Contemporary Mathematics
Volume 300, 2002
Ramification Theory for Higher Dimensional Local Fields
Victor Abrashkin
To 60th birthday of A.N.Parshin.
ABSTRACT.
The paper contains a construction of ramification theory for higher
dimensional local fields K provided with additional structure given by an in-
creasing sequence of their "subfields of i-dimensional constants", where 0
~
i
~
nand n is the dimension of K. It is also announced that a local analogue
of the Grothendieck Conjecture still holds: all automorphisms of the abso-
lute Galois group of K, which are compatible with ramification filtration and
satisfy some natural topological conditions appear as conjugations via some
automorphisms of the algebraic closure of K.
0. Introduction
This paper deals with the formalism of ramification theory of higher dimen-
sional local fields. It comes from I.Zhukov's approach
[Zh], [Ab5]
to such a theory
in the case of 2-dimensional local fields K, which is based on the introduction of
the additional structure on K given by its closed 1-dimensional local subfield Kc
of dimension 1 - "the subfield of 1-dimensional constants". Then the filtration
of
rK
=
Gal(Ksep/K) by its ramification subgroups appears in the form of de-
creasing filtration of rK by normal subgroups {r~l}}EJ( 2 ). Here J(2)
=
J
1
II
J
2
,
where J
1 =
{j
E
Q
I
j
~
0},
h
=
{j
E
Q2
I
j
~
(0, 0)} (with respect to the
lexicographical ordering on
Q2
),
and by definition each element of J2 is greater
than every element of J1. For
j
E
J1, the groups r~l appear as the preimages of
the classical ramification subgroups of r
Kc
=
Gal(Kc,sep/ Kc) with respect to the
natural projection 7r from r
K
to r
Kc.
The "2-dimensional part" of ramification
filtration of rK appears as a decreasing filtration {r~lhEJ2 of rK
=
Ker7r and its
definition can be given in terms of semistable reduction of the arithmetical scheme
SpecOK--+ SpecOKc attached to the field extension K
::J
Kc (here
OK
and
OKc
are corresponding valuation rings).
The above interpretation of Zhukov's approach admits a direct generalization
to the case of local fields of arbitrary dimension n, which are supposed to be pro-
vided with an additional F-structure given by increasing sequence of subfields of
2000 Mathematics Subject Classification. Primary 11815, 11820, Secondary 11870.
Key words and phrases. local fields, ramification, anabelian conjecture.
©
2002 American Mathematical Society
1
http://dx.doi.org/10.1090/conm/300/05140
Previous Page Next Page