14

VICTOR ABRASHKIN

PROPOSITION 12.

p*

E HomFPG(rL,

rK ).

PROOF.

If p

comes from a separable field embedding of

K

into

L

then our

proposition follows from Prop.7. Therefore, by Prop.ll we can assume that

p

is

isomorphism and we must prove that for any

j

E

J(n),

it holds p*(r~l)

=

r~l.

Suppose that L' is a finite Galois extension of L, then K'

:=

P(L') is Galois

over K and we obtain induced group isomorphism

p* :

r

L' I L

---t

r

K' I K.

We must

verify that for any

j

E

J ( n),

(8)

The compatibility of

p

with

F

-structures on

K

and

L

implies that for all 0

~

i

~

n, p*

is compatible with natural projections r

L' I L

---t

r

L~

J

Lc;

and r

K' I K

---t

r

K' . IK . ,

and induces group isomorphisms

C't C't

'Pc*•:

rL'

-IL ·

---t

rK'

-IK ··

" c

't

c' c

1.

c

1.

(9)

We can assume by induction that

p~

i

are compatible with ramification filtration

for all

i

n and, therefore, it is sufficient to prove (8) only for all

j

E Jn. Notice

also that ( 9) implies that e( L'

I

L)

=

e( K'

I

K).

Choose a finite extension M of L~,n- 1 such that if

l/

=

L' M, L

=

LM,

K

=

,p(i)

and

K'

=

,p(l/),

then:

a) there is L~ with standard F-structure such that L'

c

L~

c

l/;

b) there is K~ with standard F -structure such that K'

C

K~

C

K'.

Then there are

fh

E L~ and

OK

E K~ such that

OL,

=

OL[OL]

and

OK,

OJ[OK], cf.

Remark in n.4. Therefore,

oi'R(L'

l

=

oLR(L _

l[OL]

=

oLR(L _

J['P(OK)].

c,n-1 c,n 1 c,n 1

(10)

LEMMA.

If VL' and vK' are valuations of rank n from Prop.3, then p*vL' =

VK', i.e. for any z

E

C(n)p, VK'(z)

=

vL'('P(z)).

PROOF OF LEMMA. Because

e(l/ I

L')

=

e(K' I

K'), it will be sufficient to prove

that

p*vL,

=

vK''

By induction we can assume also that

p*vL,

and

Vf,

coincide

when being restricted to K~,n- 1 .

Notice, that any system of local parameters of K~,n-

1

being completed by

OK

gives a system of local parameters of

K'.

So, we must prove only that

vL, (p(OK))

=

VJ,(OK)·

From the definition of valuations

vL,

and

vi,

it follows

V£, (OL)

=

ey) 1 L~ V£~ (OL)

=

(0, ... , 0, 1) and, similarly,

Vf, (OK)

=

(0, ... , 0, 1), i.e.

Vf! (OK) =VI,, (OL)·

It

remains only to note that

OK

and

OL

appear as lifts of uniformizing elements

of the (n- 1)-th residue fields of the fields

l/

R(L~

n- 1

)

and

K'

R(K~

n-1

),

which

are isomorphic under

p.

Therefore,

VI,,('P(OK))

=

vi;,(OL). '

The lemma is proved.

From (10) it follows that we can use

p(OK)

instead of

OL

to compute rami-

fication invariants of

rL'IL·

So, for any

T

E

rL'IL

=

ri'IL'

it holds

iL'IL(r)

=

V£1 ( r( p(OK))- p( OK)) -VL' ( p(OK))

=

VK' ( p* ( r)OK -OK) -VK' (OK)

=

iK' 1 K( p*T ).

The proposition is proved.