Timeline for Torsionfree crystalline cohomology implies torsionfree etale cohomology?
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| Feb 12, 2017 at 7:25 | comment | added | R. van Dobben de Bruyn | 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. | |
| Feb 12, 2017 at 7:24 | comment | added | R. van Dobben de Bruyn | 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. | |
| Feb 11, 2017 at 20:49 | comment | added | R. van Dobben de Bruyn | @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 $H^2_{\operatorname{crys}}$. | |
| Feb 11, 2017 at 14:37 | comment | added | pbelmans | Your question is closely related to the first project outline in Bhatt's lecture series for this year's Arizona Winter School, see swc.math.arizona.edu/aws/2017/2017BhattOutline.pdf. | |
| Feb 11, 2017 at 14:31 | history | edited | Monsie | CC BY-SA 3.0 |
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| Jan 29, 2017 at 14:46 | comment | added | Jim Humphreys | @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. | |
| Jan 29, 2017 at 10:08 | comment | added | Jason Starr | 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. | |
| Jan 29, 2017 at 9:37 | review | First posts | |||
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| Jan 29, 2017 at 9:37 | history | asked | Monsie | CC BY-SA 3.0 |