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. 



Coherent Sheaves on Noetherian schemes
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](http://math.stanford.edu/~vakil/216blog/FOAGjan1412public.pdf). 


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...)


