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I asked this question on math.stackexchange but nobody answers, so I try here even if I'm not sure my question is a research level one..

Let $X$ be a scheme over a number field $k$. Feel free to add any hypothesis you need or to enlarge the setting (for example if there is an answer for any site over $X$).

Which are the known relations between respectively

  1. The categories of etale smooth sheaves $\mathbb{Sh}_{\text{smooth}}(\text{Et}/X),\mathbb{Sh}_{\text{smooth}}(\text{Et}/X\times\overline{k})$ and $\mathbb{Sh}_{\text{smooth}}(\text{Et}/X\times \mathbb{C})$,

  2. The derived categories $D(\mathbb{Sh}_{\text{smooth}}(\text{Et}/X)),D(\mathbb{Sh}_{\text{smooth}}(\text{Et}/X\times\overline{k}))$ and $D(\mathbb{Sh}_{\text{smooth}}(\text{Et}/X\times \mathbb{C}))$,

  3. The etale cohomology groups associated with a complex of the previous derived categories.

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    $\begingroup$ There are pullback maps from $X$ to $X_{\overline{k}}$ and from $X_{\overline{k}}$ to $X_{\mathbb C}$, which induce pullbacks in the derived category, and the second functor induces an isomorphism on étale cohomology of sheaves with torsion coefficients. What types of relations are you looking for beyond these ones? $\endgroup$
    – Will Sawin
    Commented Oct 16, 2023 at 19:10
  • $\begingroup$ Hi @WillSawin thanks, I wasn't sure if the second pullback gives us an isomorphism, that's the main reason I asked this question. Do you have a reference on this? I wrote this question very general and vague in case there are other important relationships. In particular, I have no idea what the other six operations give us (except using the adjoint relation perhaps). $\endgroup$
    – Marsault Chabat
    Commented Oct 16, 2023 at 19:33
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    $\begingroup$ Lei Fu, étale cohomology, Corollary 7.7.3 is a reference. Pullback from a field to its separable closure is a limit of étale morphisms and $f^*$ and $f^!$ agree for étale morphisms so probably agree for pullback from a field to its algebraic closure. But pullback from $\overline{K}$ to $\mathbb C$ is a limit of smooth morphisms of increasing dimension, and the comparison between $f^*$ and $f^!$ for smooth morphisms involves a shift, so I think if $f^!$ is defined for that one it would have to be zero. $\endgroup$
    – Will Sawin
    Commented Oct 16, 2023 at 19:49
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    $\begingroup$ Pushforward tends to produce enormous sheaves. The pushforward of the constant sheaf along an extension of fields has rank equal to the degree of the extension, so infinite rank for infinite extensions. $\endgroup$
    – Will Sawin
    Commented Oct 16, 2023 at 19:52

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