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Let $X$ be a scheme which is smooth and quasi-projective over $\operatorname{Spec} \mathbf{Z}[1/N]$, and let $\ell$ be a prime dividing $N$ (hence invertible on $X$). Then then there is a regulator map $$ H^i_{\mathrm{mot}}(X, \mathbf{Q}(j)) \otimes \mathbf{Q}_\ell \to H^i_{\mathrm{et}}(X, \mathbf{Q}_\ell(j))$$ for any integers $i \ge 0$ and $j \in \mathbf{Z}$, where the left-hand side is motivic cohomology and the right is etale cohomology.

Is it expected that this map should always be injective? If so, is there a published reference where this conjecture is written down explicitly? I'm principally interested in the case when $j < i \le 2j$.

(EDIT. I found a version for projective varieties, avoiding the awkward boundary case $i = 2j$, in Bloch and Kato's article in the Grothendieck Festschrift. I'm still keen to know of any references treating the non-projective case.)

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  • $\begingroup$ I'm not sure if the following helps. Frédéric Déglise once explained me how to get a long exact sequence relating the motivic cohomology of $X$ with that of its generic fiber and that of its special fibers. I think there should be a corresponding long exact sequence in continuous étale cohomology. Now if you assume bijectivity of the regulator map for the generic fiber, and injectivity for the special fibers (note that I'm not so familiar with these conjectures), then standard diagram chasing gives you injectivity of the regulator map for $X$. $\endgroup$
    – François Brunault
    Commented Apr 12, 2018 at 13:47
  • $\begingroup$ I don't think bijectivity of the regulator for the generic fibre is expected to hold, is it? The etale cohomology has $H^1(k, H^{i-1}(X_{\overline{K}}, \mathbf{Q}_\ell(j)))$ as a subquotient, and if $i$ is large relative to $j$, there is lots of junk in that which is not de Rham at the primes above $\ell$. $\endgroup$
    – David Loeffler
    Commented Apr 12, 2018 at 14:45
  • $\begingroup$ You're right about the generic fiber I think. The injectivity you want could still be obtained if motivic cohomology of the special fibers happen to vanish, I still have to check for which range it conjecturally vanishes. $\endgroup$
    – François Brunault
    Commented Apr 12, 2018 at 15:37
  • $\begingroup$ Hi David, sorry for the late comment. I'm sure you know this already but this is part of Beilinson's conjectures on mixed motives, in particular that the $\ell$-adic realisation functor should be faithful and exact whenever your coefficients are a $\mathbb{Q}$-algebra. So a reference is Section 5.A of Beilinson, A., Height pairing between algebraic cycles, K-theory, arithmetic and geometry (Moscow, 1984–1986), 1–25. Lecture Notes in Math., 1289 .Springer-Verlag, Berlin, 1987. $\endgroup$
    – Oli Gregory
    Commented Oct 17, 2023 at 9:30
  • $\begingroup$ I really like the discussion in Part II Section 11 of Uwe Jannsen's Mixed motives and algebraic K-theory book, and in his Motivic sheaves and filtrations on Chow groups article in the Motives volumes. $\endgroup$
    – Oli Gregory
    Commented Oct 17, 2023 at 9:31

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