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For a morphism $f$ of varieties over a field of characteristic $\neq l$ I can consider the functors $Rf_*$, $f^\ast$, $f_!$, and $f^!$ both for the corresponding derived categories of 'all' (constructible?) etale sheaves, or for the categories of $\mathbb{Z}/l\mathbb{Z}$-(constructible?) etale sheaves. There are also the 'forgetful' functors from (the derived categories of) $\mathbb{Z}/l\mathbb{Z}$-sheaves to 'all' (constructible?) sheaves; I guess that these functors possess adjoints.

My question is: do the 'image functors' $Rf_*$, $f^\ast$, $f_!$, and $f^!$ commute with the 'coefficient change' functors mentioned? What are the best references here?

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What do you call 'coefficient change' ? The forgetful functor or its adjoint ? And what is the category of 'all' etale sheaves ? Is it $\mathbb{Z}_\ell$-sheaves or torsion sheaves ? –  Alex Feb 7 '12 at 11:04
I am more interested in the forgetful functor and torsion sheaves; yet certainly all the versions of my question are closely related. So, I would be deeply grateful for an answer to any of the versions! –  Mikhail Bondarko Feb 7 '12 at 11:07

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