Let $(X,\mathcal{O}_X)$ be a quasi-compact ringed space which is locally isomorphic to an affine variety. Then is $\Gamma(X,\mathcal{O}_X)$ Noetherian?
I've tried to prove that the result is true. Take an ascending chain $A_1\subseteq A_2\subseteq\cdots$ of ideals of $\Gamma(X,\mathcal{O}_X)$. Then let $U_1,\ldots, U_n$ be an open affine cover of $X$. We can take $n$ to be finite since $X$ is quasi-compact.
Since each $U_i$ is affine, $\Gamma(U_i,\mathcal{O}_X)$ is Noetherian, and so the chain $A_1\hspace{-0.14cm}\mid_{U_i}\subseteq A_2\hspace{-0.14cm}\mid_{U_i}\subseteq\cdots$ of ideals of $\Gamma(U_i,\mathcal{O}_X)$ must stabilise for each $i$.
Since we have a finite number of such $U_i$, there exists some $m\geq1$ such that $A_{m+j}\hspace{-0.14cm}\mid_{U_i}=A_m\mid_{U_i}$ for any $j\geq0$ and $1\leq i\leq n$.
Take any $f\in A_{m+j}$ for any $j\geq0$. Then for each $i$ there exists some $g_i\in A_m$ such that $g_i\hspace{-0.14cm}\mid_{U_i}=f\mid_{U_i}$.
However I can't then deduce that $f\in A_m$, since stitching together the $g_i\mid_{U_i}$ might not leave us in $A_m$. On the other hand, I can't come up with a counterexample.
Any help would be much appreciated.