  For $n\geq 3$ and $r\geq n$, we show that there are rank-$r$ vector bundles
on $\mathbb{P}^n$ with arbitrary homological dimension. We apply the
Bernstein-Gel'fand-Gel'fand correspondence to translate the vector bundle
question into a problem on modules over the exterior algebra. Then, we use
linear algebra to construct the desired modules.
