Suppose $X$ is a smooth scheme, not necessarily projective, over $\mathbb{Z}[1/N]$ for some integer $N\neq 0$. I would like to understand the cohomology groups $H^i_{ét}(X_{\overline{\mathbb{F}}_p}, \mathbb{Q}_{\ell})$ in terms of the singular cohomology of the complex points $X(\mathbb{C})$, but I've run in to some technical issues I don't know how to deal with.

Here is what I know. If we assume additionally that $X$ is projective, then as I understand it there are isomorphisms

$$(*) \ \ \ H^i_{ét}(X_{\overline{\mathbb{F}}_p}, \mathbb{Q}_{\ell}) \cong H^i_{ét}(X_{\overline{\mathbb{Q}}}, \mathbb{Q}_{\ell})$$

induced by specialization provided $p$ does not divide $\ell$. Artin's comparison theorem plus smooth base change then gives isomorphisms $H^i_{ét}(X_{\overline{\mathbb{Q}}}, \mathbb{Q}_{\ell})\cong H^i_{sing.}(X(\mathbb{C}), \mathbb{Q}_{\ell})$ and we're done. The isomorphisms $(*)$ seem very reasonable to me because a smooth proper morphism is something like a fiber bundle in topology, and in that setting any two fibers have isomorphic cohomology.

I have heard that in the non-projective case the isomorphisms $(*)$ still hold provided we omit finitely many primes $p$ (where the set of omitted primes depends on $i$). However, I have not been able to prove this myself or locate a suitable reference. I think I can prove a version for $\mathbb{Z}/\ell^n\mathbb{Z}$ coefficients but I don't see how to get the result upon passage to the limit, since this might involve omitting all primes. I am mostly interested in the case of $X$ a smooth affine scheme over $\mathbb{Z}[1/N]$.

$\textbf{Question}$: If $X$ is a smooth affine scheme over $\mathbb{Z}[1/N]$, are there isomorphisms $H^i_{ét}(X_{\overline{\mathbb{F}}_p}, \mathbb{Q}_{\ell}) \cong H^i_{ét}(X_{\overline{\mathbb{Q}}}, \mathbb{Q}_{\ell})$ for all but finitely many primes $p$?

  • 4
    $\begingroup$ This is a rather direct consequence of Deligne’s generic base change theorem (in SGA 4 1/2, Finitude), and of the fact that any constructible sheaf if locally constant on a dense open subscheme. There is no need of smoothness: we may work with constructible coefficients over any scheme of finite type over $\mathbf{Z}[1/N]$ and have the same conclusion. $\endgroup$ Jul 21, 2018 at 20:03
  • $\begingroup$ Great -- thank you for the reference. My background is in topology, so part of the difficulty for me is knowing where to find these things. $\endgroup$
    – K.K.
    Jul 21, 2018 at 20:08
  • $\begingroup$ @Denis-CharlesCisinski: SGA 4 1/2 only deals with torsion sheaves, and does not address the limit issue. This is the whole point of the question. $\endgroup$ Jul 21, 2018 at 20:10
  • 6
    $\begingroup$ @R.vanDobbendeBruyn. The limit issue is only psychological: an $\ell$-adic sheaf is constructible (resp. smooth) iff its reduction modulo $\ell$ is constructible (resp. locally constant). Hence the only non trivial part of the statement is the torsion coefficient case. $\endgroup$ Jul 21, 2018 at 20:13
  • 1
    $\begingroup$ Without looking in SGA4.5, my inclination is to replace $X$ with a smooth proper simplicial scheme $Y$ with boundary $D$ (using Nagata and Hironaka). Cohomology depends only on a finite truncation of $(Y, D)$, and then you can find a model over $\mathbf{Z}[1/NM]$ for some $M$ and use purity (or log etale cohomology). $\endgroup$ Jul 22, 2018 at 12:22


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