Let k be a field. Is there a realization functor
$DM_{gm}(k,\mathbb{Z}/n)^{op} \to D^b_c(k, \mathbb{Z}/n)$
from category of motives to category of complexes of étale sheaves of $\mathbb{Z}/n$ modules with bounded constructible cohomology sheaves?
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$\begingroup$ What do you mean by realization? With the little explanation you offer, I feel tempted to answer: the trivial functor. $\endgroup$– Fernando MuroCommented Feb 6, 2013 at 16:21
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A similar functor, but restricted to the effective part of $DM_{gm}$, is established in Voevodsky's paper on $DM_{gm}$. Actually it is an equivalence (if $n$ is prime to the characteristic of $k$). Also look at Ayoub's paper on etale realization.
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$\begingroup$ Why only effective? The target category seems to me to admit Tate twists. $\endgroup$ Commented Mar 20, 2013 at 15:25
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$\begingroup$ For this, you need to consider the étale version of effective motives with $\mathbb{Z}/n\mathbb{Z}$-coefficients ($n$ prime to $char(k)$). Then the equivalence with the derived category of Galois representations with $\mathbb{Z}/n\mathbb{Z}$-coefficients (which is reformulation of Suslin-Voevodsky's rigidity theorem) imply that the Tate twist is already invertible in the triangulated category of effective étale motives with $\mathbb{Z}/n\mathbb{Z}$-coefficients (so that the adjective 'effective' may be dropped after all). But there is no need to restrict to geometric motives though. $\endgroup$ Commented Apr 6, 2013 at 23:44