Simplicial resolutions of the quadratic power of monomial ideals

Given any monomial ideal I minimally generated by q monomials, we define a simplicial complex Mq² that supports a resolution of I². We also define a subcomplex M²(I), which depends on the monomial generators of I and also supports the resolution of I². 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 I², which are significantly tighter than those determined by the Taylor resolution of I². Moreover, we introduce the permutation ideal Tq which is generated by q monomials. For any monomial ideal I with q generators, we establish that β(I²) ≤ β(Tq²). We show that the simplicial complex Mq² supports the minimal resolution of Tq². In fact, Mq² is the Scarf complex of Tq².