2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/62354We 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.8 pages, LaTeX2e using amsart documentclassAlgebraic GeometryCommutative AlgebraPrimary 14Q10; Secondary 13D02, 14Q05, 65D17text