Minimal cellular resolutions of monomial ideals with five generators and their Artinian reductions

TL;DR

This paper proves that monomial ideals with up to five generators and their Artinian reductions possess minimal cellular resolutions, extending previous results and confirming the existence of such resolutions for this class of ideals.

Contribution

It establishes the existence of minimal cellular resolutions for monomial ideals with five or fewer generators and their Artinian reductions, expanding the class of ideals known to have such resolutions.

Findings

Monomial ideals with ≤5 generators have minimal generalized Barile-Macchia resolutions.

These ideals also have minimal cellular resolutions, extending prior results.

Independent confirmation by alternative methods using pruned resolutions.

Abstract

We prove that monomial ideals with at most five generators and their Artinian reductions have minimal generalized Barile-Macchia resolutions. As a corollary, these ideals have minimal cellular resolutions, extending a result by Faridi, D.G, Ghorbanic, and Pour. This corollary is independently obtained by Montaner, Garc\'{i}a, and Mafi, using a different class of cellular resolutions called pruned resolutions.