Lattices and Hypergraphs associated to square-free monomial ideals

TL;DR

This paper explores the relationship between LCM-lattices and hypergraphs associated with square-free monomial ideals, providing conditions to simplify hypergraphs without changing the projective dimension and offering algorithms for computation.

Contribution

It establishes a connection between LCM-lattices and hypergraphs, introduces a condition to remove higher-dimensional edges without affecting projective dimension, and develops algorithms for computation.

Findings

Connected LCM-lattice and hypergraph structures.

Provided a sufficient condition to remove hypergraph edges impacting projective dimension.

Developed algorithms for computing projective dimension of specific monomial ideals.

Abstract

Given a square-free monomial ideal $I$ in a polynomial ring $R$ over a field $\mathbb{K}$, one can associate it with its LCM-lattice and its hypergraph. In this short note, we establish the connection between the LCM-lattice and the hypergraph, and in doing so we provide a sufficient condition for removing higher dimension edges of the hypergraph without impacting the projective dimension of the square-free monomial ideal. We also offer algorithms to compute the projective dimension of a class of square-free monomial ideals built using the new result and previous results of Lin-Mantero.