Constructing monomial ideals with a given minimal resolution

TL;DR

This paper unifies recent methods for constructing monomial ideals with specified minimal resolutions through lattice coordinatization, and explores their applications and equivalences in special cases like trees.

Contribution

It demonstrates that various recent constructions are instances of lattice coordinatization and applies this perspective to unresolved questions in the field.

Findings

All recent constructions are instances of lattice coordinatization.

Application of lattice coordinatization to unresolved cases by Faridi and Fløystad.

Equivalence between maximality notions in trees and Betti strata.

Abstract

This paper gives a description of various recent results which construct monomial ideals with a given minimal free resolution. We show that these are all instances of coordinatizing a finite atomic lattice as defined by Mapes. Subsequently, we explain how in some of these cases (in one case work by Faridi, and in another work by Fl\oystad), where questions still remain, this point of view can be applied. We also prove an equivalence in the case of trees between the notion of maximal defined by Fl\oystad and a notion of being maximal in a Betti stratum.