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

V£

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·