Cohomological criterion for being a vector bundle 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.
 A: As Schemer said in the comments, if your sheaf $F$ is supported on a point then all intermediate cohomology vanishes, but it is not a vector bundle. However, this is practically the only problem that can occur.
Specifically, Hartshorne, in Lemma 6.3 Chapter 3 of Ample Subvarieties of Algebraic Varieties, gives the following result:

Let $E$ be a coherent sheaf on $\mathbb{P}=\mathbb{P}^n_k$. Then there exist integers $m_i$ such that $E\cong\sum_{i}\mathscr{O}_{\mathbb{P}}(m_i)$ if and only if
i) $H^{0}(\mathbb{P},E(m))=0$ for all $m\ll 0$, and
ii) $H^{i}(\mathbb{P},E(m))=0$ for all $m$ and $0<i<n$.

A related result is the following:

If $X$ is a smooth projective variety over an algebraically closed field then a coherent sheaf $E$ is locally free if and only if $H^{i}(X,E(j))=0$ for all $i=0,1,\ldots,n-1$ and $j\ll 0$.

One direction is from Serre Duality.
The other direction is a proof by contradiction. Specifically, assume that $\mathscr{E}xt^{i}_{X}(E,\mathscr{O}_{X})\neq 0$ for some positive $i$.
Then Serre Duality and Beilinson's Spectral Sequence give you a contradiction.
Abe and Yoshinaga explicitly write the proof in Lemma 3.2 of Splitting criterion for reflexive sheaves (the arXiv version - not the published version!)
