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I am having trouble proving the following claim and would be glad if someone could help me out.

Claim: Let $\mathbb P$ denote n-dimensional projective space, and let $F$ be a coherent sheaf on $\mathbb P$. Then there exists some integer k such that $F\otimes O(k)$ is generated by a finite number of global sections.

I think I know how to begin: Let $\mathbb P^n=\bigcup U_i$ be the usual cover by open affines, and observe that $F\restriction U_i$ is finitely generated (as a coherent sheaf on an affine variety).

Now I would like to choose generators of all the $F\restriction U_i$ and tensor them with elements of $O(k)(U_i)$ (for some k) to obtain sections of $F\otimes O(k)$ which lift to global sections generating this sheaf.

The problem is that I don't understand well enough what sections of $F\restriction U_i$ look like, so I don't know what to tensor with.

Thanks in advance!

Roy

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1 Answer 1

up vote 3 down vote accepted

The key requirement is to to do things in an orderly fashion!

1) First choose for each $i$ finitely many sections $s_i^\alpha \in \Gamma(U_i,F)$ which generate every fiber $F_x$ $(x\in U_i)$. This is possible by Theorem A for affine schemes.
[You write "The problem is that I don't understand well enough what sections of $F\restriction U_i$ look like". Actually, you don't have to !]

2) Now we'll consider a variable integer $k$ and the key observation is that a global section $s\in \Gamma(X,F(k))$ is a collection of sections $s_i \in \Gamma(U_i,F)$ satisfying $s_i=\frac {x_j^k}{x_i^k}s_j $ on $U_i\cap U_j$.
The global sections $\Gamma(X,F(k))$ will generate all the fibers $F_x$ if we can find sections $s^\alpha\in \Gamma(X,F(k))$ whose restrictions to $U_i$ are $s_i^\alpha$.

3) Finally prove that given $s_i \in \Gamma(U_i,F)$ there exists $k_0$ such that for all $k\geq k_0$ there exists $s\in \Gamma(X,F(k))$ given on $U_i$ by $s_i$.
This a little technical. The main ingredient for that last result is

4) Reminder: Given an affine scheme $X$ , a coherent sheaf $F$ on $X$ and $f\in \mathcal O(X)$, then for any global section $s\in\Gamma(X, F)$, zero on $D(f)$, there exists $N$ such that $f^N\cdot s=0\in \Gamma(X,F)$.

I've learned all this in Serre's FAC. Here is an opinion on that paper .

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Thank you very much! –  Roy Ben-Abraham Jul 13 '12 at 11:44

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