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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:

  1. Why is $F(r+p)$ generated by its global sections if $p$ is large enough?
  2. Why does the surjectivity of $\phi_{r,p}$ imply that $F(r)$ is also generated by global sections?
The Thin Whistler
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  • The solution to part 2 is handled by the linked duplicate. The solution to part 1 is also mentioned in the comments there: by a theorem of Serre (available in Hartshorne as theorem II.5.17), for any projective scheme $X$ over a noetherian ring $A$ with very ample line bundle $\mathcal{O}_X$, if $\mathcal{F}$ is a coherent sheaf on $X$, then there is some $n_0$ so that for all $n\ge n_0$, we have that $\mathcal{F}(n)$ is generated by finitely many global sections. – KReiser Jun 03 '21 at 18:33
  • @KReiser Part 2 is NOT handled by the linked duplicate. The answer there just says "..... therefore $H^{0}(\mathbb{P}^{n}{k},F(r))\otimes H^{0}(\mathbb{P}^{n}{k},\mathcal{O}{\mathbb{P}^{n}{k}}(p))\longrightarrow H^{0}(\mathbb{P}^{n}_{k},F(r+p))$ is surjective and therefore $F(r)$ is generated by global sections. It does NOT explain this implication. – The Thin Whistler Jun 03 '21 at 19:19
  • That quote does not appear anywhere in the linked duplicate (even adjusting for substituting $r$ and $p$ for $m$ and $l$) - are we sure that we're talking about the same post? I do not understand your objection - in the linked answer, I show that the cokernel of $H^0(\mathcal{F}(m))\otimes\mathcal{O}_{\Bbb P^n} \to \mathcal{F}(m)$ is zero, which is exactly the statement that $\mathcal{F}(m)$ is generated by global sections. – KReiser Jun 03 '21 at 19:32
  • @KReiser I am terribly sorry. You are absolutely right, I just got confused. – The Thin Whistler Jun 05 '21 at 16:37

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