Minimal Resolutions of Monomial Ideals

It has been an open problem since the early 1960s to construct free resolutions of monomial ideals. Beginning with the 1966 Taylor resolution, the first resolution for arbitrary monomial ideals, there have since been many constructions of free resolutions of monomial ideals, satisfying some of the following properties: canonical, minimal, universal, closed form, and combinatorial. The goal is for such a construction to satisfy all of the desired properties. The constructions given so far each satisfy some subset of these properties. This dissertation gives a full solution to the problem,satisfying all of the desired properties, over a field of characteristic 0 and most positive characteristics, with these positive characteristics depending on the ideal. The differential is a weighted sum over lattice paths in $\mathbb{Z}^n$ that come from analogues of spanning trees in simplicial complexes that are indexed by the lattice. Over a field of any characteristic, noncanonical sylvan resolutions are defined. The noncanonical resolutions are minimal, universal, closed form, and combinatorial. The differentials sum over choices of these generalized spanning trees at points along the lattice paths. Finally, a combinatorial description of the canonical three-variable case is given, and noncanonical sylvan resolutions are used to produce planar graph resolutions in the three-variable case and a minimal resolution of the Stanley–Reisner ideal of the minimal triangulation of $\mathbb{RP}^2$ in characteristic 2. Substantial portions of the results are based on joint work with John Eagon and Ezra Miller.