Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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!


share|improve this question
add comment

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 .

share|improve this answer
Thank you very much! –  Roy Ben-Abraham Jul 13 '12 at 11:44
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.