  In this paper I investigate minimal surfaces of general type with p_g=5, q=0
for which the 1-canonical map is a birational morphism onto a surface in P^4
(so called canonical surfaces in P^4) via a structure theorem for the Hilbert
resolutions of the canonical rings of the afore-mentioned surfaces, viewed as
Gorenstein algebras of codimension 2 over the homogeneous coordinate ring of
P^4. I discuss how the ring structure of such an algebra is encoded in its
resolution. Among other things I show how this method can be applied to analyze
the moduli space of canonical surfaces with p_g=5, q=0, K^2=11, thus recovering
a result previously obtained by D. Rossberg with different techniques.
