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Let $k$ be a finite field and $l$ be different from characteristic of $k$. Is there a realization functor from the Voevodsky's category with $\mathbb Q$ coefficients to the constructible mixed étale category of sheaves over $k$ such that $M_{gm}(X)\mapsto \mathcal Rp_*p^*1_k$ for any scheme $X$ (of finite type, seperated over $k$)?

(This is true for $X$ smooth by work of Florian Ivorra -- "Realisation l-Adique des Motifs Triangules Geometriques, I." and using deJong's resolutions and Galois descent, I suspect this should be true for all $X$. However, there is a issue related to "functoriality" of the cone which seems to be an obstacle for deducing this directly in the triangulated categories and I am stuck).

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  • $\begingroup$ Can you work with the underlying model categories (where the cone IS functorial)? $\endgroup$
    – name
    Commented Jul 6, 2012 at 16:18
  • $\begingroup$ @name I don't know. I am not very comfortable with the whole jargon of model categories et al and hence was expecting a concrete answer in this context. [ The actual object I am interested in is this: $Hom_{DM_{gm}(k)}(M_{gm}(X)[r],1_k)$. For $X$ smooth, this is related to motivic cohomology/higher chow groups and is non-zero only for $r=0$ where it is $\mathbb Q$ if one is working with rational coefficients. For $X$ singular, I want to compare this with étale cohomology via realizations and conclude a similar result for some schemes which perhaps are not smooth. ] $\endgroup$
    – vava
    Commented Jul 6, 2012 at 16:33
  • $\begingroup$ You don't need the Jargon of model categories, as everything ends up being contained in DM_{gm} which is obtained in a straight-forward way from the category of chain complexes in the additive category SmCor(k). $\endgroup$
    – name
    Commented Jul 6, 2012 at 16:46
  • $\begingroup$ If you are willing to assume resolution of singularities, what you want would result from the analogue in the derived category of l-adic sheaves of the blow-up distinguished triangle mentioned in Section 2.2 of "Triangulated categories of motives over a field" (If you check Proposition 4.1.3 $X_Z \to X$ is any proper birational morphism, not necassarily a blowup). $\endgroup$
    – name
    Commented Jul 6, 2012 at 17:01
  • $\begingroup$ and I guess this follows from proper base change and the conservativity of $(i^*, j^*)$ where $(i, j)$ is a closed-open immersion complementary pair. $\endgroup$
    – name
    Commented Jul 6, 2012 at 17:05

1 Answer 1

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In principle, the answer is yes.

The question is what do you mean by $M_{gm}(X)$ if $X$ is not smooth. For smooth $X$, $M_{gm}(X)$ can be defined as $f_\sharp f^* 1$. Here $f: X \to k$ is the structural map and $1$ is the unit (=motive of $k$). This expression is isomorphic to $f_! f^! 1$ by relative purity, provided that $f$ is smooth. If you define the motive with the latter formula in general, then the question becomes whether the realization functor converts $f_!$ to $(R)f_* $ and $f^!$ to $f^* $. Such statements are proven by Ayoub in his paper on etale realizations. (I don't remember exactly, it is possible that Ayoub requires the map to be quasi-projective. Also note that his paper mostly studies a covariant realization functor, in which case $f^*$ commutes with the realization functor. The change to $f^!$ comes when you switch to the more classical language of contravariant realization functors.)

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