When are Morse resolutions polyhedral?

TL;DR

This paper investigates when Morse resolutions of monomial ideals are polyhedral, proving that ideals with up to four generators can have polyhedral Morse complexes, but larger ideals may not support such structures.

Contribution

It establishes conditions under which Morse complexes are polyhedral, showing that up to four generators allow polyhedral resolutions, while six generators can prevent this.

Findings

Morse complexes are polyhedral for monomial ideals with up to four generators.

Counterexample exists for six generators where no polyhedral Morse resolution is possible.

Some monomial ideals cannot have any polyhedral minimal free resolution.

Abstract

It is known that the chain complex of a simplex on $q$ vertices can be used to construct a free resolution of any ideal generated by $q$ monomials, and as a direct result, the Betti numbers always have binomial upper bounds, given by the number of faces of a simplex in each dimension. It is also known that for most monomials the resolution provided by the simplex is far from minimal. Discrete Morse theory provides an algorithm called \say{Morse matchings} by which faces of the simplex can be removed so that the chain complex on the remaining faces is still a free resolution of the same ideal. An immediate positive effect is an often considerable improvement on the bounds on Betti numbers. A caveat is the loss of the combinatorial structure of the simplex we started with: the output of the Morse matching process is a cell complex with no obvious structure besides an \say{address} for each cell. The main question in this paper is: which Morse matchings lead to Morse complexes that are polyhedral cell complexes? We prove that if a monomial ideal is minimally generated by up to four generators, then there is a maximal Morse matching of the simplex such that the resulting cell complex is a polyhedral cell complex. We then give an example of a monomial ideal minimally generated by six generators whose minimal free resolution is supported on a Morse complex and the Morse complex cannot be polyhedral no matter what Morse matching is chosen, and we go further to show that this ideal cannot have any polyhedral minimal free resolution.