The aim of global class field theory is the description of abelian extensions of a finitely generated field k in terms of arithmetic invariants attached to k. The solution of this problem in the case of fields of dimension 1 was one of the major achievements of number theory in the first part of the previous century. In the 1980s, mainly due to K. Kato and S. Saito [KS2], a generalization to higher dimensional fields has been found. The description of the abelian extensions is given in terms of

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... a generalized idèle class group, whose rather involved definition is based on Milnor K-sheaves. However, if one restricts attention to unramified extensions (with respect to a regular, projective model X of k), then class field theory has a nice geometric description using the Chow group CH 0 (X) of zero-cycles modulo rational equivalence (see [KS1] , [Sa]). The objective of this paper is to give a description of a similar geometric flavor of the tamely ramified abelian extensions of k (w.r.t. a finite set Y 1 , . . . , Y n of prime divisors ofX). We do this in the case of positive characteristic, where the role of CH 0 (X) is taken over by the 0th singular homology group (introduced by A. Suslin) h 0 (X), where X =X − {Y 1 ∪ · · · ∪ Y n }. The arithmetic case, i.e. if k is a finitely generated extension of Q, will be considered in a separate paper. LetX be a smooth, projective and geometrically connected variety over a finite field and let X be an open subvariety ofX. Let π t (X) be the tameétale fundamental group of X (cf. [GM]), i.e. π t (X) classifiesétale coverings of X which are tamely ramified over all prime divisors ofX not lying on X. Our main result is the existence of a natural map (reciprocity homomorphism) which is injective with dense image and which induces an isomorphism on the profinite completions. Passing to small open subschemes, this completely de-