Combinatorial Minimal Free Resolutions of Ideals with Monomial and Binomial Generators

In recent years, the combinatorial properties of monomials ideals [7, 10, 14] and binomial ideals [9, 10] have been widely studied. In particular, combinatorial interpretations of minimal free resolutions have been given in both cases. In this present work, we will generalize existing techniques to obtain two new results. The first is S[Λ]-resolutions of Λ-invariant submodules of k[Z] where Λ is a lattice in Z satisfying some mild conditions. A consequence will be the ability to resolve submodules of k[Z/Λ], and in particular ideals J of S/IΛ, where IΛ is the lattice ideal of Λ. Second, we will provide a detailed account in three dimensions on how to lift the aforementioned resolutions to resolutions of ideals in k[x, y, z] with monomial and binomial generators.