Let $X$ be a smooth projective variety over an algebraically closed field $k$. Are there examples of $X$ and $\ell\neq char(k)$ for which homological equivalence with coefficients $\mathbb Z_{\ell}$ on algebraic cycles depends on $\ell$? For example, different characteristics of Griffiths group with $\ell$.
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2$\begingroup$ The canonical divisor of an Enriques surface is $2$-torsion (at least if ${\rm char}(k)\neq 2$), so it is $\mathbf{Z}_2$-homologically nontrivial but $\mathbf{Z}_\ell$-homologically trivial for $\ell\neq 2$. With $\mathbf{Q}_\ell$ coefficients, such an example would contradict the Standard Conjecture D. $\endgroup$– Piotr AchingerCommented Apr 29, 2017 at 12:03
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$\begingroup$ This has to do with torsion in $\operatorname{Pic}/\operatorname{Pic}^0$. The torsion is finitely generated, so some primes do and some do not occur in it. $\endgroup$– R. van Dobben de BruynCommented Apr 29, 2017 at 18:11
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$\begingroup$ Also, in characteristic $0$ the result with $\mathbb Q_\ell$-coefficients is known unconditionally, by comparison with singular cohomology. An unconditional proof in characteristic $p > 0$ would be very interesting; this is not known. $\endgroup$– R. van Dobben de BruynCommented Apr 29, 2017 at 18:20
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$\begingroup$ mathoverflow.net/questions/223975/… $\endgroup$– SashaPCommented Apr 29, 2017 at 19:27
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