  We examine the dependence on all gauge parameters in the example of the
Abelian Higgs model by applying a general algebraic method which roots in an
extension of the usual Slavnov-Taylor identity. This method automatically
yields all information about the gauge parameter dependence of Green functions
and therefore especially allows to control the range of ``good'' normalization
conditions. In this context we show that the physical on-shell normalization
conditions are in complete agreement with the restrictions dictated by the
enlarged Slavnov-Taylor identity and that the coupling can be fixed in an
easily handleable way on the Ward identity of local gauge invariance. As an
application of the general method we also study the Callan-Symanzik equation
and the renormalization group equation of the Abelian Higgs model.
