Covering data and higher dimensional global class field theory

For a connected regular scheme X, flat and of finite type over Spec(Z), we construct a reciprocity homomorphism , which is surjective and whose kernel is the connected component of the identity. The (topological) group CX 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.