Divisibility Relations and $\mathcal{D}$-Extremal ideals
Susan M. Cooper, Sabine El Khoury, Sara Faridi, Susan Morey, Liana M. Şega, Sandra Spiroff
TL;DR
The paper introduces a formal language of divisibility relations among generators of square-free monomial ideals, encoding these relations as pairs $(b,B)$ and developing closures $\overline{\mathcal{D}}$ to capture all consequences. It then defines $\mathcal{D}$-extremal ideals $\mathcal{E}_{\mathcal{D}}$, proving they satisfy exactly the relations deduced from $\mathcal{D}$ (no extras) and that their powers provide sharp Betti-number bounds for any ideal sharing those relations. A key result is that ${\sf Div}(\mathcal{E}_{\mathcal{D}})=\overline{\mathcal{D}}$, and the minimal generating set for $\mathcal{E}_{\mathcal{D}}^r$ consists of products of $r$ generators, paving the way for extremal bounds on free resolutions via maps to general ideals. The framework generalizes extremal ideals by incorporating divisibility relations, yielding tighter bounds on Betti numbers and enabling transfer of resolution structure through the LCM lattice.
Abstract
A divisibility relation between the generators of a square-free monomial ideal formally encodes the situation when one generator divides the least common multiple of some other generators. The divisibility relations contribute to the deletion of some parts of the Taylor resolution of the ideal, and therefore lead to finding a resolution closer to the minimal one. Motivated by this observation, for a given set $\mathcal{D}$ of divisibility relations, we study all square-free monomials satisfying the relations in $\mathcal{D}$. We define a class of square-free monomial ideals called $\mathcal{D}$-extremal ideals $\mathcal{E}_\mathcal{D}$ , and show it is optimal in the sense that it is an ideal satisfying exactly those divisibility relations coming from $\mathcal{D}$, and no others. We then show that $\mathcal{E}_\mathcal{D}$ is extremal in the sense that the resolution and betti numbers of the powers of any square-free monomial ideal satisfying the relations in $\mathcal{D}$ are bounded by those of the same powers of $\mathcal{E}_\mathcal{D}$.