Simplicial resolutions for the second power of square-free monomial ideals

TL;DR

This paper introduces a simplicial complex construction that provides explicit free resolutions for the second power of square-free monomial ideals, leading to sharp bounds on their Betti numbers.

Contribution

It defines a new simplicial complex-based method to resolve and analyze the algebraic properties of the second power of square-free monomial ideals.

Findings

Constructs a simplicial complex supporting a free resolution of I^2.

Provides sharp upper bounds on Betti numbers of I^2.

Enhances understanding of the algebraic structure of monomial ideals.

Abstract

Given a square-free monomial ideal $I$, we define a simplicial complex labeled by the generators of $I^2$ which supports a free resolution of $I^2$. As a consequence, we obtain (sharp) upper bounds on the Betti numbers of the second power of any square-free monomial ideal.