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).