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Let $X$ be a smooth projective variety over an algebraically closed field $k$ of characteristic $p>0$. Let $W=W(k)$ be the ring of Witt vectors of $k$.

Assume that the crystalline cohomology $H^2_{crys}(X/W)$ is a torsionfree $W$-module.

Question. Does it follow that $H^2_{et}(X,\mathbb Z_\ell)$ is a torsionfree $\mathbb Z_\ell$-module for all prime numbers $\ell$ invertible in $k$?

Do we "expect" the answer to this question to be positive or negative (assuming some standard conjectures)?

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    $\begingroup$ It is certainly not true that $NS(X)$ is torsionfree if it has no nontrivial $p$-torsion. For instance, Enriques surfaces in characteristic $p\neq 2$ have $2$-torsion in $NS(X)$, yet they have no $p$-torsion, cf. Cossec-Dolgachev. $\endgroup$
    – Jason Starr
    Commented Jan 29, 2017 at 10:08
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    $\begingroup$ @Monsie: Are there specific examples of varieties $X$ you have in mind? Of course, the question may be reasonable to ask in general, but it's natural to wonder what motivates it. $\endgroup$
    – Jim Humphreys
    Commented Jan 29, 2017 at 14:46
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    $\begingroup$ @pbelmans: I think the OPs question is very different, because $p \neq \ell$ (in $p$-adic Hodge theory, one studies the $p$-adic étale cohomology!). Jason's examples would give a negative answer to the question if one can relate the $p$-torsion in $NS$ to the $p$-torsion in $H^2_{\operatorname{crys}}$. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Feb 11, 2017 at 20:49
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    $\begingroup$ Ah, in Cossec–Dolgachev Prop 1.4.4, they also prove that $H^2_{\operatorname{crys}}$ is torsion free for $p \neq 2$, so @Jason's answer fully settles the question. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Feb 12, 2017 at 7:24
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    $\begingroup$ Note however that even for complex varieties, $p$-torsion and $\ell$-torsion have nothing to do with each other if $p \neq \ell$. Much more subtle are questions in integral $p$-adic Hodge theory where one has different $p$-adic theories, some of which may have torsions while others do not. This is what @pbelmans was referring to. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Feb 12, 2017 at 7:25

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