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I'm wondering how one could prove or disprove that any non-Noetherian UFD is a filtered colimit of Noetherian UFDs. This would allow for some absolute Noetherian approximation to be applied for results in algebraic geometry and commutative algebra about possibly non-Noetherian UFDs.

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    $\begingroup$ My first thought for a potential counterexample is $k[[x_1, x_2, \ldots]]$ but I'm not sure how you'd go about showing it's not a filtered colimit. We'd want some kind of finitary invariant of UFDs with nice properties for noetherian ones, I guess? $\endgroup$
    – Brendan Murphy
    Commented Aug 20 at 3:00

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