Corresponding notion of unramified for motives (or de Rham cohomology) - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T01:54:58Z http://mathoverflow.net/feeds/question/118158 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/118158/corresponding-notion-of-unramified-for-motives-or-de-rham-cohomology Corresponding notion of unramified for motives (or de Rham cohomology) Masse 2013-01-05T22:03:51Z 2013-01-05T22:03:51Z <p>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 $O_K[1/S]$.</p> <p>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.</p> <p>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?</p> <p>What about the motive? </p>