Morse resolutions of powers of square-free monomial ideals of projective dimension one

TL;DR

This paper uses Discrete Morse theory to construct minimal free resolutions of powers of square-free monomial ideals with projective dimension one, providing explicit descriptions of the supporting CW complexes.

Contribution

It introduces a method to explicitly describe minimal free resolutions of powers of such ideals using acyclic matchings on Taylor complexes.

Findings

Constructed CW complexes support minimal free resolutions

Explicit acyclic matchings on Taylor complexes are described

Resolutions are minimal and combinatorially explicit

Abstract

Let $I$ be a square-free monomial ideal $I$ of projective dimension one. Starting with the Taylor complex on the generators of $I^r$, we use Discrete Morse theory to describe a CW complex that supports a minimal free resolution of $I^r$. To do so, we concretely describe the acyclic matching on the faces of the Taylor complex.