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I am interested in the map from etale motivic cohomology of a smooth and projective variety over a field $K$ to the Galois invariants of etale motivic cohomology over the algebraic closure $\bar K$:

$$H^i_{et}(X,Z(n))\to H^i_{et}(X\times_K\bar K,Z(n))^G.$$

Does anybody have an example where the cokernel is infinite? (I am mostly interested in local fields or global fields.)

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  • $\begingroup$ I have two ideas on this matter; sorry of one or both of them are stupid. So: 1) Can the question be reduced to the study of etale motivic cohomology with torsion coefficients? 2) Does the latter differ from the corresponding etale cohomology? $\endgroup$
    – Mikhail Bondarko
    Commented Jan 30, 2017 at 13:53
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    $\begingroup$ @MikhailBondarko The answer to both your questions is no (but for (1) you can reduce to torsion and rational coefficients, the latter of which are pretty much rational algebraic K-theory). The equivalence of torsion étale cohomology and torsion Lichtenbaum cohomology (away from the characteristic of the field, of course) is a theorem by Cisinski and Deglise. $\endgroup$
    – Denis Nardin
    Commented Feb 17, 2018 at 12:06
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    $\begingroup$ @DenisNardin The map is an isomorphism with rational coefficients by a transfer argument. But to know it rationally and with torsion coefficients is not enough as the cokernel with integral coefficients can non-trivially map to the kernel for torsion coefficients in the coefficient sequence. Torsion etale motivic cohomology Z/m(n) has been known to agree with etale cohomology with roots of unity coefficients (away from the characteristic) and logarithmic de Rham-Witt coefficients (at the characteristic) long before Cisinki's and Deglise's work. $\endgroup$
    – Thomas Geisser
    Commented Feb 19, 2018 at 6:07

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