Large rank simple bundles of all homological dimensions

For $n\ge 3$ and $r\ge n$, 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.