Counterexamples for Cohen-Macaulayness of Lattice Ideals

TL;DR

This paper constructs infinitely many examples of lattice ideals and their associated toric ideals in various codimensions, demonstrating cases where one is Cohen-Macaulay and the other is not, highlighting nuanced differences in their algebraic properties.

Contribution

It introduces a method to generate infinite pairs of lattice and toric ideals with contrasting Cohen-Macaulay properties across all codimensions.

Findings

Infinite examples of lattice and toric ideal pairs with differing Cohen-Macaulayness

Demonstrates the nuanced relationship between lattice ideals and their saturation

Highlights the complexity of Cohen-Macaulay property in lattice-based ideals

Abstract

Let $\mathscr{L}\subset \mathbb{Z}^n$ be a lattice, $I$ its corresponding lattice ideal, and $J$ the toric ideal arising from the saturation of $\mathscr{L}$. We produce infinitely many examples, in every codimension, of pairs $I,J$ where one of these ideals is Cohen--Macaulay but the other is not.