TITLE: When is the Galois representation on the étale cohomology unramified/Hodge-Tate/de Rham/crystalline/semistable?
QUESTION [13 upvotes]: Let $X/K$ be a variety over a global field $K$. When (and why) is the Galois representation $H^i_{et}(X \times_K \bar{K}, \mathbf{Q}_\ell)$ unramified at a place $v$ of $K$?
I guess this is true if $X$ has a model smooth (or regular?) over $v$ by using base change theorems.
Same question for Hodge-Tate/de Rham/crystalline/semistable.

REPLY [2 votes]: By asking "when is it ramified," you might be also be asking whether there are any other conditions under which it's ramified, i.e. a converse to Emerton's answer. For abelian varieties, the converse is true, and this is known as N\'eron-Ogg-Shafarevich. There are $p$-adic Hodge theory versions as well.
However, in general, a variety with Galois representation unramified at $v$ need not have good reduction at $v$. In all cases where this has been proven, one relates the Galois representation in question to the representations of auxiliary varieties that do have good reduction at $v$. As a simple example, there are smooth projective curves with bad reduction at $v$ but whose Jacobian has good reduction at $v$.
One might conjecture that any Galois representation coming from a variety actually comes in some fashion from varieties with good reduction over $v$. More precisely, any motive over a global field $K$ whose associated $\ell$-adic Galois representation is unramified at a finite place $v$ (whose residue characteristic $p$ is not $\ell$) is the base change of a motive (of smooth proper varieties) over $\mathcal{O}_{K,v}$, the local ring at $v$. One can make a similar conjecture for $\ell=p$ if one assumes the representation is crystalline. One can even make a modification by defining a category of motives using semistable varieties. I don't know if this is written down anywhere, but I and other graduate students I know have wondered this.