TITLE: Sheaves created by global sections and their cohomology
QUESTION [17 upvotes]: I've got a question concerning sheaves created by global sections. Serre's vanishing theorem says:
Let $X$ be a projective scheme over a noetherian ring $A$ with a very ample invertible sheaf $\mathcal{O}_X(1)$ on $X$ over $\text{Spec}\ A$. Let $\mathcal{F}$ be a coherent sheaf. Then $H^i(X,F(n))=0$ for $i\ge 1$ and sufficiently large $n$.
There is another theorem by Serre saying that:
If $X$ is a projective scheme over a noetherian ring $A$, let $\mathcal{O}_X (1)$ denote a very ample invertible sheaf on $X$. Then there is a $d_0$ for each coherent sheaf $\mathcal{F}$ on $X$ so that $F(d)$ is generated by
its global sections, whenever $d ≥ d_0$ .
Are these theorems related in the way that one of them follows from the other one? Especially: Is it true that for every sheaf created by global sections  the higher cohomology vanishes?
Thanks
Betti

REPLY [7 votes]: The first result implies the second one. You can find a proof in Hartshorne, III, Prop. 5.3 ((ii) implies (i)).  Also the second result characterizes the ample sheaves (see the same reference).