Simplicial Resolutions of the Quadratic Power of Monomial Ideals

TL;DR

The paper extends beyond the Taylor resolution by constructing the simplicial complex $\mathbb{M}_q^2$ (and its labeled subcomplex $\mathbb{M}^2(I)$) to support free resolutions of $I^2$ for any monomial ideal $I$ generated by $q$ monomials, providing sharper bounds on projective dimension and Betti numbers. It further introduces the permutation ideal $\mathcal{T}_q$ and proves that $\operatorname{Scarf}(\mathcal{T}_q^2)=\mathbb{M}_q^2$, establishing that $\mathbb{M}_q^2$ supports the minimal free resolution of $\mathcal{T}_q^2$. Consequently, $\beta(I^2)\le}\beta({\mathcal{T}_q}^2)$ for any $I$ with $q$ generators, yielding universal generator-based bounds that improve on Taylor-based estimates. The results emphasize that polarization may fail to preserve homological invariants for powers and highlight a structural, combinatorial route to understanding resolutions of quadratic powers of monomial ideals with broad applicability, including precise bounds and a Scarf-based minimal resolution framework.

Abstract

Given any monomial ideal $ I $ minimally generated by $ q $ monomials, we define a simplicial complex $\mathbb{M}_q^2$ that supports a resolution of $ I^2 $. We also define a subcomplex $\mathbb{M}^2(I)$, which depends on the monomial generators of $I$ and also supports the resolution of $ I^2 $. As a byproduct, we obtain bounds on the projective dimension of the second power of any monomial ideal. We also establish bounds on the Betti numbers of $ I^2 $, which are significantly tighter than those determined by the Taylor resolution of $ I^2 $. Moreover, we introduce the permutation ideal $\mathcal{T}_q$ which is generated by $q$ monomials. For any monomial ideal $I$ with $q$ generators, we establish that $β(I^2) \leq β({\mathcal{T}_q}^2)$. We show that the simplicial complex $\mathbb{M}_q^2$ supports the minimal resolution of ${\mathcal{T}_q}^2$. In fact, $\mathbb{M}_q^2$ is the Scarf complex of ${\mathcal{T}_q}^2$.

Figures (4)

• Figure 1: A $3$-dimensional simplicial complex $\Delta$.
• Figure 2: $\mathbb{M}^2_3$ (on the left) is a simplicial tree with six vertices, whereas the complex $\mathbb{L}_3^2$ (on the right) is not a simplicial tree.
• Figure 3: Simplicial complexes $\mathbb{M}_2^2$ (on the left) and $\mathbb{M}_3^2$ (on the right).
• Figure 4: For the ideals in \ref{['example31']}: $\mathbb{M}^2(I)$ (left) and $\mathbb{M}^2(J)$ (right).

Theorems & Definitions (41)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Theorem 2.4: Criterion for supporting a free resolution
• Remark 2.5
• Definition 2.6: Scarf Complex
• Lemma 2.7
• Example 3.1: An example where $\mathbb{L}_q^2$ is not large enough
• Definition 3.2
• Proposition 3.3