Over an (algebraically closed) characteristic $p$ field, is it known that the cohomological equivalence of cycles relation (with respect to $\mathbb{Q}_l$-adic \'etale cohomology) does not depend on the choice of a prime $l$ (distinct from $p$). At least, is it known that: if the numerical equivalence of cycles relation coincides with the cohomological one for one value of $l$, this is also true for all other $l$'s?
I believe that in order to have Kunneth decompositions for cohomological (pure) motives and the Standard Lefshetz conjecture is suffices for the numerical equivalence of cycles relation to coincide with the cohomological one. Is this true, or does the Hodge Standard conjecture play some role here (or in the first question)?
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asked Jan 16, 2011 at 17:43
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$\begingroup$ More than 10 years late to the party, but as you may be aware I showed that question 1 is equivalent to independence of $\ell$ of Betti numbers; see arXiv:1808.00119. $\endgroup$– R. van Dobben de BruynCommented Apr 1, 2022 at 14:57
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$\begingroup$ No, I was not.:) Thank you! $\endgroup$– Mikhail BondarkoCommented Apr 2, 2022 at 15:02
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1 Answer
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The answer to both questions in 1 is NO. For example, Clozel has shown that for an abelian variety over the algebraic closure of a finite field, there are infinitely many l for which numerical and homological equivalence coincide, but this doesn't help with proving the statement for all l (or even the independence of homological equivalence from l, for all l).
The answer to question 2 can be found in Kleiman's articles.
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1$\begingroup$ Thanks; yet could you say more one question 2? I have some references, that contradictы my own knowledge on the subject. $\endgroup$ Commented Jan 16, 2011 at 19:14