In his essay on the proof of the representability of the $\operatorname{Quot}$ functor, Nitsure proves that if $m\in\mathbb{N}$, $k$ is a field, $F$ is coherent over $\mathbb{P}^{n}_{k}$ and $m$-regular (meaning $\forall i\geq 1:H^{i}(\mathbb{P}^{n}_{k},F(m-i))=0$), then for $r\geq m$ and any $p\in\mathbb{N}$ the map $\phi_{r,p}:H^{0}(\mathbb{P}^{n}_{k},\mathcal{F}(r))\otimes H^{0}(\mathbb{P}^{n}_{k},\mathcal{O}_{\mathbb{P}^{n}_{k}}(1))\longrightarrow H^{0}(\mathbb{P}^{n}_{k},\mathcal{F}(r+p))$ is surjective.
So far, so good.
Nitsure then proceeds to claim that for $p>>0$ the sheaf $F(r+p)$ is generated by its global sections and from this and from the surjectivity of $\phi_{r,p}$ it follows that $F(r)$ is again generated by its global sections.
QUESTIONS:
- Why is $F(r+p)$ generated by its global sections if $p$ is large enough?
- Why does the surjectivity of $\phi_{r,p}$ imply that $F(r)$ is also generated by global sections?