Minimal free resolutions of monomial ideals and of toric rings are supported on posets

TL;DR

This paper introduces the concept of resolutions supported on posets, extending cellular resolutions to all monomial ideals and demonstrating applications to toric rings and Alexander duality.

Contribution

It defines homology CW-posets supporting minimal free resolutions for all monomial ideals, generalizing cellular resolutions and applying to toric rings and duality theories.

Findings

Every monomial ideal has a homology CW-poset supporting its minimal resolution.

Minimal resolutions of toric rings are supported on toric hcw-posets.

Provides a new combinatorial proof relating Artinianizations and Alexander duality.

Abstract

We introduce the notion of a \emph{resolution supported on a poset}. When the poset is a CW-poset, i.e. the face poset of a regular CW-complex, we recover the notion of cellular resolution as introduced by Bayer and Sturmfels. Work of Reiner and Welker, and of Velasco, has shown that there are monomial ideals whose minimal free resolutions are not cellular, hence cannot be supported on any CW-poset. We show that for any monomial ideal there is a \emph{homology CW-poset} that supports a minimal free resolution of the ideal. This allows one to extend to every minimal resolution, essentially verbatim, techniques initially developed to study cellular resolutions. As two demonstrations of this process, we show that minimal resolutions of toric rings are supported on what we call toric hcw-posets, and we give a new combinatorial proof of a fundamental result of Miller on the relationship between Artininizations and Alexander duality of monomial ideals.