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I have been told several times, at least implicitly, that motivic cohomology should be universal with respect to Bloch-Ogus cohomology theories. Is it proved somewhere or is it just some folk theorem?

More generally, what can be said if one allows general schemes over a sufficiently good base $S$ (probably something like geometrically unibranch?) or change the topology (e.g., étale motivic cohomology)?

Still more generally, what can be said if one uses Fulton-Macpherson's bivariant theories or Mixed Weil cohomology theories?

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    $\begingroup$ For Mixed Weil cohomology theories see Theorem 1 here : math.univ-toulouse.fr/~dcisinsk/mwc.pdf $\endgroup$
    – user123627
    Nov 8, 2018 at 6:23
  • $\begingroup$ I suspect that it would be rather difficult to formulate a (correct) general universality statement of this sort. This is rather a "rule" for "natural" cohomology theories that can probably be justified under certain additional assumptions. $\endgroup$
    – Mikhail Bondarko
    Nov 8, 2018 at 6:40
  • $\begingroup$ @Gasterbiter Sorry, but maybe I'm missing something trivial. Why this implies universality? $\endgroup$
    – user40276
    Nov 8, 2018 at 19:28
  • $\begingroup$ @user40276 well what do you mean by universality? Usually people want cycle class maps from motivic cohomology to other theories. That's what part 2 of theorem 1 is. The statement's rational of course (Weil cohomologies are considered with char 0 coefficients) $\endgroup$
    – user123627
    Nov 9, 2018 at 20:15
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    $\begingroup$ Note that the cycle class map is just induced from realization functors from the category of motives to E-modules for a mixed Weil cohomology theory E, which commute with the six functors. More generally any time you have such a realization functor from DM to some category of coefficients you'll get some correpsonding cycle class maps in motivic cohomology to the corresponding cohomology theory. Such realization functors exist in abundance but I dont know if there's some sort of universal property of DM that gives them more generally than the mixed Weil case $\endgroup$
    – user123627
    Nov 9, 2018 at 20:19

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I think a proof that motivic Borel-Moore homology is a universal Borel-Moore homology theory could be deduced from work of Bloch, Geisser-Levine, and Voevodsky, at least for varieties over fields. I have never seen the details of this written down.

Bloch sketches the fact that higher Chow groups have a cycle class map to certain cohomology theories in "Algebraic cycles and the Beilinson conjectures." A paper of Geisser-Levine elaborate on this, in the case of etale cohomology. I think that these methods can be used to show that higher Chow groups are the universal Borel-Moore homology theory.

Voevodsky proves that motivic Borel-Moore homology for smooth varieties over a field are isomorphic to higher Chow groups. See "Lectures on Motivic Cohomology" by Mazza-Weibel-Voevodsky. It should be possible to verify that this isomorphism is induced by the map from higher Chow groups constructed via the method of Bloch and Geisser-Levine.

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  • $\begingroup$ Thanks for the answer. I will try to look at these references. $\endgroup$
    – user40276
    Nov 8, 2018 at 19:38

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