When $X$ is a smooth proper variety over $\mathbb F_q$, we know by Deligne's theory of weights that the dimension of $H^i_{\operatorname{\acute et}}(\bar X, \mathbb Q_\ell)$ does not depend on $\ell$. In fact, we even get that Frobenius acts with the same characteristic polynomial.

Do we have a similar theorem when $X$ is no longer smooth and proper (e.g. $X$ smooth affine, or proper but singular)? How about if we assume resolution of singularities? (We could also assume other things like the Tate conjecture or a suitable subset of the standard conjectures, but that feels a bit like cheating.)

A related question: in the smooth proper case, do we know that there is some sort of canonical isomorphism $$H^i(\bar X,\mathbb Q_\ell) \otimes_{\mathbb Q_\ell} A \stackrel \sim \to H^i(\bar X,\mathbb Q_{\ell'}) \otimes_{\mathbb Q_{\ell'}} A$$ for some big 'period' ring $A$ containing $\mathbb Q_\ell$ and $\mathbb Q_{\ell'}$?

**Remark.** (Also remarked by Ben Webster) In characteristic $0$, the result is known by comparison to the singular cohomology of the associated analytic space. However, in positive characteristic there cannot exist a $\mathbb Q$-valued Weil cohomology theory to compare with. In fact, there does not even exist an $\mathbb R$-valued or a $\mathbb Q_p$-valued Weil cohomology theory. The argument is outlined in 2.2 of these notes, and is attributed to Serre.

An obvious workaround would be to lift everything to characteristic $0$ and use comparison there, but again no luck: Serre gave an example of a smooth projective variety that does not lift to characteristic $0$.

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