Rigid invariance as derived from BRS invariance: The abelian Higgs model

Consequences of a symmetry, e.g.\ relations amongst Green functions, are renormalization scheme independently expressed in terms of a rigid Ward identity. The corresponding local version yields information on the respective current. In the case of spontaneous breakdown one has to define the theory via the BRS invariance and thus to construct rigid and current Ward identity non-trivially in accordance with it. We performed this construction to all orders of perturbation theory in the abelian Higgs model as a prelude to the standard model. A technical tool of interest in itself is the use of a doublet of external scalar ``background'' fields. The Callan-Symanzik equation has an interesting form and follows easily once the rigid invariance is established.