Simplicial Resolutions of the Quadratic Power of Monomial Ideals

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