TITLE: 'Cohomologically approximating' a $\mathbb{Q}[[t]]$-scheme by a one over the henselization of $\mathbb{Q}[t]$?
QUESTION [7 upvotes]: For certain matters the henselization $R$ of $\mathbb{Q}[t]$ at $0$ is a 'reasonable approximation' for  $\mathbb{Q}[[t]]$ (Artin's approximation theorem and so on). Now, I would like to study certain (\'etale) cohomological properties of a finite type $\mathbb{Q}[[t]]$-scheme $S$; I would like to say that there exists an $R$-scheme $S'$ that is 'very similar to $S$' ('from the cohomological point of view'). What are the possible methods for doing so? I guess that I should embedd $S$ into a 'family' and then apply either Artin's approximation (so, this a sort of deformation), or smooth base change, or both. Yet I would certainly be deeply grateful for any hints (and references); in particular, where can I find an argument for presenting $S$ as a 'member of a family'?  
I can certainly say more on these cohomological issues; yet they are rather specific.
They are somewhat related (yet more complicated) to the weight-monodromy conjecture; cf. http://www.jstor.org/stable/40067932
Upd. As we discussed in my previous question, $\mathbb{Q}[[t]]$ is the limit of its finitely generated subalgebras. Yet the corresponding etale pullback functors are not very nice, since $\mathbb{Q}[[t]]$ is not 'pro-smooth' over these subalgebras. So I suspect that 'approximating' $\mathbb{Q}[[t]]$ by $R$ is more convenient for my purposes.

REPLY [6 votes]: For the last question, the standard approach is to use the results in EGA IV.8, in particular Theorem 8.8.2: there is a finitely generated $\mathbb{Q}$-subalgebra $A\subset\mathbb{Q}[[t]]$ and a scheme of finite type $S_0$ over $U=\mathrm{Spec}(A)$ such that $S\to \mathrm{Spec}\,\mathbb{Q}[[t]]$ is induced from $U$ by the natural base change $i:\mathrm{Spec}\,\mathbb{Q}[[t]]\to U$. You may impose on $S_0$ various properties of $S$ (see e.g. EGA IV, (8.10.5)). 
Then, as you say, one may apply Artin's (in fact, Greenberg's) approximation theorem to $i$, to get an $R$-point of $U$ arbitrarily close to $i$, in the $t$-adic sense. By base change you get $S'\to \mathrm{Spec}(R)$. 
It remains to use the étale cohomology toolbox (constructibility theorem?) to see that $S$ and $S'$ have "similar" cohomologies.