RAMIFICATION THEORY FOR HIGHER DIMENSIONAL LOCAL FIELDS
Now we can set RF
to obtain the functor RF
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
rK is an open subgroup then
and Hab is provided with the ?-topological structure
coming from ?-topology on
by Witt-Artin-Schreier duality. Clearly, RF p( cp) is
a morphism of the category FPGP(n).
The functor RF
is faithful. This follows from the faithfulness of action of
the group of all ?-continuous field automorphisms of
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
THEOREM 1 '. Suppose that K is an n-dimensional local field of characteris-
tic p, K' is its subfield of (n- !)-dimensional constants and
"n-dimensional part of ramification filtration" of
Then any continuous group
for any j
b) for any open subgroup H of
is induced by a ?-continuous field automorphism
The proof follows the strategy from the proof of the corresponding !-dimensional
and will appear in
for the case of 2-dimensionallocal
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-
6.5. The functor RF0
DVFop(n). As earlier, K is provided with
the canonical F-structure and RF0 (K)
rK is an object of the category FPGP(n).
is a finite extension
and the corresponding group embedding
gives the morphism
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
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.
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
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).