The Theorem of Horrocks sais that a locally free sheaf $\mathcal{F}$ on $\mathbb{P}^n$ splits into a direct sum of line bundles if and only if all the intermediate cohomologies vanish, i.e. $H^i(\mathbb{P}^n,\mathcal{F}(j))=0$ for all $0<i<n$ and all $j$.
Is this also true if $\mathcal{F}$ is not assumed to be a vector bundle but rather a coherent sheaf? So my question is: Is a coherent sheaf on $\mathbb{P}^n$ whose intermediate cohomology groups vanish necessarily a vector bundle?
If yes, I would in particular be interested in a citable reference.