Grade three perfect ideals and length four self-dual resolutions

TL;DR

The paper addresses translating the structure theory between grade-three perfect ideals and grade-four Gorenstein ideals by constructing, in a reversible way, a self-dual length-4 resolution from a grade-3 resolution. It develops both a forward A→B construction (using a DG-algebra structure and a compensating piece C) and a backward B→A reconstruction (via spinor geometry and isotropic decompositions), establishing conditions under which the resulting complexes are acyclic and resolve the desired ideals. Key contributions include an explicit acyclicity criterion tied to a grade-4 locus $J_C$, a criteria for the existence of a suitable $C$ when the input ideal is generically Gorenstein, and a spinor-based inverse that recovers a grade-3 perfect resolution from a self-dual length-4 complex, illustrated in a Gulliksen–Negård-type example. The framework paves the way for transferring structure-theoretic insights between grade-3 and grade-4 families and informs the classification of low-generator grade-four Gorenstein ideals.

Abstract

Starting with a grade three perfect ideal $I \subset R$, we demonstrate how to produce the a self-dual resolution of length four using the resolution of the original ideal. This process is also reversible. The main case of interest is when the grade three perfect ideal has type two, so the output complex resolves $R/J$ for a grade four Gorenstein ideal $J$. This suggests that the structure theory of these two families of ideals should be closely related.

Theorems & Definitions (31)

• Theorem 2.1: Buchsbaum-Eisenbud73
• Example 3.1
• Remark 3.2
• Lemma 3.3
• proof
• Theorem 3.4
• proof
• Lemma 3.5
• proof
• Proposition 3.6