I am trying to prove the following Weak-Lefschetz-type statement for motivic cohomology: if $Z$ is a smooth hyperplane section of a smooth projective $X$ of dimension $d$ (say, over the field of complex numbers), then for any integral $r\ge 0$ and $l>1$ the homomorphism $H^i(X,\mathbb{Z}/l\mathbb{Z}(r))\to H^i(Z,\mathbb{Z}/l\mathbb{Z}(r))$ is injective for $i=d-1$ and is bijective for $i\le d-2$. In particular, $Chow^j(X, \mathbb{Z}/l\mathbb{Z})$ should be isomorphic to $Chow^j(Z, \mathbb{Z}/l\mathbb{Z})$ if $2j\le d-2$.

Could this statement be true? Are there any counterexamples known, or is there any evidence supporting 'my' conjecture? I would be interested in any comments! Note that for motivic cohomology with rational coefficients the analogue of 'my' conjecture easily follows from the (conjectural!) existence of a 'reasonable' motivic $t$-structure (for Voevodsky's motives). Yet motivic conjectures tell almost nothing on motives and cohomology with torsion coefficients. Still, if they are true, it would be natural to conjecture that there exists a bound on the exponent of the kernel and the cokernel of the homomorphism $H^i(X,\mathbb{Z}(r))\to H^i(Z,\mathbb{Z}(r))$ for $i\le d-2$ (and this bound only depends on $d$?). Could this be true?

More generally, I would certainly like to understand which motivic conjectures become wrong for integral (and torsion coefficients), and 'how much' they are wrong.

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    $\begingroup$ is there any motivation for this conjecture? after all, the weak Lefschetz theorem is not true with integer coefficients over a general field; see Deligne's Weil II or his papers in the "Motives" volume where he comments that Lefschetz theorems are usually rational: his Lefschetz decomposition of $hX$ in the derived category of motives will not work integrally. Also, should $l$ be different from the characteristic of the base field? probably, the case of $r=1$ is easier and perhaps already known. anyway, this old paper may be relevant projecteuclid.org/euclid.dmj/1077305050 $\endgroup$ – SGP Jan 18 '12 at 4:06
  • $\begingroup$ My motivation is that I have some ideas towards the proof of the statement.:) In my question I told that the base field is complex numbers; in the general case I should certainly demand that $l$ is prime to the base field characteristic. Did Deligne really give some counterexample to Weak Lefschetz with integral coefficients? Then I should look at the paper mentioned (once more)! $\endgroup$ – Mikhail Bondarko Jan 18 '12 at 6:00
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    $\begingroup$ @ SGP & Mikhail Bondarko : What does "weak Lefschetz with integral coefficients" mean over a field of positive characteristic ? (there is no cohomology theory with integral coefficients that is available). If you mean integral $l$-adic étale cohomology (ie coefficients in the $l$-adic integers), then the weak Lefschetz theorem holds. See for instance the book by Freitag and Kiehl, Cor. 9.4, p. 106. Note that the strong Lefschetz theorem is not proved with coefficients in the $l$-adic integers (see Deligne's Weil II). $\endgroup$ – Damian Rössler Jan 18 '12 at 10:04
  • $\begingroup$ Well, I am currently studying weak Lefschetz for motivic cohomology (with $\mathbb{Z}_l$-coefficients?). At the moment, I am not really interested in Hard Lefschetz. $\endgroup$ – Mikhail Bondarko Jan 18 '12 at 16:03

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