  So far only six families of smooth irregular surfaces are known to exist in
P^4 (up to pullbacks by suitable finite covers of P^4). These are the elliptic
quintic scrolls, the minimal abelian and bielliptic surfaces (of degree 10),
two different families of non-minimal abelian surfaces of degree 15, and one
family of non-minimal bielliptic surfaces of degree 15.
  The main purpose of the paper is to describe the structure of the
Hartshorne-Rao modules and the syzygies for each of these smooth irregular
surfaces in P^4, providing at the same time a unified construction method (via
syzygies) for these families of surfaces.
