Ramification theory for higher dimensional local fields.

Abstract

The paper contains a construction of ramification theory for higher dimensional local fields $K$ provided with additional structure given by an increasing sequence of their "subfields of i-dimensional constants", where $0\le i\le n$ and $n$ is the dimension of $K$. It is also announced that a local analogue of the grothendieck Conjecture still holds: all automorphisms of the absolute 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 $K$.