THE LCM-LATTICE in MONOMIAL RESOLUTIONS

Describing the properties of the minimal free resolution of a monomial ideal I is a difficult problem posed in the early 1960’s. The main directions of progress on this problem were: • constructing the minimal free resolutions of special monomial ideals, cf. [AHH, BPS] • constructing non-minimal free resolutions; for example, Taylor’s resolution (cf. [Ei, p. 439]) and the cellular resolutions • the Stanley-Reisner theory for computing the Betti numbers of I by simplicial complexes; it has a long tradition and has led to important results in combinatorics and commutative algebra [St]. Working with special monomial ideals or with non-minimal resolutions simplifies the problem significantly because it removes the main difficulty (finding minimal generators of the homology in general). In this paper we obtain results on the minimal free resolution of an arbitrary monomial ideal. We introduce a new approach inspired by the topological theory of subspace arrangements. Some of the best results in this theory show that the cohomology algebra of the complement of a complex subspace arrangement is independent of the geometric position of the subspaces and is determined by the structure of a certain lattice. Inspired by this, we introduce the lcm-lattice of a monomial ideal and show how its structure relates to the Betti numbers, the maps in the minimal free resolution, and the structure of the Tor-algebra for the ideal. This is outlined more precisely below: The intersection lattice L of a complex subspace arrangement plays a significant role in describing the cohomology of the complementM of the arrangement: