Rigid monomial ideals

Abstract

In this paper we investigate the class of rigid monomial ideals and characterize them by the fact that their minimal resolution has a unique $\mathbf{Z}^d$-graded basis. Furthermore, we show that certain rigid monomial ideals are lattice-linear, so their minimal resolution can be constructed as a poset resolution. We then give a description of the minimal resolution of a larger class of rigid monomial ideals by appealing to the structure of $\mathcal{L}(n)$, the lattice of all lcm-lattices of monomial ideals on $n$ generators. By fixing a stratum in $\mathcal{L}(n)$ where all ideals have the same total Betti numbers, we show that rigidity is a property which propagates upward in $\mathcal{L}(n)$. This allows the minimal resolution of any rigid ideal contained in a fixed stratum to be constructed by relabeling the resolution of a rigid monomial ideal whose resolution has been constructed by other methods.