Rigid monomial ideals

TL;DR

This paper studies rigid monomial ideals, characterizing their minimal free resolutions, especially for certain subclasses, and demonstrates how their rigidity influences the structure and construction of these resolutions.

Contribution

It provides a characterization of minimal free resolutions for subclasses of rigid monomial ideals and links rigidity to lattice structures and Betti numbers.

Findings

Certain subclasses are lattice-linear and have poset resolutions.

Rigidity is upward closed within the lattice of lcm-lattices.

Resolutions of rigid ideals in a fixed stratum are isomorphic to constructed resolutions.

Abstract

In this paper we investigate the class of rigid monomial ideals. We give a characterization of the minimal free resolutions of certain classes of these ideals. Specifically, we show that the ideals in a particular subclass of rigid monomial ideals are lattice-linear and thus their minimal resolution can be constructed as a poset resolution. We then use this result to give a description of the minimal free resolution of a larger class of rigid monomial ideals by using $\mathcal{L}(n)$, the lattice of all lcm-lattices of monomial ideals with $n$ generators. By fixing a stratum in $\mathcal{L}(n)$ where all ideals have the same total Betti numbers we show that rigidity is a property which is upward closed in $\mathcal{L}(n)$. Furthermore, the minimal resolution of all rigid ideals contained in a fixed stratum is shown to be isomorphic to the constructed minimal resolution.