Pruned cellular free resolutions of monomial ideals

TL;DR

This paper introduces an algorithm using discrete Morse theory to prune the Taylor resolution of monomial ideals, producing more efficient cellular free resolutions, including simplicial ones, and offers new insights into splitting theory.

Contribution

The authors develop a novel pruning algorithm that refines cellular resolutions of monomial ideals, extending to simplicial resolutions and connecting with Lyubeznik and splitting resolutions.

Findings

The pruning algorithm effectively reduces the complexity of Taylor resolutions.

The method produces simplicial resolutions with slight modifications.

A new approach to the theory of splitting monomial ideals is presented.

Abstract

Using discrete Morse theory, we give an algorithm that prunes the excess of information in the Taylor resolution and constructs a new cellular free resolution for an arbitrary monomial ideal. The pruned resolution is not simplicial in general, but we can slightly modify our algorithm in order to obtain a simplicial resolution. We also show that the Lyubeznik resolution fits into our pruning strategy. We finally use our methods to give a different approach to the theory of splitting of monomial ideals.