Lattice complements and the subadditivity of syzygies of simplicial forests

Abstract

We prove the subadditivity property for the maximal degrees of the syzygies of facet ideals of simplicial forests. We do this by interpreting the subadditivity property for any monomial ideal as a property of homologies of its lcm lattice. For an ideal $I$ that is the facet ideal of a simplicial forest, 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 number. The subadditivity formula follows from this fact.