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The etale cohomolgoy of a variety $X$ over a number field $K$ is a Galois representation of $\mathrm{Gal}(\overline K/K)$ with some properties coming from $X$, e.g., it is unramified outside $S$ if $X$ has a smooth model over $O_K[1/S]$.

Now, one can also consider the de Rham cohomology (resp. "motive") of $X$. I'm wondering whether there is an analogous notion of "unramified" for this. I don't know what I really mean by this, though.

In some sense, etale cohomology is seeing some of the arithmetic properties of $X$ (such as its reduction behaviour of $O_K$). Does de Rham cohomology also see something similar?

What about the motive?

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    $\begingroup$ I would say that De Rham cohomology has too little structure to 'see' anything like that. On the other hand, motives have etale realizations; hence the motive of X over K would 'see' everything that can be seen via etale cohomology. $\endgroup$ – Mikhail Bondarko Jan 5 '13 at 23:01
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    $\begingroup$ If $K$ was a function field, de Rham cohomology would come equipped with a Gauss-Manin connection, whose singularities would tell us about bad reduction. I don't know if there is a good analogue for number fields... $\endgroup$ – Piotr Achinger Jan 6 '13 at 10:21
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    $\begingroup$ I just noted that you wanted some more information than just 'yes' or 'no'.:) There are now some categories of relative motives (studied by Ayoub, Cisinski, and Deglise); for any morphism of schemes $f$ you obtain the corresponding $f^*,f_*,f^!,f_!$. Then it is no problem to say what is a model of $M(X_K)$ over $O_K$; you can also express this condition in terms of certain functors from motives over $K$ to motives over points in $S$. $\endgroup$ – Mikhail Bondarko Jan 7 '13 at 22:11
  • $\begingroup$ @Mikhail. Thank you for your comment. $\endgroup$ – Masse Jan 8 '13 at 8:11
  • $\begingroup$ @Piotr. That's very useful! I didn't realize that. $\endgroup$ – Masse Jan 8 '13 at 8:11

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