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
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