# Toward Free Resolutions Over Scrolls

## TL;DR

This paper computes Betti numbers and minimal free resolutions of the ground field over coordinate rings of rational normal scrolls, advancing understanding of their algebraic structure.

## Contribution

It provides explicit Betti numbers for the ground field over these rings and the minimal free resolution when the scroll is a 2-scroll.

## Key findings

- Betti numbers of the ground field over $R$ are explicitly computed.
- Minimal free resolution of ${k}$ over $R$ is given for the case $k=2$.
- Enhanced understanding of the algebraic properties of rational normal scrolls.

## Abstract

Let $R = k[x]/I$ where $I$ is the defining ideal of a rational normal $k$-scroll. We compute the Betti numbers of the ground field $\mathbb{k}$ as a module over $R$. For $k = 2$, we give the minimal free resolution of $\mathbb{k}$ over $R$.