Combinatorial minimal free resolutions of ideals with monomial and binomial generators

In recent years, the combinatorial properties of monomials ideals and binomial ideals 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. If Lambda is an integer lattice in the n-dimensional integers satisfying some mild conditions, S is the polynomial ring with n variables and R is the group algebra of S[Lambda], then the first result is resolutions of Lambda-invariant submodules of the Laurent polynomial ring in n variables as R-modules. A consequence will be the ability to resolve submodules of the polynomial ring with variables that a Lambda-cosets of the n-dimensional integers modulo Lambda. In particular ideals J of S modulo the lattice ideal associated to Lambda. Second, we will provide a detailed account in three dimensions on how to lift the aforementioned resolutions to resolutions of ideals with monomial and binomial generators in the 3-dimensional polynomial ring

A Combinatorial Algorithm to Find the Minimal Free Resolution of an Ideal with Binomial and Monomial Generators

In recent years, the combinatorial properties of monomials ideals and binomial ideals have been widely studied. In particular, combinatorial interpretations of free resolution algorithms have been given in both cases. In this present work, we will introduce similar techniques, or modify existing ones to obtain two new results. The first is $S[\Lambda]$-resolutions of $\Lambda$-invariant submodules of $k[\mathbb{Z}^n]$ where $\Lambda$ is a lattice in $\mathbb{Z}^n$ satisfying some trivial conditions. A consequence will be the ability to resolve submodules of $k[\mathbb{Z}^n/\Lambda]$, and in particular ideals $J$ of $S/I_{\Lambda}$, where $I_{\Lambda}$ is the lattice ideal of $\Lambda$. Second, we will provide a detailed account in three dimensions on how to lift the aforementioned resolutions to resolutions in $k[x,y,z]$ of ideals with monomial and binomial generators.Comment: Dissertatio