A structure theorem for a class of grade three perfect ideals

Abstract

is a minimal free resolution of R/Z, then type(Z) = rank(F,,). In this paper, we prove a structure theorem for a certain class of grade 3, type 2, perfect ideals in a noetherian local ring R. If Z is such an ideal, the structure theorem consists of an explicit minimal free resolution for R/Z and hence, a minimal set of generators for I. We employ a numerical invariant of an ideal Z in a noetherian local ring R to describe the class completely. Let (IF, D) be a minimal free resolution of R/Z. Let C= Im(D,) and let K be the submodule of C which is generated by the Koszul relations on the entries of D,. Note that if Z is minimally generated by r, ,..., rn and {e, ,..., e,,} is a basis of F,, then K is the module generated by the set {r,ei + riej: 1 d i < j< n}. Define