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?](https://mathoverflow.net/questions/204745/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](https://en.wikipedia.org/wiki/Popescu%27s_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!



Open affine subscheme of a direct limit of smooth algebras
Turning the comments into an answer (CW). Write $A=\varinjlim\_{i\in I} A\_i$ and let $X=\operatorname{Spec}(A)$, $X\_i=\operatorname{Spec}(A\_i)$ and $U=\operatorname{Spec}(B)\subseteq X$. Every point $x\in U$ has an open neighborhood of the form $\operatorname{Spec}(A[f^{-1}])\subseteq U$ for some $f\in A$. Since $U$ is quasi-compact, we have $U=\bigcup\_{j=1}^n \operatorname{Spec}(A[f^{-1}\_j])$ for some $f\_1, \ldots, f\_n\in A$. In particular, we have an exact sequence
$$ 
0\to B \to \prod\_j A[f\_j^{-1}] \to \prod\_{j,k} A[(f\_j f\_{k})^{-1}]. 
$$
Changing the index set $I$, we may assume that it has a smallest element $0$ and that $f\_1, \ldots, f\_n\in A\_0$. Let $U\_i\subseteq X\_i$ denote the union of the opens $\operatorname{Spec}(A\_i[f\_j^{-1}])$ for $j=1, \ldots, n$. Writing $B\_i=\mathcal{O}(U\_i)$, we then have short exact sequences  

$$
0\to B\_i \to \prod\_j A\_i[f\_j^{-1}] \to \prod\_{j,k} A\_i[(f\_j f\_{k})^{-1}]. 
$$
Since for $f\in A\_0$, we have $A[f^{-1}]= \varinjlim\_i A\_i[f^{-1}]$, and because colimit is exact, taking the colimit of the above exact sequences and comparing with the previous one we obtain
$$
B \simeq \varinjlim B\_i.
$$
Now each $U\_i$ is smooth and the maps $U\_i\to U\_{i'}$ are etale for $i\geq i'$. So we can conclude if we show that the $U\_i$ are affine for $i\gg 0$. But this follows from [SP Tag 01Z6](https://stacks.math.columbia.edu/tag/01Z6).


