Characteristic Independence of Betti Numbers of Monomial Ideals in Five Variables

Abstract

Alesandroni proved that Betti numbers of monomial ideals in at most four variables are independent of the characteristic of the base field, while characteristic-dependent Betti numbers occur in six variables. We prove that the five-variable case is characteristic-independent. More precisely, if $S=k[x_1,\ldots,x_5]$ and $M\subseteq S$ is a monomial ideal, then the multigraded, graded, and total Betti numbers of $S/M$ are independent of $\operatorname{char}(k)$. The proof reduces arbitrary monomial ideals to squarefree twin ideals and then applies Hochster's formula. The topological input is that simplicial complexes on at most five vertices have torsion-free integral homology. The six-variable example arising from the six-vertex triangulation of $\mathbb{RP}^{2}$ shows that the bound is sharp. We also record a computation of the graded Betti tables of squarefree monomial ideals in five variables up to relabeling.