12
VICTOR ABRASHKIN
6. A local analogue of the Grothendieck Conjecture
6.1. The category
FPG(n), n
EN. The objects of this category are profinite
groups
G
with decreasing filtration by its normal closed subgroups {GUl}jEJ(n)·
Suppose
H
is an open subgroup of
G.
Define "the vector ramification in-
dex"
eaH
= (el, ... ,en)
E
zn,
where el =
(G(
0
)H: G(O,O)H), ... ,
En-1
=
(G(0n-l)H:
G(0n)H),
en=
(G(0n)H:
H).
Define also "the inverse Herbrand function"
'1/JaH : J(n)
-----
J(n)
by setting
'1/JaH(c)
= c and
where j E
Ji,
1 ~ i ~
n, H(o,)
:=
H
n G(o,) and, as earlier, the vector eaH,~i E
zi
consists of the first i coordinates of the vector
eaH.
If G1
,
G2 are objects of the category
FPG(n),
then the set of morphisms
HomFPG(n)(G1, G2) consists of open embeddings i : G1
-+
G2 such that for any
j
E
J(n),
it holds
i(Gi/Ja
2
i(G
1
J(j)))
=i(Gl)nG~j).
(7)
Following arguments from [Ab4], n.l.2, one can verify that the composition of any 2
morphisms in
FPG(n)
is again a morphism in
FPG(n)
(what is actually equivalent
to the composition property of the above inverse Herbrand function). Therefore,
FPG(n)
is a category.
Define also the category FPGP(n). Its objects are objects
G
of the category
FPG( n) provided with additional structure given by some topology on the maximal
abelian quotient Hab of every open subgroup
H
of
G.
These topologies must be
compatible with natural maps
HJ:b
-----
Hab, where H
1
is another open subgroup
of
G
such that H
1
C
H.
Morphisms in FPGP(n) are morphisms
7f :
G1
-+
G2
from
FPG(n)
such that for any open subgroup H of G1 the corresponding map
1rJj' :
Hab
-+
1r(H)ab is continuous with respect to the corresponding topologies of
these abelian subquotients.
6.2. The category
DVFp(n).
Choose a basic n-dimensionallocal field
L
0
=
lF
P ( (
tn)) ... ( (h))
with standard F -structure {
Loi
I
0
~
i
~
n} associated to the
system of local parameters
h, ... ,
tn.
Let
Lo
be an algebraic closure of
Lo.
The
direct limit of ?-topologies of all finite extensions of L
0
gives the ?-topological
structure on
L
0 .
Denote by
C(n)p
the completion of
Lo
with respect to its first
valuation
v6
= pr1 (vL
0
).
The ?-topological structure on
C(n)p
appears as v6-adic
topology associated with P-topology of
L
0 .
For 0
~
i
~
n,
denote by
C(i)p
the
completion of the algebraic closure of
Loi
in
C(n)p·
Notice that we have the induced
?-topological structures on the fields
JFP
=
C(O)p
C
C(1)p
C · · · C
C(n)p·
Objects of the category
DVFp(n)
are finite extensions
K
of
L
0
in
C(n)p·
Any
such field
K
is provided with induced F-structure
{Kci
I 0
~
i
~
n}, where
Kci
=
K
n
C(i)p·
Notice that
C(n)~K
=
R(K)-
the radical closure (=the completion of
the maximal purely non-separable extension) of Kin
C(n)p·
Similarly, for 0
~
i
~
n,
it holds that
C(i)~K
=
R(Kci)·
Suppose K, L
E
DVF
P (
n). Then the corresponding set of morphisms
HomnvF(K,L) in the category
DVFp(n)
consists of all ?-continuous field mor-
phisms
p: C(n)p-+ C(n)p
such that for 1
~
i
~
n,
a)
p(C(i)p)
=
C(i)p;
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