Mumford's regularity theorem

Question

I am reading the section on Castelnuovo-Mumford regularity from Lazarsfeld's Positivity in algebraic geometry.

Theorem 1.8.3 in the book reads as follows:

Let $F$ be an $m$-regular sheaf on $P^n$. Then for every $k \geq 0$
(i). $F(m+k)$ is generated by its global sections.
(ii). The natural maps $$H^0(P^n, F(m))\otimes H^0(P^n,O_{P^n}(k))\rightarrow H^0(P^n,F(m+k))$$ are surjective.
(iii). $F$ is $(m + k)$-regular.

The proof claims (i) is a consequence of (ii): it says that, since for $l\gg 0$, $F(m+l)$ is globally generated, the surjectivities in (ii) imply that $F(m)$ itself must be globally generated. I am not able to understand this statement.

Choose $l\gg 0$ such that $F(m+l)$ itself is globally generated. Now consider the morphism $$H^0(F(m))\otimes O_{P^n}\rightarrow F(m)\,.$$ We need to prove that this is surjective.

Tensoring by $O(l)$, we get $$H^0(F(m))\otimes O_{P^n}(l)\rightarrow F(m+l)\,.$$

A priori we do not know the injectiveness or surjectiveness of the above morphism. But if we take global sections, by (ii), the global sections morphism is surjective.

Does this mean $H^0(F(m))\otimes O_{P^n}(l)\rightarrow F(m+l)$ is surjective? If so, I can tensor back by $O(-l)$ and get my required result.

Thanks in advance.

Answer 1

Lemma: If $\mathcal{F}_1\to\cdots\to\mathcal{F}_n$ is an exact sequence of coherent sheaves on $\Bbb P^n$, then there exists an $l_0>0$ so that for all $l\geq l_0$, the sequence of global sections $\mathcal{F}_1(l)(\Bbb P^n)\to\cdots\to\mathcal{F}_n(l)(\Bbb P^n)$ is exact.

Proof: Saying the initial sequence is exact means we have short exact sequences $0\to\mathcal{G}_i\to\mathcal{F}_i\to\mathcal{G}_{i+1}\to 0$ for all $i$, where $\mathcal{G}_i=\ker(\mathcal{F}_i\to\mathcal{F}_{i+1})=\operatorname{im}(\mathcal{F}_{i-1}\to\mathcal{F}_i)$ are coherent. By a result of Serre, there exists an $l_i>0$ so that for all $l\geq l_i$ we have $H^1(\mathcal{G}_i(l))=0$, or that the sequence of global sections $0\to\mathcal{G}_i(l)(\Bbb P^n)\to\mathcal{F}_i(l)(\Bbb P^n)\to\mathcal{G}_{i+1}(l)(\Bbb P^n)\to 0$ is exact. Letting $l_0=\max(l_i)$ and stitching together the exact sequences on global sections, we see the result. $\blacksquare$

Now let $\mathcal{K}$ and $\mathcal{R}$ be the kernel and cokernel of $H^0(\mathcal{F}(m))\otimes\mathcal{O}_{\Bbb P^n} \to \mathcal{F}(m)$, respectively. Consider the exact sequence $$0\to \mathcal{K}(l)\to H^0(\mathcal{F}(m))\otimes\mathcal{O}_{\Bbb P^n}(l) \to \mathcal{F}(m+l)\to \mathcal{R}(l)\to 0.$$ As $\mathcal{K}$ and $\mathcal{R}$ are coherent, there exists an $l_0$ so that for all $l\geq l_0$ we have that the sequence of global sections is exact and the middle map is surjective. Therefore $\mathcal{R}(l)$ has vanishing global sections for all large enough $l$, which means $\mathcal{R}=0$. Therefore the map $H^0(\mathcal{F}(m))\otimes\mathcal{O}_{\Bbb P^n} \to \mathcal{F}(m)$ is surjective and $\mathcal{F}(m)$ is globally generated.