  We formulate lattice perturbation theory for gauge theories in noncommutative
geometry. We apply it to three-dimensional noncommutative QED and calculate the
effective action induced by Dirac fermions. In particular "parity invariance"
of a massless theory receives an anomaly expressed by the noncommutative
Chern-Simons action. The coefficient of the anomaly is labelled by an integer
depending on the lattice action, which is a noncommutative counterpart of the
phenomenon known in the commutative theory. The parity anomaly can also be
obtained using Ginsparg-Wilson fermions, where the masslessness is guaranteed
at finite lattice spacing. This suggests a natural definition of the
lattice-regularized Chern-Simons theory on a noncommutative torus, which could
enable nonperturbative studies of quantum Hall systems.
