  We investigate on the plane the axial anomaly for euclidean Dirac fermions in
the presence of a background Aharonov--Bohm gauge potential. The non
perturbative analysis depends on the self--adjoint extensions of the Dirac
operator and the result is shown to be influenced by the actual way of
understanding the local axial current. The role of the quantum mechanical
parameters involved in the expression for the axial anomaly is discussed. A
derivation of the effective action by means of the stereographic projection is
also considered.
