TITLE: Coherent Sheaves on Noetherian schemes
QUESTION [7 upvotes]: Let $X$ be a Noetherian scheme (in particular, we assume that it has only finitely many irreducible components). Is it true that for any open set $U$, the ring $\Gamma(U, \mathscr{O}_X)$ is a Noetherian ring. Let $\mathscr{F}$ be a coherent sheaf on $X$. Is it true that for any open set $U$, $\Gamma(U,\mathscr{F})$ is a finitely generated $\Gamma(U,\mathscr{O}_X)$ module.

REPLY [10 votes]: There exists a noetherian scheme , which is even a variety over a field $k$, such that 
$\Gamma(X, \mathscr{O}_X)$ is not a Noetherian ring.
 It is given as  Exercise 21.9. D. in Ravi Vakil's wonderful online book.   
Ravi takes for $X$ the total space of the vector  bundle associated to a locally free sheaf  $\mathcal E$ of rank 2 on an elliptic curve $E$.
The locally free sheaf is the direct sum $\mathcal E=\mathcal P\oplus \mathcal N$ of a an invertible sheaf  of positive degree   $\mathcal P$  and of a non-torsion  invertible sheaf $\mathcal N$ of degree $0$ on $E$.     
(Full disclosure: I have just discovered that exercise and I have not yet checked  it and the  ones  leading to it in detail. But to say that I trust Ravi is a vast understatement...)