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.
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