 
Summary: CHARACTERISTIC p CASE OF THE GROTHENDIECK
CONJECTURE FOR 2DIMENSIONAL LOCAL FIELDS
Victor Abrashkin
In the case of local fields K of characteristic p > 0 a local analogue of the Grothen
dieck Conjecture appears as a characterization of ``analytical'' automorphisms of the
Galois group #K of K, i.e. the automorphisms of #K , which come from conjugation
by extensions of field automorphisms of K to its algebraic closure •
K. It was proved
earlier by the author that the compatibility with the ramification filtration of #K
will be su#cient for that characterization in the case of 1dimensional local fields of
characteristic p # 3. In this paper it is shown that for higher dimensional local fields
the compatibility with ramification filtration together with some natural topological
conditions is still su#cient to characterize the ``analytical'' automorphisms of #K .
Throughout all this paper p is a prime number # 3.
Let K be a 1dimensional local field, i.e. a complete discrete valuation field with
finite residue field. Suppose K is of characteristic p. Then K is isomorphic to the
field of formal Laurent series k((t 0 )) with a finite field of coe#cients k. Choose an
algebraic closure •
K of K and denote by Iso(K, •
K) the group of all continuous field
