Let $R$ be (assumed to be commutative, Noetherian) a regular local ring. Let $A$ be a direct limit of $R$-smooth algebras, such that the transition maps are $R$-étale.

Let $U= Spec(B)$ be an affine open subscheme of $Spec(A)$.

Further, assume that A and B are Noetherian (since it might happen that A is not necessarily Noetherian as noted at Are essentially smooth schemes noetherian?).

Is it true that $B$ can be written as a direct limit of $R$-smooth algebras with transition maps $R$-étale.

By Popescu's desingularization theorem, it follows that $B$ is a direct limit of $R$-smooth algebras. But I suppose $R$-étale transition maps may not be guaranteed.

Also, can we put further restrictions on the base ring $R$, so that such a statement as above would be true?

Any comments are much appreciated!

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