE,;L(j))
=
'1/J'e,;L(j)'l/J'e,;E ('l/JEIL(j))-
1
=
'1/J'eiL(j).
The proposition is proved.
Notice that the left-continuity property of the ramification filtration implies
PROPOSITION
8.
For any finite extension E I L of n-dimensional local fields,
there is a maximal j(EIL)
E
J(n), such that f~)L acts non-rivially onE if and
only if j
~
j(EIL).
One must be a bit careful about the definition of edge points in
Oi
E
Ji for
1
~
i
~
n.
They should correspond to tamely ramified sub-extensions. Anyway,
if
E
I
L
is a p-extension, then such sub-extensions doesn't exist, and we have the
following important property.
PROPOSITION
9.
If E I L is a p-extension, then all edge points of the Herbrand
function correspond to the jumps of the ramification filtration, and the point
(pE~L(j(EIL)),j(EIL))
is the last edge point of'PEIL·
In the paper
[Ab5]
the definition of ramification filtration was
give~
in slightly
different terms: when giving the definition of ramification subgroups f
E
I
L,j
from
n.3 we used the extension of a chosen from the very beginning valuation
v
of the
basic field
L
0
instead of the canonical valuation
VE.
Actually, if
v
=
VL
then the
both definitions of ramification filtration for the Galois group f
L
coincide. So, the
main result from
[Ab5]
gives an explicit description of the ramification filtration of
M
the groups fLifi C3 (fL), where M;?: 1, C3 (fL) is the subgroup of commutators
of order
;?:
3 and L is a 2-dimensional local field of characteristic
p
provided with
a standard F-structure. This result admits a direct generalization to the case of
n-dimensional local fields and plays a crucial role in the proof of a local analogue
of the Grothendieck Conjecture, cf. n.6 below.
As usually, let L be an n-dimensionallocal field with the subfield of i-dimensional
constants
Lc
i
and the i-th residue field
£(i),
0
~
i
~
n. Then there are natural
group epimorphisms 'lri : f
L
--+
f
Lc
i
and 7r(i) : f
L
--+
f
L(il.
By the use of the
relation between
VL, VLc,
and
vL'l
we obtain the following property.
PROPOSITION
10.
a) If j
E
J1
c
J(n) then 1ri(f~))
=
e if l
i
and 1ri(f~))
=
f~l, if l ~
i;
b)
If L is provided with a standard F-structure and j
E
J(i) then 7r(i)(f~(j)))
=
f~(il, where ~ : J(i)
---
J(n) is such that for
0
~ s ~
i
and j
E
J
8
,
it holds
~(j)
=
On-i
X
j
E
Js+n-i·
Previous Page Next Page