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As described in the title, is any normal domain a filtered colimit of Noetherian normal domains? It will be great if one can show this, even with additional conditions, or if one can provide a counterexample.

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    $\begingroup$ Every commutative unital ring is the filtered colimit of its unital subrings that are finitely generated over $\mathbb{Z}$. By the Noether normalization theorem, the normalization of each such subring is finite over the subring, hence itself finitely generated over $\mathbb{Z}$. $\endgroup$
    – Jason Starr
    Commented Sep 4 at 22:54

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