RAMIFICATION THEORY FOR HIGHER DIMENSIONAL LOCAL FIELDS

13

b)

p(Kei)

C

LeiR(Le,i-d-

the closure of the composite of

Lei

and

R(Le,i-d

in

C(n)p;

c)

LeiR(Le,i-d

is separable over

p (KeiR(Ke,i-1)).

Notice that for all i,

Kei, Lei

E DVFp(i) and 'PIKe, E HomovF(Kei,

Lei)·

If 'Y : K

-----+

L is a separable field embedding then it induces a morphism in

DVFp(n),

which we denote by the same symbol 'Y·

It is easy to see that

p

E HomovF(K,

L)

is isomorphism if and only if

LR(Le,n-d

=

p(KR(Ke,n-1)).

This implies that

LeiR(Le,i-1)

=

p(KeiR(Ke,i-1)),

i.e. 'PIKe, is

an isomorphism in DVFp(i) and

R(Lei)

=

p(R(Kei))

for all1

~

i

~

n.

PROPOSITION 11.

Any p

E HomovF(K,L)

is uniquely decomposed into the

composition of a field embedding and an isomorphism.

PROOF. The proof can be obtained by the use of the following lemma, which

is a consequence of Krasner's Lemma and results from [Jac], n.8.7.

LEMMA.

Suppose K' is a closed sub field of K

E DVF

p( n). Then for any finite

separable extension M

of K R( K') of some degree d, there is a unique separable

extension M of K of degree d such that M

=

MR(K'). In addition, if M' is

the algebraic closure of K' in M then R(M')

=

M' R(K') and, therefore, M

=

MR(M').

6.3. The category

DVFop(n).

Choose a basic n-dimensionallocal field

L

0

=

Qp { { tn}} ... { { t2}}

with the standard F -structure {

Loi

I

0

~

i

~

n} associated to

the system of local parameters

p

=

h, t

2 , ... ,

tn.

Let

L0

be an algebraic closure

of

L

0 .

Denote by

IC(n)p

the completion of

L0

with respect to its first valuation.

For 0

~

i

~

n,

denote by IC(i)p the completion of the algebraic closure of

Loi

in

IC(n)p-

As earlier, the ?-topological structure of finite extensions of

L

0

induces

?-topological structures on the fields

Qp,ur

=

IC(O)p

C

IC(1)p

C · · · C

IC(n)p-

The objects of the category

DVFop(n)

are finite extensions

K

of

L

0

in

IC(n)P"

Any such field

K

is provided with the induced F-structure {

Kei

I

0

~

i

~

n },

where

Kei

=

K

n

IC(i)p·

Notice that

IC(n)~K =

K

cf. [Hy] and, similarly, for all

1 ~ i ~

n,

IC(i)~K

=

Kei·

Suppose

K, L

E

DVFop(n).

Then the corresponding set of morphisms

HomovF(K,

L)

in the category

DVFop(n)

consists of all ?-continuous field mor-

phisms

p:

IC(n)p ____, IC(n)p

such that for 1

~

i

~

n,

a) cp(IC(

i)p)

=

IC(

i)p;

b) p(Kei)

C

Lei·

6.4. The functor

RFp.

Let

K

E

DVFp(n).

Then

K

is provided with canon-

ical F-structure and, therefore,

RFp(K)

:=

fK

=

Gal(Ksep/K), where Ksep is the

separable closure of Kin

C(n)p,

being provided with the corresponding ramification

filtration becomes an object of the category

FPG(n).

Let

L

E

DVFp(n)

and

p

E HomovF(K,L). By Lemma from n.6.2 the cat-

egories of separable extensions of

L

and of

K

are equivalent to the categories of

separable extensions of

LR(Le,n- 1)

and, respectively, of

KR(Ke,n-d·

Therefore,

the separable field embedding

p : K R(Ke,n-d

-----+

LR(Le,n-d

gives rise to the

embedding

rp

of the first category into the second and we obtain an open embedding

of topological groups

p* :

f

L

-----+

f

K.