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].
Previous Page Next Page