Tame coverings of arithmetic schemes

The objective of this paper is to investigate tame fundamental groups of schemes of finite type over Spec(Z). More precisely, let X be a connected scheme of finite type over Spec(Z) and let X̄ be a compactification of X, i.e. a scheme which is proper and of finite type over Spec(Z) and which contains X as a dense open subscheme. Then the tame fundamental group of X classifies finite étale coverings of X which are tamely ramified along the boundary X̄−X, in particular, the tame fundamental group π 1(X̄, X̄ − X) is a quotient of the étale fundamental group π1(X). Our interest in the tame fundamental group arises from the observation that it seems to be the maximal quotient of π1(X) which is ‘visible’ via class field theory by algebraic cycle theories (see [S-S1], [S] and [S-S2] for more precise statements on ‘tame class field theory’).