Resolving a coherent sheaf $\mathcal F$ over $\mathbb P^n_S$ via sheafs of the form $\oplus^p\mathcal O_{\mathbb P^n_S}(n)$, where $S$ is Noeth., aff.

Question

I am studying Nitsure's exposé on the representability of the Hilbert functor. On p. 14, he makes the claim that if $S$ is an affine Noetherian scheme and $\mathcal{F}$ is a coherent sheaf over $\mathbb{P}^{n}_{S}$, then there exist integers $a,b\in\mathbb{Z}$ and natural numbers $p,q\in\mathbb{N}$ such that \begin{equation*} \oplus^{p}\mathcal{O}_{\mathbb{P}^{n}_{S}}(a)\longrightarrow\mathcal{O}_{\mathbb{P}^{n}_{S}}(b)\longrightarrow\mathcal{F}\longrightarrow 0 \end{equation*} is exact. Why is that so?