  We study the syzygies of a codimension two ideal I = <f_1,f_2,f_3> in
k[x,y,z]. Our main result is that the module of syzygies vanishing
(scheme-theoretically) at the zero locus Z = V(I) is generated by the Koszul
syzygies iff Z is a local complete intersection. The proof uses a
characterization of complete intersections due to Herzog. When I is saturated,
we relate our theorem to results of Weyman and of Simis and Vasconcelos. We
conclude with an example of how our theorem fails for four generated local
complete intersections in k[x,y,z] and we discuss generalizations to higher
dimensions.
