# Linkage classes of grade 3 perfect ideals

## TL;DR

This paper explores the linkage properties of grade 3 perfect ideals in regular local rings, showing they are linked to either complete intersections or Golod ideals, and uses this to analyze Cohen-Macaulay ring structures.

## Contribution

It proves that grade 3 perfect ideals are linked to complete intersections or Golod ideals, extending linkage theory beyond grade 2 cases.

## Key findings

- Grade 3 perfect ideals are linked to complete intersections or Golod ideals.
- Linkage reveals finer structures of Cohen-Macaulay rings of codimension 3.
- Homological classification aids in understanding ideal linkages.

## Abstract

While every grade 2 perfect ideal in a regular local ring is linked to a complete intersection ideal, it is known not to be the case for ideals of grade 3. We soften the blow by proving that every grade 3 perfect ideal in a regular local ring is linked to a complete intersection or a Golod ideal. Our proof is indebted to a homological classification of Cohen-Macaulay local rings of codimension 3. That debt is swiftly repaid, as we use linkage to reveal some of the finer structures of this classification.