Structure sheaf consists of noetherian rings Let $X\subseteq \mathbb{A}^n$ be an affine variety. The ring $k[x_1,\ldots,x_n]$ is noetherian because of Hilbert's basis theorem. 
The coordinate ring $k[X]=k[x_1,\ldots,x_n]/I(X)$ is noetherian because ideals of $k[X]$ are of the form $J/I(X)$, where $J\supseteq I(X)$ is an ideal of $k[x_1,\ldots,x_n]$.
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 $k[X]$, and the ideals of a ring of fractions $S^{-1}A$ are of the form $S^{-1}J$, where $J$ is an ideal of $A$.
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?

There is a counterexample in section 19.11.13 of Ravi Vakil's Foundations of Algebraic Geometry https://math216.wordpress.com/
