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?
