Why are the global sections of a coherent sheaf on a projective variety finite dimensional?

I have read the following:

Suppose $F$ is a vector bundle on $X$. Then $X$ is an affine scheme, then a quasi-coherent sheaf is a vector bundle iff its global sections are a projective variety $\Gamma(O_x)$-module [of finite dimension].

I have no idea how this answers my question? $\Bbb{P}^1=\text{Proj}(\Bbb{C}[x,y])$ is not an affine scheme, and so surely I can't just take my projective variety to be an affine scheme? Does the highlighted text answer my question? If not, how does one show this?

(Edit: This is precisely how it was written, although it seems like broken english perhaps)


1 Answer 1


Here is a correct version of my previous answer, which was totally wrong as @Johann noticed.

First it is enough to prove it for $X = \Bbb P^n$. Indeed, we have $H^0(X,F) = H^0(\Bbb P^n, i_*F)$ where $i : X \to \Bbb P^n$ is a closed immersion (and $i_*F$ is coherent in this case). We need to know the "Serre computation" written in FAC which computes $H^i(\Bbb P^n, \mathcal O(m))$ for any $i,n,m$. It is always finitely generated.

Now one can prove that $H^i(\Bbb P^n,F) = 0$ by descending induction on $i$. For $i = n$ it's clear because the long exact sequence $0 \to K \to \bigoplus_{i} \mathcal O(d_i) \to F \to 0$ gives a surjection $H^n(\Bbb P^n, H) \to H^n(\Bbb P^n,F)$ where $H := \bigoplus_i \mathcal O(d_i)$.

Now $H^i(\Bbb P^n,F)$ is between $H^i(\Bbb P^n, H)$ and $H^{i+1}(\Bbb P^n, K)$ in the long exact sequence and they are both finitely generated, the first one because of Serre's computation and the second one by induction hypothesis. It follows that $H^i(\Bbb P^n, F)$ is finitely generated for all $i \in \Bbb Z$, in particular for $i = 0$.

  • $\begingroup$ I don't think it is that clear that this surjection on global section exists - you get a surjection of sheaves by definition, but the first cohomology of the kernel might not vanish. Indeed, iirc the claim in the question is usually proved by descending induction for all cohomology groups. $\endgroup$ Oct 18, 2017 at 8:20
  • $\begingroup$ @Johann : of course you are right. I'll delete my answer. $\endgroup$ Oct 18, 2017 at 9:03
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    $\begingroup$ @Johann : I edited my answer and did realize that even the fact that there is a surjection $O_X^m \to F$ is wrong, for example taking $F = \mathcal O(-1)$ on $\Bbb P^1$. Everything should be correct now. $\endgroup$ Oct 18, 2017 at 22:23

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