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Let $C$ be the category of quasi-projective schemes over a base field $k$ (maybe I will need some assumptions on $k$). Let $Ab(C_{\tau})$ be the category of Abelian sheaves on a site $C_{\tau}$, where $\tau$ is a Grothendieck topology on $C$. We have a canonical functor $$\mathbb{Z}_{\tau}(-):C\rightarrow Ab(C_{\tau}).$$ Let $ch\big(Ab(C_{\tau})\big)$ be the category of unbounded chain complexes on $Ab(C_{\tau})$, and let $D\big(Ab(C_{\tau})\big)$ be its derived category. Since the $qfh$-topology is finer than the Nisnevich topology, we have an adjunction $Ab(C_{Nis})\rightleftarrows Ab(C_{qfh})$.

Let $X$ be any $k$-scheme in $C$.

  • QUESTION

Is the canonical morphism $$\mathbb{Z}_{Nis}(X)\rightarrow \mathbb{Z}_{qfh}(X)$$ viewed as a chain complex concentrated in degree zero, an isomorphism in the derived category $D\big(Ab(C_{Nis})\big)\otimes\mathbb{Q}$ with rational coefficients?

Thanks in advance.

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  • $\begingroup$ No.:) I am too lazy to think of a counterexample; yet you certainly cannot obtain Voevodsky's finite correspondences sheaf represented by $X$ (that is given by $Z_{qfh}(X)$) by simple Nisnevich sheafification. $\endgroup$ Feb 4, 2015 at 20:42
  • $\begingroup$ Dear Mikhail thank you for your comment. My question arose to understand what happen when I consider that canonical morphism in the above derived category with rational coefficients, in parallel situation of the equivalence of categories of $DM^{eff}_{-}(k)\otimes\mathbb{Q}$ and $DM_h(k)\otimes\mathbb{Q}$, when $k$ admits resolution of singularities (See Voevodsky. Triangulated categories of motives over a field, Th.4.1.12), but without transfers. One can alternatively ask the same question for $\mathbb{Z}_{Nis}(X)\rightarrow\mathbb{Z}_{h}(X)$. But it is not clear to me either. $\endgroup$ Feb 4, 2015 at 23:22

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