Covering data and higher dimensional global class field theory

Abstract

For a connected regular scheme X, flat and of finite type over Spec ( Z ) , we construct a reciprocity homomorphism ρ X : C X → π 1 ab ( X ) , which is surjective and whose kernel is the connected component of the identity. The (topological) group C X is explicitly given and built solely out of data attached to points and curves on X. A similar but weaker statement holds for smooth varieties over finite fields. Our results are based on earlier work of G. Wiesend.