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Let $X\subseteq \mathbb{A}^n$ be an affine variety.

The local ring of $X$ at $p\in X$, given by $\mathcal{O}_{X,p}=\{f\in k(X):f \text{ regular at } p\}$ is noetherian because it is a localization of $k[X]$.

If $U\subseteq X$ is open, let $\mathcal{O}_X(U)=\bigcap_{p\in U}\mathcal{O}_{X,p}$. Is this ring noetherian as well?

Marco Flores
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    $O_X(U)$ is also a localization of $k[X]$. – YCor Mar 17 '16 at 18:23
  • @DenisNardin Are you saying $\mathcal{O}_X(U)={\frac{f}{g}\mid f,g\in k[X], g(p)\neq 0\forall p\in U}$?. Because this is not true. Consider $X=V(xy-zw)\subseteq\mathbb{A}^4$, and $U=U_y\cup U_w$, where $U_y={(x,y,z,w)\mid y\neq 0}$ and $U_w={(x,y,z,w)\mid w\neq 0}$. Then there is $h\in\mathcal{O}_X(U)$ such that $h=\frac{z}{y}$ in $U_y$ and $h=\frac{x}{w}$ in $U_w$, but there is no global expression for $h$. – Marco Flores Mar 17 '16 at 18:59
  • Why the downvote? – Fan Zheng Mar 18 '16 at 01:04
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    @Fan Zheng is absolutely right: this is a very interesting and highly non-trivial question. I'm upvoting it right now. – Georges Elencwajg Mar 27 '16 at 10:47
  • I don't believe that $\mathcal O_X(U) $ is noetherian. Beware that the accepted answer below is false. – Georges Elencwajg Mar 27 '16 at 12:31
  • @GeorgesElencwajg I had already asked this question at mathSE, but couldn't get an answer. I asked it then at MO, where everyone regarded it as trivial. Now that they migrated this, there are two copies of the same question, both with valuable comments. What should I do? – Marco Flores Mar 27 '16 at 15:51
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    I started a bounty for my original question: http://math.stackexchange.com/questions/1696350/structure-sheaf-consists-of-noetherian-rings I am going to delete this version of the question in a couple of days. – Marco Flores Mar 27 '16 at 16:41
  • Dear Marco, the question is definitely not trivial and unfortunately both sites are infested with incompetent, arrogant users who seem to take pleasure in closing and (in the case of MO) migrating questions to a "lesser" site. In your case one of the migrators is a highly competent algebraic geometer but I still think he was wrong to vote for migrating the question. For what it's worth, let me repeat that I am almost (but not quite!) sure that there exist cases where $\mathcal O_X(U)$ is not noetherian. – Georges Elencwajg Mar 27 '16 at 16:45
  • Dear Marco, I think the best you can do is to evoke your problem on metaMO and hope that someone will help you there. – Georges Elencwajg Mar 27 '16 at 18:39
  • Dear @GeorgesElencwajg You were right. I asked professor Ravi Vakil, he told me he gives a counterexample in section 19.11.13 of his notes: https://math216.wordpress.com/ – Marco Flores Mar 30 '16 at 01:18
  • Dear Marco, Professor Vakil's example is very interesting but his $X$ is not affine, contrary to what you require.. Could you ask him if his example can be modified (maybe by deleting one or several points in the elliptic curve $E$) so as to obtain an $X\subset \mathbb A^n$ ? – Georges Elencwajg Mar 30 '16 at 05:26
  • @GeorgesElencwajg Is it possible to migrate it back to mathoverflow? – Fan Zheng Apr 07 '16 at 05:51
  • Dear @Fan Zheng: I am all for that but I have no power to implement that. I think Marco should try to do that. – Georges Elencwajg Apr 07 '16 at 06:07

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For the sake of completeness, let me just note that this question has been re-crossposted on Mathoverflow and got an answer there.

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