Free resolutions of Dynkin format and the licci property of grade 3 perfect ideals

TL;DR

This paper explores the connection between Dynkin diagram structures in free resolutions of grade 3 perfect ideals and their linkage class, proposing a conjecture linking these algebraic objects.

Contribution

It introduces a conjecture that grade 3 perfect ideals with Dynkin-type resolutions are in the linkage class of complete intersections.

Findings

Proposes a conjecture relating Dynkin diagrams to linkage classes.

Highlights the significance of graph structures in free resolutions.

Suggests a classification framework for grade 3 perfect ideals.

Abstract

Recent work on generic free resolutions of length 3 attaches to every resolution a graph and suggests that resolutions whose associated graph is a Dynkin diagram are distinguished. We conjecture that in a regular local ring, every grade 3 perfect ideal whose minimal free resolution is distinguished in this way is in the linkage class of a complete intersection.