An ADE correspondence for grade three perfect ideals

TL;DR

This paper extends the Buchsbaum-Eisenbud structure theorem to classify grade three perfect ideals with small type and deviation, establishing an ADE correspondence and analyzing Betti table restrictions.

Contribution

It introduces a new ADE correspondence framework for grade three perfect ideals, generalizing existing theorems and providing classification results.

Findings

Classification of grade 3 perfect ideals with small type and deviation

Establishment of an ADE correspondence for these ideals

Restrictions on Betti tables in the graded setting

Abstract

Using the theory of "higher structure maps" from generic rings for free resolutions of length three, we give a classification of grade 3 perfect ideals with small type and deviation in local rings of equicharacteristic zero, extending the Buchsbaum-Eisenbud structure theorem on Gorenstein ideals and realizing it as the type D case of an ADE correspondence. We also deduce restrictions on Betti tables in the graded setting for such ideals.