Finite atomic lattices and resolutions of monomial ideals

Abstract. In this paper we primarily study monomial ideals and their minimal free resolutions by studying their associated LCM lattices. In particular, we formally define the notion of coordina-tizing a finite atomic lattice P to produce a monomial ideal whose LCM lattice is P, and we give a complete characterization of all such coordinatizations. We prove that all relations in the lattice L(n) of all finite atomic lattices with n ordered atoms can be re-alized as deformations of exponents of monomial ideals. We also give structural results for L(n). Moreover, we prove that the cel-lular structure of a minimal free resolution of a monomial ideal M can be extended to minimal resolutions of certain monomial ideals whose LCM lattices are greater than that of M in L(n). 1