# Structural decomposition of monomial resolutions

## TL;DR

This paper introduces a structural decomposition method for monomial resolutions, expressing Betti numbers in terms of simpler ideals, enabling new minimal resolutions and projective dimension computations.

## Contribution

It provides a novel decomposition approach for multigraded Betti numbers of monomial ideals, facilitating resolution construction and dimension analysis.

## Key findings

- Decomposition expresses Betti numbers via basic ideals
- Constructs minimal resolutions for certain monomial classes
- Computes projective dimensions efficiently

## Abstract

We express the multigraded Betti numbers of an arbitrary monomial ideal in terms of the multigraded Betti numbers of two basic classes of ideals. This decompo- sition has multiple applications. In some concrete cases, we use it to construct minimal resolutions of classes of monomial ideals; in other cases, we use it to compute projective dimensions. To illustrate the effectiveness of the structural decomposition, we give a new proof of a classic theorem by Charalambous.