  We present perturbative calculations made with domain-wall fermions which
possess a finite number of points $N_s$ in the extra fifth dimension. We have
derived the required propagator functions, investigated the one-loop properties
of quark amplitudes at finite $N_s$ and evaluated three quantities that can
provide insights on chirality-breaking effects from the perturbative side.
  First we have computed the residual mass for various choices of $N_s$ and of
the domain-wall height $M$. We have found that this radiatively induced mass
approaches zero reasonably fast with the extent of the fifth dimension,
depending on $M$ and on a lesser extent on the coupling $g_0$. We have also
computed the differences of the renormalization constants of the vector and
axial-vector currents and of the scalar and pseudoscalar densities. Finally we
have calculated the chirally-forbidden mixing (which at finite $N_s$ is
suppressed only partially) of an operator which describes the lowest moment of
the $g_2$ structure function. In general we see that at $M=1.8$, where
simulations are usually performed, values of $N_s=20$ or larger would be
desirable in order for chiral violations to be negligible.
  The quantities that we have studied turn out to lose gauge invariance when
$N_s$ is not infinite. We have also found that anomalous dimensions of
operators at finite $N_s$ generally depend on $N_s$ and $M$. In particular, the
vector and axial-vector currents have in general a nonzero anomalous dimension
at finite $N_s$.
