Rigid monomial ideals

In this paper we investigate the class of rigid monomial ideals and characterize them by the fact that their minimal resolution has a unique 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 L(n), the lattice of all lcmlattices of monomial ideals on n generators. By xing a stratum in L(n) where all ideals have the same total Betti numbers, we show that rigidity is a property which propagates upward in L(n). This allows the minimal resolution of any rigid ideal contained in a xed stratum to be constructed by relabeling the resolution of a rigid monomial ideal whose resolution has been constructed by other methods.