Betti numbers of monomial ideals via facet covers

TL;DR

This paper provides a combinatorial characterization of Betti numbers, projective dimension, and regularity of facet ideals of simplicial forests using minimal facet covers, offering both necessary and sufficient conditions.

Contribution

It introduces a combinatorial criterion based on minimal facet covers that precisely characterizes Betti numbers, projective dimension, and regularity for facet ideals of simplicial forests.

Findings

A sufficient condition for nonzero Betti numbers in multidegrees.

For simplicial forests, the condition is also necessary.

Characterization of algebraic invariants via combinatorial structures.

Abstract

We give a sufficient condition for a monomial ideal to have a nonzero Betti number in each multidegree. In the case of facet ideals of simplicial forests, this condition becomes a necessary one and it allows us to characterize Betti numbers, projective dimension and regularity of such ideals combinatorially. Our condition is expressed in terms of minimal facet covers of simplicial complexes.