Is the $\ell$-adic cohomology of a non-proper variety unramified at good primes? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T18:01:32Z http://mathoverflow.net/feeds/question/99923 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/99923/is-the-ell-adic-cohomology-of-a-non-proper-variety-unramified-at-good-primes Is the $\ell$-adic cohomology of a non-proper variety unramified at good primes? David Loeffler 2012-06-18T17:18:12Z 2012-06-18T22:31:05Z <p>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 &eacute;tale cohomology groups <code>$H^i_{et}(\overline{X}, \mathbb{Z}_{\ell})$</code> are unramified at any prime <code>$\mathfrak{p} \nmid \ell$</code> at which $X$ has good reduction (and in fact are isomorphic as representations of <code>$\operatorname{Gal}(\overline{K}_\mathfrak{p} / K_\mathfrak{p})$</code> to the &eacute;tale cohomology groups of the special fibre).</p> <p>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 <code>$H^i_{et}(\overline{X}, \mathbb{Z}_{\ell})$</code> is unramified almost everywhere?</p> <p>(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.)</p> http://mathoverflow.net/questions/99923/is-the-ell-adic-cohomology-of-a-non-proper-variety-unramified-at-good-primes/99951#99951 Answer by David Helm for Is the $\ell$-adic cohomology of a non-proper variety unramified at good primes? David Helm 2012-06-18T22:31:05Z 2012-06-18T22:31:05Z <p>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 $k$.</p> <p>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.</p> <p>David</p>