6

VICTOR ABRASHKIN

PROPOSITION 3.

Any n-dimensional local field L with an F -structure can be

provided with a unique valuation v L of rank n such that

a)

if Lhasa standard F-structure and

t1, ... ,

tn is a corresponding system of local

parameters, then for alll

~

i

~

n, VL(ti)

=

(8il, ... ,

8in), where 8 is the Kronecker

symbol;

b)

if

[E: L]

oo, where E has standard F-structure, then VL

=

eEfLVE.

PROOF. a) Clearly, the values

vL(ti), 1

~

i

~

n, determine

VL

uniquely and it

is easy to see that for any other corresponding system oflocal parameters

u1. ... , un,

it holds

vL(ui)

=

vL(ti), i

=

1, ... ,n.

b) It will be sufficient to verify that if E and L are provided with standard

F

-structures, then

(3)

Suppose

h, ... ,

tn

is a corresponding system of local parameters in

L

and u1

, ... ,

Un

is a corresponding system of local parameters in E. Then relation (3) easily follows

from the fact that for any 1

~

j

~

n,

u1, ... , Uj-1,

ti

is a system oflocal parameters

of

Lc i Ec,j _

1

and u1

, ... ,

Uj

is a system of local parameters of

Ec i.

The proposition

is proved.

Notice that the above valuation

VL

is automatically compatible with given

F-

structure on Land for any finite extension

E

of

L,

it holds

VE = eEfLV£.

Besides,

for any

1

~

i

~

n,

VL

induces the valuation

VLc;

when being restricted to

Lei·

On the other hand, if £(i) is the i-th residue field of

L,

where 1

~ i ~

n, then

VL

generally does not induce the valuation

V£(iJ

on £(i). But this will be true if

e.g. a given

F

-structure of

L

is standard.

4. Subgroups

rE/L

and its ramification subgroups

Let

L

0

be a local field of dimension

n

with F-structure. Choose an algebraic

closure

Lo

of

Lo

and suppose everywhere below that any algebraic extension

L

of

Lo

is chosen inside

Lo

and is provided with the induced

F

-structure {

Lc

i

I

0

~

i

~

n}.

For any finite normal extension

E

of

L,

set

rE/L =

Gal(E/L(i)), where L(i)

is the maximal purely non-separable extension of

L

in

E.

Notice that

rE/L

is

identified also with the Galois group of the maximal separable extension

E(s)

of

L

in

E, cf.

[Jac], n.8.7.

Wit~

the above agreement use the induced F-structure on

E

to introduce the group

rE/L

:=

rE/LEc,n-1'

Clearly, there is a natural exact

sequence

1----+ fE/L----+ fE/L----+ fEc,n-1/Lc,n-1 ----+

1.

Let

Jn

=

{j

E

Qn

I

j

~

On}, where

Qn

is provided with the lexicographical

ordering. Consider a finite extension

M'

of

Ec,n-

1

in

L

0

such that the induced

F

-structure {

Ec,i

I

1 ~

i

~ n} of

E

:=

M' E

is standard, cf. Prop.

1.

Then

any system of local parameters

t

1 , ... ,

tn-

1

of

Ec,n-

1

=

M'

can be extended to

a system of local parameters

h, ... ,

tn-1.

(}of

E

= EM'.

Let

L

= LM'.

Then

the extension of !-dimensional complete discrete valuation fields _E(n-

1)

;Dn-1) is

totally ramified and

0

mod m ~- 1 ) is uniformizing element of _E(n-

1).

This implies

that

h, ... ,

tn-1 N E/L(}

is a system of local parameters of

L

and we obtain very

important property of monogeneity OE

=

Oz[O].