Generic models of licci ideals parametrized by Schur functors

TL;DR

This paper provides a combinatorial parametrization of Herzog classes of licci ideals of codimension 3 using Schur functors, leading to new classification results and structure theorems.

Contribution

It introduces a novel combinatorial framework to classify licci ideals via Schur functors and extends these results to higher codimensions under conjectural assumptions.

Findings

Parametrization of Herzog classes using pairs of partitions

Description of the graph of Herzog classes and links

New results on Tor algebra multiplications and licci ideal structures

Abstract

Let R be a commutative Noetherian ring. Licci ideals are the ideals of R that can be linked in a finite number of steps to a complete intersection. Each licci ideal admits a rigid deformation, and two licci ideals are in the same Herzog class if they have a common deformation. In this work, we show how all the Herzog classes of licci ideals of codimension 3 can be parametrized in terms of pairs of partitions associated to Schur functors. This fact allows to describe in a purely combinatorial way the (infinite) graph whose vertices correspond to Herzog classes and edges represent direct links between representatives of such classes. As applications, we obtain results on the classification of multiplications in Tor Algebras, and new structure theorems for families of licci ideals. In the final section, we extend many of these results to arbitrary codimension, but under some conjectural assumptions.