  We analyze gauge parameter dependence by using an algebraic method which
relates the gauge parameter dependence of Green functions to an enlarged
Slavnov-Taylor identity. In the course of the renormalization it turns out that
gauge parameter dependence of physical parameters is already restricted at the
level of Green functions. In a first step we consider the on-shell conditions
which we find to be in complete agreement with these restrictions to all orders
of perturbation theory. The fixing of the coupling, however, is much more
involved outside the complete on-shell scheme. In the Abelian Higgs model we
prove that this fixing can be properly chosen by requiring the Ward identity of
gauge invariance to hold in its tree form to all orders of perturbation theory.
