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;