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Let $X$ be a scheme of finite type over $\mathrm{Spec}(A)$, where $A$ is a commutative ring. Let $U\subset X$ be any open subset, is $\Gamma(U,\mathscr{O}_{X})$ a finitely generated $\Gamma(X,\mathscr{O}_{X})$-algebra?

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  • $\begingroup$ (off topic) I stumbled upon the "with 1". The name ring include this. See arxiv.org/abs/1404.0135. $\endgroup$ Nov 15, 2022 at 9:35
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    $\begingroup$ Welcome new contributor. This is a perennial question on MathOverflow. In addition to the great answer below by user @Gro-Tsen, you can also refer to the following MathOverflow answer: mathoverflow.net/questions/240170/… There are many other examples. $\endgroup$ Nov 15, 2022 at 12:15

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No. The following counterexamples are part of the folklore:

(To be clear, here, “variety over $k$” := “reduced scheme of finite type over $k$”.)

This other question, which also links to the same two counterexamples, is also relevant. See also the context of Hilbert's 14th problem.

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