Rigid monomial ideals

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In this paper we investigate the class of rigid monomial ideals. We give a characterization of the minimal free resolutions of certain classes of these ideals. Specifically, we show that the ideals in a particular subclass of rigid monomial ideals are lattice-linear and thus their minimal resolution can be constructed as a poset resolution. We then use this result to give a description of the minimal free resolution of a larger class of rigid monomial ideals by using L(n), the lattice of all lcm-lattices of monomial ideals with n generators. By fixing a stratum in L(n) where all ideals have the same total Betti numbers we show that rigidity is a property which is upward closed in L(n). Furthermore, the minimal resolution of all rigid ideals contained in a fixed stratum is shown to be isomorphic to the constructed minimal resolution.