Tamely ramified covers of varieties and arithmetic schemes

Let Y -> X be an etale cover of regular varieties or arithmetic schemes. We give the following intrinsic characterisation of tameness: Y -> X is tamely ramified (outside X), iff for every irreducible curve C subset of X with normalisation (C) over tilde the induced cover Y x(X) (C) over tilde -> (C) over tilde is tamely ramified (outside (C) over tilde).