Projective dimension of Hypergraphs

TL;DR

This paper investigates how the projective dimension of square-free monomial ideals associated with hypergraphs is affected by the structure of the hypergraph, especially focusing on cases where the 1-skeleton is a string or cycle.

Contribution

It provides explicit computations of the projective dimension for hypergraphs with specific 1-skeletons and analyzes the effect of adding higher dimensional edges.

Findings

Higher dimensional edges either do not affect the projective dimension or increase it by one.

Explicit formulas are derived for hypergraphs with string or cycle 1-skeletons.

The impact of higher dimensional edges on projective dimension is characterized.

Abstract

Given a square-free monomial ideal $I$, satisfying certain hypotheses, in a polynomial ring $R$ over a field $\mathbb{K}$, we compute the projective dimension of $I$. Specifically, we focus on the cases where the 1-skeleton of an associated hypergraph is either a string or a cycle. We investigate the impact on the projective dimension when higher dimensional edges are removed. We prove that the higher dimensional edge either has no effect on the projective dimension or the projective dimension only goes up by one with the extra higher dimensional edge.