Cellular resolutions of monomial ideals and their Artinian reductions

TL;DR

This paper investigates which monomial ideals possess minimal cellular resolutions, showing that ideals with up to four generators, especially those generated by two monomials, have such resolutions and providing detailed descriptions and invariants.

Contribution

It establishes that monomial ideals with at most four generators, including all two-generator cases, have minimal cellular resolutions and characterizes their algebraic properties.

Findings

Ideals with up to four generators have minimal cellular resolutions.

Explicit CW-complex descriptions for two-generator monomial ideals.

Computed Betti numbers, Cohen-Macaulay type, and levelness conditions.

Abstract

The question we address in this paper is: which monomial ideals have minimal cellular resolutions, that is, minimal resolutions obtained from homogenizing the chain maps of CW-complexes? Velasco gave families of examples of monomial ideals that do not have minimal cellular resolutions, but those examples have large minimal generating sets. In this paper, we show that if a monomial ideal has at most four generators, then the ideal and its (monomial) Artinian reductions have minimal cellular resolutions. When the ideal is generated by two monomials, we can give a precise description of the CW-complex supporting minimal free resolution of the ideal and its Artinian reduction. Also, in this case, we compute the multigraded Betti numbers, Cohen-Macaulay type and determine when the corresponding algebra is a level algebra.