Lattice Complements and the Subadditivity of Syzygies of Simplicial Forests

TL;DR

This paper proves the subadditivity of syzygy degrees for facet ideals of simplicial forests by analyzing the lattice structure of their Betti numbers, revealing new combinatorial properties.

Contribution

It introduces a novel lattice-theoretic approach to establish subadditivity of syzygies in simplicial forest ideals, extending understanding of their algebraic structure.

Findings

Subadditivity holds for maximal degrees of syzygies in simplicial forest ideals.

Existence of complementary monomials in the lattice supporting nonvanishing Betti numbers.

Lattice structure explains the subadditivity property.

Abstract

We prove the subadditivity property for the maximal degrees of the syzygies of facet ideals simplicial forests. For such an ideal $I$, if the $i$-th Betti number is nonzero and $i=a+b$, we show that there are monomials in the lcm lattice of $I$ that are complements in part of the lattice, each supporting a nonvanishing $a$-th and $b$-th Betti numbers. The subadditivity formula follows from this observation.