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.


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.

  • $\begingroup$ This was my previous version.:) The problem is that the morphism $S\to S_0$ does not have to be pro-smooth; hence apriori this base change 'changes the cohomology'. Possibly, I can get away with certain 'continuity' of cohomology (actually, I am interested in perverse sheaves); yet I would like to learn about other possibilities. $\endgroup$ – Mikhail Bondarko Jan 22 '14 at 9:30
  • $\begingroup$ Anyway, thank you for the references! I will have a look at them. $\endgroup$ – Mikhail Bondarko Jan 22 '14 at 9:48

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