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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.

Dave
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  • Can you make explicit what you mean by "affine variety"? I'm guessing affine + finite type over a field or something like that, but it matters, because for example Spec of any non-noetherian ring will give a negative answer. – KReiser Jun 25 '19 at 03:59
  • Apologies, yes, I take an affine variety to be a ringed space $(V,\mathcal{O}_V)$, where $V$ is an affine algebraic set over some field $k$ and $\mathcal{O}_V$ is the structure sheaf, so $\Gamma(V,\mathcal{O}_V)$ is a reduced $k$-algebra of finite type – Dave Jun 25 '19 at 04:05
  • Note that $A_m|_{U_i}$ need not even be an ideal, since the restriction map $\Gamma(X,\mathcal{O}_X)\to\Gamma(U_i,\mathcal{O}_X)$ may not be surjective. – Eric Wofsey Jun 25 '19 at 04:21
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    Also https://math.stackexchange.com/questions/1696350/structure-sheaf-consists-of-noetherian-rings – Eric Wofsey Jun 25 '19 at 04:32
  • Thank you, I’ll mark my question as a duplicate of the first you mentioned – Dave Jun 25 '19 at 05:14

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