  The $k$th module of a surface-knot of a genus $g$ in the 4-sphere is the
  $k$th integral homology module of the infinite cyclic covering of the
surface-knot
  complement. The reduced first module is the quotient module of the first
module by the finite
  sub-module defining the torsion linking. It is shown that the reduced first
module for every
  genus $g$ is characterized in terms of properties of a finitely generated
module. As a
  by-product, a concrete example of the fundamental group of a surface-knot of
genus $g$
  which is not the fundamental group of any surface-knot of genus $g-1$ is
given for every
  $g>0$. The torsion part and the torsion-free part of the second module are
determined by
  the reduced first module and the genus-class on the reduced first module. The
third module
  vanishes. The concept of an exact leaf of a surface-knot is introduced, whose
linking is an orthogonal sum of the torsion linking and a hyperbolic linking.
