By the seminal Popescu's theorem, $R=\mathbb{Q}[[t]]$ is a filtered colimit of smooth $\mathbb{Q}$-algebras. Could you give me a hint: which $\mathbb{Q}$-algebras can yield such a colimit? My problem is that subalgebras of $R$ seem to be 'usually' singular. 


Upd. 1. I realized that $R$ actually contains plenty of smooth $\mathbb{Q}$-subalgebras; yet I am not sure that I can prove that $R$ can be presented as their filtered colimit. Is this true; are there any nice references on these matters? 


Upd. 2. Possibly, it suffices to consider henselizations of freely generated $\mathbb{Q}$-subalgebras of $R$ here. In any case, I would like to understand presentations of regular complete local rings as 'smooth colimits' better.



Presenting $\mathbb{Q}[[t]]$ as an explicit colimit of smooth $\mathbb{Q}$-algebras: an explicit example for the Popescu's theorem
One can prove Popescu's theorem directly in this special case (due to the two strong assumptions -- characteristic $0$ and one-dimensionality -- present here). The basic point is that any 'singular' subalgebra $A \subset R$ may be resolved by a proper birational map $X \to \mathrm{Spec}(A)$ by Hironaka, and that the map $\mathrm{Spec}(R) \to \mathrm{Spec}(A)$ lifts to $X$ by the valuative criterion. 




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Write $R$ as a filtered colimit of finitely generated $\mathbb{Q}$-subalgebras $A\_i \subset R$, so $\mathrm{Spec}(R) = \lim \mathrm{Spec}(A\_i)$. Each $A\_i$ is a domain, and the map $\mathrm{Spec}(R) \to \mathrm{Spec}(A\_i)$ maps the generic point to the generic point. For each $i$, consider the cofiltered projective system $ C\_i := \{X\_{i,j} \to \mathrm{Spec}(A\_i)\}$ of all proper maps which are isomorphisms over the generic point of the target. Pullback along $\mathrm{Spec}(A\_{i'}) \to \mathrm{Spec}(A\_i)$ gives a functor $C\_i \to C\_{i'}$, so we get a two-variable cofiltered projective system $\{X\_{i,j}\}$ of schemes. Also, the map $\mathrm{Spec}(R) \to \mathrm{Spec}(A\_i)$ admits a unique lift to $\mathrm{Spec}(R) \to X\_{i,j}$ for each $j$ by the valuative criterion for properness. Let $B\_{i,j}$ be the local ring of $X\_{i,j}$ at the image of the closed point of $\mathrm{Spec}(R)$, so there is a natural map $B\_{i,j} \to R$ extending the given map $A\_i \to R$. Varying over the projective system, we get a filtered system $\{B\_{i,j}\}$ of rings with a map to $R$. One checks that $\mathrm{colim} B\_{i,j} \simeq R$; the surjectivity is completely trivial, while injectivity comes from the fact that $B\_{i,j}$'s are domains birational to $A\_i$ for a cofinal set of $j$'s. Moreover, by resolutions, for a fixed $i$, the $B\_{i,j}$'s are smooth over $\mathbb{Q}$ for a cofinal set of $j$'s. Thus, we have expressed $R$ as a filtered colimit of smooth $\mathbb{Q}$-algebras.


