  We give a perturbative proof that U(1) lattice gauge theories generate the
axial anomaly in the continuum limit under very general conditions on the
lattice Dirac operator. These conditions are locality, gauge covariance and the
absense of species doubling. They hold for Wilson fermions as well as for
realizations of the Dirac operator that satisfy the Ginsparg-Wilson relation.
The proof is based on the lattice power counting theorem. The results
generalize to non-abelian gauge theories.
