Simplicial Resolutions of the Quadratic Power of Monomial Ideals

TL;DR

This paper introduces simplicial complexes that support resolutions of the square of monomial ideals, providing tighter bounds on Betti numbers and projective dimension, and establishing connections with permutation ideals and the Scarf complex.

Contribution

It defines new simplicial complexes supporting resolutions of $I^2$, offers improved bounds on Betti numbers, and relates these to permutation ideals and the Scarf complex.

Findings

Supports resolution of $I^2$ using $ ext{M}_q^2$ and $ ext{M}^2(I)$.

Provides tighter bounds on Betti numbers and projective dimension.

Shows $ ext{M}_q^2$ is the Scarf complex of $ ext{T}_q^2$.

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 $\beta(I^2) \leq \beta({\mathcal{T}_q}^2)$. 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 ${\mathcal{T}_q}^2$.