A construction of covers of arithmetic schemes

Let X be a regular arithmetic scheme, i.e. a regular integral separated scheme flat and of finite type over Spec Z. Assume that for all closed irreducible subschemes C subset of X of dimension 1 with normalisation (C) over tilde there are given open normal subgroups N-C of pi(1) ((C) over tilde), which fulfil the following compatibility condition: For all (x) over tilde is an element of (C) over tilde (1) x (X) (C) over tilde (2) the pre-images of N-C1 and N-C2 in pi(1) ((x) over tilde) coincide. If the indices of the N-C are bounded, then these data uniquely determine an open normal subgroup of pi(1) (X), whose pre-image in pi(1) ((C) over tilde) is N-C for all C. (C) 2006 Elsevier Inc. All rights reserved.