Multidimensional theorem of shafarevich and serre

Abstract

Let H be the group obtained by taking the product of n copies of the maximal ideal of the ring of integers 0 of a local field of characteristic 0 with an algebraically closed residue field k of characteristic p>0, and let the composition law be defined as for an n-parametric commutative formal group over 0. Let the kernel of multiplication by p in H be finite. A filtration pmH (m≥0 is an integer) in H is introduced whose properties allow us to obtain an exact sequence of proalgebraic groups 0→Z p r →Ws→H→ 0, where Zp and W are the additive groups of p-adic integers and Witt vectors of infinite length over k, respectively; r≥0 and s>0 are integers.