# Betti numbers of monomial ideals in four variables

## TL;DR

This paper provides a formula for Betti numbers of monomial ideals in four variables and proves their independence from the base field using a specific class of 66 squarefree ideals.

## Contribution

It introduces a new method to compute Betti numbers in four variables and establishes their field independence, advancing understanding of monomial resolutions.

## Key findings

- Betti numbers can be expressed via 66 specific squarefree ideals.
- Monomial resolutions in four variables are field-independent.
- A general formula for Betti numbers in four variables is provided.

## Abstract

We express the multigraded Betti numbers of monomial ideals in 4 variables in terms of the multigraded Betti numbers of 66 squarefree monomial ideals, also in 4 variables. We use this class of 66 ideals to prove that monomial resolutions in 4 variables are independent of the base field. In addition, we give a formula for the Betti numbers of an arbitrary monomial ideal in 4 variables.