Cellular resolutions of second powers of square-free monomial ideals with divisibility relations

TL;DR

This paper investigates the structure of the second power of square-free monomial ideals using divisibility relations and discrete Morse theory to construct minimal cellular resolutions, providing bounds on projective dimension.

Contribution

It introduces a method to describe divisibility relations in $I^2$ and constructs minimal cellular resolutions using discrete Morse theory for certain extremal ideals.

Findings

Describes divisibility relations between generators of $I^2$

Constructs minimal cellular free resolutions for specific ideals

Provides bounds on the projective dimension of $I^2$

Abstract

Using divisibility relations between the generators of a square-free monomial ideal $I$, we describe divisibility relations between the generators of the second power $I^2$. We then employ discrete Morse theory to produce a cellular free resolution of $I^2$ which is minimal for specific ideals that are extremal with respect to a given divisibility relation. In particular, we provide sharp bounds on the projective dimension of $I^2$ when the generators of $I$ satisfy at least one divisibility relation.