The local correspondence over absolute fields: an algebraic approach

Abstract

Let K,K′ be infinite fields which are finitely generated over their prime fields. Pop proved using model-theoretic methods that any isomorphism of the absolute Galois groups of K and K′ maps the decomposition groups of the Zariski prime divisors on K bijectively onto the decomposition groups of the Zariski prime divisors on K′ (relative to the separable closures). This was a main ingredient in his proof of the 0-dimensional case of Grothendieck's anabelian conjecture. In this paper we give a simplified and purely algebraic proof of this fact.