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This question consists of two related sub-questions.

  1. Let $X$ be a Noetherian integral affine scheme. Under what assumptions on $X$ does every open subscheme of $X$ have a Noetherian ring of global sections?
  2. Let $k$ be a field, $X$ be an integral affine scheme of finite type over $k$. Under what assumptions on $X$ does every open subscheme of $X$ have a finitely generated $k$-algebra as the ring of global sections? I think $X$ factorial should be enough (because if there is anything of codimension$\geq 2$ in the complement, we can throw it out by Hartogs, and the complement of a pure codimension $1$ subset in a factorial affine scheme is affine).
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