A structure theorem for type 3, grade 3 perfect ideals

Abstract

In 1977 Buchsbaum and Eisenbud gave a complete characterization of grade 3 perfect ideals I of type 1 (i.e., Gorenstein), in a noetherian local ring R. Exploiting the fact that a minimal free resolution for R/I has a structure of associative, commutative, differential graded algebra they also studied successfully the structure of grade 3 almost complete intersections ideals. In 1981 and 1982 Kustin and Miller studied the structure of Gorenstein ideals of grade 4, but they introduced a new variable to the problem. They defined the concept of defect of an ideal and they found that for grade 4 Gorenstein ideals, the structures vary, in part, according to the value of d( I), the defect of I. Later in 1984, Ann Brown in her Ph.D. thesis gave a structure theorem for grade 3 perfect ideals of type 2 and defect positive. She also proved that for grade 3 perfect ideals of type 2, d( I) can be either 0 or 1. The purpose of this paper is to give a structure theorem for grade 3 perfect ideals of type 3 and defect 2 or greater. Furthermore, this structure theorem permits us to establish that for grade 3, type 3 perfect ideals, d( I) can be either 0 or 1 or 2 or 4.