# Covering a coherent sheaf on $\mathbb{P}^n$ with a sum of invertibles.

Let $F$ be a coherent sheaf on $\mathbb{P}^n$. Why do there exist integers $N$ and $p$ such that there is a surjection $$\mathcal{O}_{\mathbb{P}^n}(p)^{\oplus N}\rightarrow F\rightarrow 0\;?$$

I might be misinterpreting a book, so if the above is false, then what would a similar true statement be?

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Do you know the theorem of Serre that says for a coherent $F$, there exists some $n_0$ such that for all $n\geq n_0$ we have $F(n)$ is generated by global sections? –  Matt Nov 29 '12 at 18:44
Deleted previous comments because I was wrong. I must also admit I found a proof of my statement in the same book. I'm terribly sorry! –  Igor Nov 29 '12 at 18:55
Since I'm new here I don't know what the right way to handle this is. But if necessary this question can be marked as "closed" or something to that effect. –  Igor Nov 29 '12 at 19:00
You could write the answer yourself and then accept it... –  Matt Nov 29 '12 at 20:35
Well... Just in case this'll be helpful to someone, the above is corollary 5.4.3 on page 62 (page 36 of the pdf) in George Kempf's book: bib.tiera.ru/b/72466. –  Igor Dec 11 '12 at 19:37