Structure theory for some classes of grade three perfect ideals

We show that the Buchsbaum–Eisenbud structure theorem for almost complete intersections of grade 3 can be characterized by the almost complete matrix f of grade 3 and its associated ideal K3(f). We also provide a simple proof of the structure theorem for some classes of perfect ideals of grade 3 which are algebraically linked to an almost complete intersection of grade 3 by a regular sequence. This contains three classes of perfect ideals of grade 3 which were determined by Buchsbaum–Eisenbud, Brown and Sanchez. Finally we give an additional proof of the Buchsbaum–Eisenbud structure theorem for Gorenstein ideals of grade 3.