RAMIFICATION THEORY FOR HIGHER DIMENSIONAL LOCAL FIELDS
15
Now we can set RF
P (
cp)
=
cp*
to obtain the functor RF
P :
DVF
P (
n) -----+
FPG(n). Actually, if K E DVFp(n) then rK can be considered naturally as an
object of the category FPGP(n). Indeed, if H
c
rK is an open subgroup then
H
=
r
E
where
[E :
K]
00
and Hab is provided with the ?-topological structure
coming from ?-topology on
E
by Witt-Artin-Schreier duality. Clearly, RF p( cp) is
a morphism of the category FPGP(n).
The functor RF
P
is faithful. This follows from the faithfulness of action of
the group of all ?-continuous field automorphisms of
K R(Kc,n-
1
)
on the Galois
group of the maximal abelian extension of K of exponent p. The proof is based on
a suitable version of Artin-Schreier theory. Actually, we have the following local
analogue of the Grothendieck Conjecture in characteristic p:
THEOREM 1. The functor RFp: DVFp(n)-----+ FPGP(n) is fully faithful.
The above formalism of ramification theory reduces the above statement to the
following result.
THEOREM 1 '. Suppose that K is an n-dimensional local field of characteris-
tic p, K' is its subfield of (n- !)-dimensional constants and
{r~l}jEJn
is the
"n-dimensional part of ramification filtration" of
r
K.
Then any continuous group
automorphism
7r :
f
K ---
f
K
such that
a)
for any j
E
Jn, 1r(f~))
=
f~);
b) for any open subgroup H of
r
K'
7fiHab
is
p
-continuous,
is induced by a ?-continuous field automorphism
cp
of
K R(K')
such that
cp(R(K'))
=
R(K').
The proof follows the strategy from the proof of the corresponding !-dimensional
property from
[Ab4]
and will appear in
[Ab6]
for the case of 2-dimensionallocal
field
K.
We use the explicit description of ramification filtration of the maximal
quotient of the Galois group of the maximal p-extension rK(P) of nilpotence class
2. Then we prove that any its group automorphism, which is compatible with
ramification filtration and ?-continuous on rK(P)ab, must satisfy very serious re-
strictions.
6.5. The functor RF0

Let K
E
DVFop(n). As earlier, K is provided with
the canonical F-structure and RF0 (K)
:=
rK is an object of the category FPGP(n).
If
L
E
DVFop(n) and
cp
E
HomnvF(K,
L)
then
cp(K)
C
L
is a finite extension
and the corresponding group embedding
f
L
C
f
p(K)
£
f
K
gives the morphism
RFo(cp)
E
HomFPGP(rL,rK)· Again RFo is a functor and we have
THEOREM 2. The functor RFo: DVFop(n) -----+ FPGP(n) is fully faithful.
The proof follows again the strategy from
[Ab4].
First of all, we adjust the
construction of the field-of-norms functor to the case of higher dimensional local
fields. Then we apply it to deduce Theorem 2 from Theorem 1.
References
[AN]
J.W.
Cassels, A.Frohlich (eds.), Algebraic number theory (Proceedings of an instructional
conference organized by the London Mathematical Society), London:Academic press, 1967.
[HLF] I. Fesenko, M. Kurihara (eds.), Invitation to Higher Local Fields. Geometry
€3
Topology
Monographs, vol. 3, 2000, p. 5-18.
[Ab1J V.A. Abrashkin, Ramification filtration of the Galois group of a local field. II, Proceedings
of Steklov Math. Inst. 208 (1995).
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