RAMIFICATION THEORY FOR HIGHER DIMENSIONAL LOCAL FIELDS
5
REMARK. If in the notation of the above proposition
M'
is a finite extension
of
E'
then the induced F-structure of
M
:=
LM'
is not generally standard. Nev-
ertheless, we have the following property:
- if
t
1
, ... ,
tn is a system of local parameters of E, which is associated with its
(standard) F-structure, and u1, ... , Un-1 is a system of local parameters of
M'
then
Ut, ... , Un-1. tn is a system of local parameters of M.
3. The valuation

A valuation of rank
n
of an n-dimensional local field L is a map
v
L
---+
Qn U { oo} such that
- viL• is a group homomorphism from L* to Qln and v(O)
=
oo;
- v(h +l2
)
?
min{
v(h), v(l2
) },
where Qln is provided with lexicographical ordering
and by definition oo is greater than any element of Qn;
-if 1
~
i
~nand
vi is the i-th coordinate function of the map v, then
(i)
1
i -
OL
=
{l ELI (v
(l), ...
,v (l))? Oi},
where here and everywhere below
Oi
:=
(0, ... , 0) E Qi.
As usually, if
E
is an algebraic extension of
L
then there is a unique valuation
v' of rank nonE such that v'IL = v. Inversely, any valuation v' of E induces the
valuation v
=
v'IL of L. (In these both situations we often use below the same
notation for v and v'.)
Suppose an n-dimensional local field L is provided with some F -structure
{ Lc
i
I
1
~
i
~
n}. A valuation v of L will be called compatible with this F-
structure if for all 1
~
i
~
n, it holds
v(L~i) C
Qi
EB On-i' i.e. for alll
E L~i'
the
last n- i components of v(l) are zeroes. Suppose
[E: L]
oo and the valuation v'
onE is the extension of v. Then the compatibility of v with some F-structure on L
is equivalent to the compatibility of v' with the corresponding induced F-structure
on
E.
PROPOSITION 2. If v and v1 are valuations of rank n on L, which are compat-
ible with its F -structure then there is dE Qln such that for any l E L, v(l)
=
dv1(l)
- the component-wise product of vectors d and v1(l).
PROOF. By Prop.1 and the uniqueness property of extension of valuations
the statement can be reduced to the case of a field L provided with a standard
F-structure. Let
t
1
, ... ,
tn be a system of local parameters associated with such
F-structure and let for 1
~
i
~
n, v(ti)
=
(a::i1, ... , O::in) E Qln and v1(ti)
=
(
a::~
1
,
... ,
a::~n)
E Qln. By the definition of valuation of rank n we have O::ij
=
a::~j
=
0
for all i j. In addition, F-compatibility of v and v1 implies O::ij
=
a::~j
=
0 for all
i j. So, we can take d = ( a::n
I
a~ 1, ... , O::nn
I
a::~n). The proposition is proved.
If
[E:
L] oo, introduce the vector ramification index
eE/L
=
(e1, ...
,en) E
.zn
by setting for 1
~
i
~
n,
ei
=
[Eci: LciEc,i-1]
=
[Eci: Lci][Ec,i-1: Lc,i-1r
1.
Notice that if
L
C
E
C
E
1
is a tower of finite extensions (with compatible
F-
structures) then
eE1 jL
=
eE1 jEeEjL·
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