Let $U$ be a normal, irreducible, quasi-affine variety over the algebraically closed field $k$ and consider the ring $A = \mathscr{O}(U)$ of regular functions on $U$. Write $Max(A)$ for the topological space of maximal ideals of $A$ (in the Zariski topology). Let $V\subset Max(A)$ be the union of all open subsets of the form $Max(A_f)$ where is $f\in A\setminus\{0\}$ is such that $A_f$ is a finitely generated $k$-algebra. Is it always the case that $V$ is quasi-affine? i.e. We know from the definition that $V$ is locally Noetherian. But is it always Noetherian?

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