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Motivic complexes are a complex of Zariski sheaves that their Zariski hypercohomology gives us the motivic cohomology groups. There are various constructions of these complexes. As far as I know they do not need to satisfy étale descent. Let's sheafify it with respect to the étale opens on some variety. Then are these complexes of étale sheaves locally constant/constructible or far from it?

Another relevant question: Locally constant/constructible sheaves are closely related to étale fundamental group. So I was wondering how much control does the étale fundamental group has over étale motivic cohomology groups? Can something interesting be said about étale motivic cohomology groups of a simply connected variety?

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    $\begingroup$ The analogue of locally constant constructible étale sheaves are strongly duallizable motivic sheaves (in the sense that in the derived category of étale sheaves of modules over a neotherian commutative ring, the strongly dualizable objects precisely are the bounded complexes with perfect stalks). Conjecturally, strongly dualizable motivic sheaves form the derived category of the abelian category of representations of a pro-algebraic group, a quotient of which is the Galois group (i.e. Grothendieck's $\pi_1(X)$). But the Galois groups does not act on motivic cohomology. $\endgroup$
    – D.-C. Cisinski
    Commented Feb 22, 2021 at 20:11
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    $\begingroup$ In the motivic context, there are very few locally constant objects: the push-forward of the constant sheaf along a smooth and proper map is strongly dualizable, but, in motives, is very far from being locally constant, unless the map has relative dimenson zero. $\endgroup$
    – D.-C. Cisinski
    Commented Feb 22, 2021 at 20:14
  • $\begingroup$ Thanks for your comment. I'll try to educate myself on these material so I can understand your comment! In the case of etale motivic cohomology with finite coefficients (prime to char), the motivic cohomology groups I think are given by etale cohomology of tensor powers of $\mu_n$. The $\mu_n$ sheaves are locally constant, so if fundamental group is trivial I think this should imply that they are given by etale cohomology with constant coefficient $\mathbb{Z}/n$. $\endgroup$
    – user127776
    Commented Feb 22, 2021 at 20:46
  • $\begingroup$ You are right: with torsion coefficients and away from the residue characteristic, motives are étale sheaves, and we have thus much more locally constant objects such as $\mu_n$. $\endgroup$
    – D.-C. Cisinski
    Commented Feb 22, 2021 at 21:18
  • $\begingroup$ But in fact, there are two flavours of motivic sheaves: a Nisnevich-local one, which does not concide with étale sheaves even with torsion coefficients, but which capture intersection theory (=Chow groups) with integral coeffients, and an étale-locale version which is the one implicitely mentioned in your previous comment. Both versions coincide with rational coefficients. Their interaction with torsion coefficients is expressed in the conjectures of Bloch-Kato and from Beilinson-Lichtenbaum proved by Rost and Voevodsky. $\endgroup$
    – D.-C. Cisinski
    Commented Feb 22, 2021 at 21:22

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