Monomial ideals with up to four generators always admit a polyhedral Morse resolution, but there is a six-generator ideal for which no Morse matching yields a polyhedral Morse complex and no minimal polyhedral resolution exists at all.
Minimal cellular resolutions of monomial ideals with five generators and their Artinian reductions
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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.
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2025 1verdicts
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When are Morse resolutions polyhedral?
Monomial ideals with up to four generators always admit a polyhedral Morse resolution, but there is a six-generator ideal for which no Morse matching yields a polyhedral Morse complex and no minimal polyhedral resolution exists at all.