When are Morse resolutions polyhedral?

TL;DR

The paper investigates when Morse resolutions of monomial ideals can be realized as polyhedral cell complexes. It applies discrete Morse theory to the Taylor complex $\mathrm{Taylor}(I)$ to obtain Morse complexes $X_M$ that still resolve $I$, proving that ideals with $\lvert \mathcal{G}\rvert \le 4$ generators admit a maximal homogeneous acyclic matching yielding a polyhedral Morse complex, while a six-generator counterexample shows that no polyhedral minimal Morse resolution need exist in general. It connects these results to the Scarf complex and homogenization, clarifying geometric obstructions to polyhedral realizations. The findings refine the understanding of when near-minimal resolutions retain polyhedral structure, with implications for explicit construction and optimization in monomial ideals.

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.

Figures (8)

• Figure 1: Figure for \ref{['example']}
• Figure 2: Homogeneous acyclic matching $M$. The vertices corresponding to the faces of $\mathop{\mathrm{Scarf}}\nolimits(I)$ are filled with black.
• Figure 3:
• Figure 4: Illustrative sub-complexes of $\mathop{\mathrm{Scarf}}\nolimits( I )$.
• Figure 5: A visualization of $\mathop{\mathrm{Scarf}}\nolimits( I )$ in $\mathbb{R}^3$. Note that three of the tetrahedra in this diagram appear twice, to avoid self-inversion. This should be thought of as a quotient relation on the space. For ease of viewing, the three pairs of $3$-cells have been highlighted in red, green, and blue, respectively. The non-repeated $3$-cells appear in dark grey, whereas the single $2$-cell which is also a facet appears in light grey.

Theorems & Definitions (14)

• Definition 2.1: convex polytopes, simplices
• Definition 2.2: polyhedral and simplicial complexes
• Definition 2.3: abstract simplicial complex
• Example 2.5
• Theorem 3.1: The Morse Complex $X_M$BW2002cellularresolution
• Lemma 3.2: Proposition 7.3, BW2002cellularresolution
• Proposition 3.3: Cells of the Morse complex
• proof
• Example 3.4
• Theorem 4.1