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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?



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2 Answers

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No, it is not true that the higher cohomology vanishes for any sheaf generated by its global sections. Consider the structure sheaf itself: for a smooth genus $g$ curve $C$, for example, $H^1(C, \mathcal{O}_C) = g$.

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Nice answer on the second question...should have known this myself... Any ideas on the first question? –  Betti Meyer Nov 6 '12 at 12:21
QiL's answer provides the relevent reference, except for a typo: it should be Hartshorne, III, Prop. 5.3. (I can only comment here...) –  Rhys Nov 7 '12 at 9:44
Thanks @Rhys for pointing out the typo! –  user18119 Nov 7 '12 at 22:26
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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).

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