Grade three perfect ideals and length four self-dual resolutions

TL;DR

This paper explores a method to construct self-dual resolutions of length four from grade three perfect ideals, revealing a close relationship between these ideals and grade four Gorenstein ideals.

Contribution

It introduces a reversible process to produce self-dual resolutions of length four from grade three perfect ideals, linking their structure to grade four Gorenstein ideals.

Findings

Constructs self-dual resolutions of length four from grade three perfect ideals.

Shows the process is reversible, connecting different ideal classes.

Highlights the structural relationship between these ideal families.

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.