Local Complete Intersections in P and Koszul Syzygies
نویسنده
چکیده
X iv :m at h/ 01 10 09 7v 1 [ m at h. A G ] 9 O ct 2 00 1 LOCAL COMPLETE INTERSECTIONS IN P AND KOSZUL SYZYGIES DAVID COX AND HAL SCHENCK Abstract. We study the syzygies of a codimension two ideal I = 〈f1, f2, f3〉 ⊆ k[x, y, z]. Our main result is that the module of syzygies vanishing (schemetheoretically) 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 [4]. When I is saturated, we relate our theorem to results of Weyman [7] and Simis and Vasconcelos [6]. 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.
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