  We study the axial anomaly defined on a finite-size lattice by using a Dirac
operator which obeys the Ginsparg-Wilson relation. When the gauge group is
U(1), we show that the basic structure of axial anomaly on the infinite
lattice, which can be deduced by a cohomological analysis, persists even on
(sufficiently large) finite-size lattices. For non-abelian gauge groups, we
propose a conjecture on a possible form of axial anomaly on the infinite
lattice, which holds to all orders in perturbation theory. With this
conjecture, we show that a structure of the axial anomaly on finite-size
lattices is again basically identical to that on the infinite lattice. Our
analysis with the Ginsparg-Wilson Dirac operator indicates that, in appropriate
frameworks, the basic structure of axial anomaly is quite robust and it
persists even in a system with finite ultraviolet and infrared cutoffs.
