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

TL;DR

This paper introduces combinatorial algorithms to compute minimal free resolutions of ideals with binomial and monomial generators, extending existing methods to lattice-invariant modules and three-dimensional cases.

Contribution

It develops new combinatorial techniques for resolving lattice-invariant modules and lifts these resolutions to ideals in three variables with mixed generators.

Findings

Resolutions of $ ext{Lambda}$-invariant submodules of $k[ ext{Z}^n]$ are constructed.

Methods to resolve ideals in $k[ ext{Z}^n/ ext{Lambda}]$ are established.

Explicit procedures for lifting resolutions to three-dimensional polynomial rings are provided.

Abstract

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.