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Let $X$ be a smooth variety of finite type over a number field $k$. Let $\overline{X} = X \times_{k} \overline{k}$, and let $\ell$ be a prime. It's well known that if $X$ is proper, then the étale cohomology groups $H^i_{et}(\overline{X}, \mathbb{Z}_{\ell})$ are unramified at any prime $\mathfrak{p} \nmid \ell$ at which $X$ has good reduction (and in fact are isomorphic as representations of $\operatorname{Gal}(\overline{K}_\mathfrak{p} / K_\mathfrak{p})$ to the étale cohomology groups of the special fibre).

Does this statement also hold if $X$ is not assumed to be proper? (I'm interested in the case of smooth affine varieties.) What about the weaker statement that $H^i_{et}(\overline{X}, \mathbb{Z}_{\ell})$ is unramified almost everywhere?

(I don't know a reference for the proof of the "well known" statement -- I couldn't find it in Milne's books or in SGA 4.5.)

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This is just in reference to your parenthetical comment. For proper smooth maps, the unramifiedness is a consequence of the general fact that local constancy for $\ell$-adic sheaves is stable under push-forward by proper smooth maps. This is Thm 20.2 in Milne's online notes. –  Keerthi Madapusi Pera Jun 18 '12 at 17:36
    
@Keerthi: Can you elaborate a little on why unramifiedness follows from this pushforward result? –  David Loeffler Jun 18 '12 at 18:42
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Thanks! This seems to use the properness in a completely essential way, though. –  David Loeffler Jun 18 '12 at 19:22
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I think the weaker statement is true, but for the main question it depends what you mean by "good reduction". Could you specify? –  Dustin Clausen Jun 18 '12 at 22:16
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Yes, you really need to say what you mean by good reduction. If you mean "the generic fiber of a smooth morphism", then every smooth variety has this property (just take a model over the good primes), so anything with ramified cohomology is a counterexample. If you mean the complement of a proper flat closed subscheme in a smooth proper scheme, then deleting a ramified closed point from the projective line should give a counterexample. If you mean the complement of a smooth proper closed subscheme in a smooth proper scheme, then it's probably true. –  JBorger Jun 19 '12 at 1:02

1 Answer 1

up vote 9 down vote accepted

I think the weaker statement should be true. Here's a sketch of an argument: by compactification theorems and resolution of singularities, there is a smooth proper scheme $Y$ over $k$ containing $X$ as an open subscheme, such that $Y \setminus X$ is a divisor $D$ with simple normal crossings. Let $D_1, \dots, D_r$ be the irreducible components of $D$. Then any $p$-fold intersection of the $D_i$'s is smooth and proper over $k$.

There should be a spectral sequence, in terms of the etale cohomology of $\overline{Y}$ and that of the intersections of the $D_i$'s, that abuts to the etale cohomology of $\overline{X}$. Thus the etale cohomology of $\overline{X}$ should be unramified at any prime of good reduction for $\overline{Y}$ and all of the intersections of the $D_i$'s. I imagine you could also use this to show that at such primes the cohomology of $\overline{X}$ was isomorphic to the cohomology of the reduction.

David

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So what's the definition of 'good reduction' used in this answer? –  JBorger Jun 19 '12 at 7:36
    
The answer only considers "good reduction" for $Y$ and the intersections of the $D_i$'s which are all proper, so we know what good reduction means there, don't we? –  David Loeffler Jun 20 '12 at 10:56
    
Oh, sorry. I missed that this was an answer to the weaker question. FYI it's also in SGA 4.5. It follows from constructibility of higher direct images. –  JBorger Jun 21 '12 at 9:44

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