# Sections of coherent sheaf on a projective variety

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

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)

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 $i = 0$.
• @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. Oct 18, 2017 at 22:23