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.
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