Finite atomic lattices and resolutions of monomial ideals

TL;DR

This paper explores the relationship between finite atomic lattices and monomial ideals, providing a complete characterization of coordinatizations and showing how lattice relations can be realized through monomial ideal deformations.

Contribution

It introduces a formal framework for coordinatizing finite atomic lattices to generate monomial ideals and characterizes all such coordinatizations, advancing the understanding of their resolutions.

Findings

Complete characterization of coordinatizations of finite atomic lattices.

All relations in L(n) can be realized as deformations of monomial ideal exponents.

Cellular structures of resolutions can be extended to related monomial ideals.

Abstract

In this paper we primarily study monomial ideals and their minimal free resolutions by studying their associated LCM lattices. In particular, we formally define the notion of coordinatizing a finite atomic lattice P to produce a monomial ideal whose LCM lattice is P, and we give a complete characterization of all such coordinatizations. We prove that all relations in the lattice L(n) of all finite atomic lattices with n ordered atoms can be realized as deformations of exponents of monomial ideals. We also give structural results for L(n). Moreover, we prove that the cellular structure of a minimal free resolution of a monomial ideal M can be extended to minimal resolutions of certain monomial ideals whose LCM lattices are greater than that of M in L(n).