Lattices and hypergraphs associated to square-free 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.

Keywords:

• Hypergraphs
• lattices
• monomial ideals
• projective dimension

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 13D02
• 05E40

Acknowledgment

The authors want to express heartfelt thanks to the reviewer for the suggestions that greatly improved this paper.