RAMIFICATION THEORY FOR HIGHER DIMENSIONAL LOCAL FIELDS 3

Only in the above two cases the absolute Galois group

rL

=

Gal(Lsep/L) is

complicated enough to be provided with interesting ramification filtration.

Let t1, ... , tn be a system of local parameters of L, i.e. for all 1

~

i

~

n,

ti

E m~) and

ti

mod m~-

1

) is uniformizing element of the complete discrete valu-

ation field

£(i-1) = or-1)

mod

mr-1).

Notice that tt, ... , tn is a system of local

parameters if and only if

h

is uniformizing element of

0_i,1),

t2, ... , tn

E

0_i,1)*

and

t2 mod m ~), ... , tn mod m ~) is a system of local parameters in

£(1).

Clearly, L can

be identified with the set of all formal Laurent series

l

=

L

[ail•••iJt~l

• • •

t~n)

(2)

h, ...

,in

where the sum is taken for all multi-indices ( i1, ... , in) such that for some (de-

pending on l) lower boundaries m, m(i1), ... , m(i1, ... ,in_1), one has i1

~

m,

i2

~

m(i1), ... , in

~

m(it, ... , in_!), and [ai

1

•..

iJ are Teichmiiller representatives

of ai

1

...

in E £(n) (if char L

=

char L(n), then the Teichmiiller representative of

a E £(n) is just a itself). This identification has been considered in basic papers on

higher dimensional local fields (A.Parshin, K.Kato) via introducing a special topol-

ogy on

L,

with respect to which (we shall call it the ?-topology) the above series

(2)

are convergent (the concept of ?-topology was analyzed and studied later by

I.Fesenko and I.Zhukov). Actually, the ?-topology brings into correlation all n dis-

crete valuation topologies of the fields L

=

£(0

), .•• ,

L(n-

1).

Notice that operations

of addition and multiplication are sequentially ?-continuous in

L. If

1

~

i

~

n

and the ring or)

c

L is provided with the induced ?-topology, then all natural

projections pri : or)

--t

£(i) are continuous. On the other hand, any choice of

local parameters t1, ... , tn gives rise to continuous sections

Si :

£{i)

--t

or) of pro-

jections pri and implies a description of elements from Las formal power series

(2).

It is also known that the ?-topology of a finite extension E of L is compatible with

that of L with respect to an identification of £-vector spaces E

~

Lm, m

=

[E: L],

given by some choice of £-basis in E. For these and related results we refer again

to the book [HLF].

So, it is natural to consider the ?-topology as an essential part of the concept

of higher dimensional local field. In other words, when working with the category

of higher dimensional local fields we shall consider only ?-continuous field mor-

phisms. For example, if t1, ... , tn is a system of local parameters in

L,

then any

'¢ E AutP-top(L) is uniquely determined by the images '¢(t1), ... , '¢(tn), which have

to form again a system of local parameters in L.

2. F -structure

If L

is an n-dimensionallocal field then its

F

-structure is given by an increasing

sequence of its closed subfields Le 1

C

Le 2

C · · · C

Len

=

L such that for all

1

~

i

~

n,

-Lei is a closed i-dimensional subfield of L;

-Lei is algebraically closed in L.

The subfields Lei may be treated as subfields of "i-dimensional constants". It will

be also convenient to introduce the subfield of 0-dimensional constants.

If

char L is

positive, then the last residue field L(n) can be naturally identified with a unique

subfield of L and of all Lei, 1

~

i

~

n.

So, L(n) may be interpreted as the subfield