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if $f:X\to Y$ is a map of small v-stacks, Scholze and Fargues define relative homology as the left adjoint to the $f^{\star}$. They say the left adjoint exist because $f$ is a slice in $v$-site (they actually take a solidification of this left adjoint). I have no idea what this means and why can't we do the same in the étale site? Perhaps we can do this but the result would not be constructible?

So my question is what is a slice in a site, why does it give a left adjoint and why can't we do this in the usual setting?

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  • $\begingroup$ A slice in a site $\mathcal C$ is just the overcategory $\mathcal C_{/X}$ for some $X\in \mathcal C$. In that case, the left adjoint to pullback is given by the obvious functor $\mathcal C_{/X}\to \mathcal C$. For the small etale site, only etale schemes over the base define slices, in which case the left adjoint is just the usual $j_!$-functor. $\endgroup$
    – Peter Scholze
    Commented Nov 4, 2021 at 8:08
  • $\begingroup$ @PeterScholze okay thanks, I wrongly though this give a right adjoint. just one question,it seems the right thing would be working on big (pro-)etale site of $X$. does it give an intersting functor in that case? for example is it has the right relation with pushforward in proper smooth case? $\endgroup$
    – ali
    Commented Nov 4, 2021 at 11:33
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    $\begingroup$ I think you at least need to version of solidification because the construction does not perserve contructible sheaves, but I like to know if you can define a concept of solid sheaf over a scheme that make everything works $\endgroup$
    – ali
    Commented Nov 4, 2021 at 11:43
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    $\begingroup$ Even in the pro-etale site, slices make sense only for pro-etale morphisms. You are right that one needs to solidify in order for homology to be related to cohomology in this generality (i.e. to get the right relation with pushforward in the proper smooth case). Solidification has the curious effect that after the fact, the pro-etale site is enough as all solid v-sheaves come from the pro-etale site. For schemes, one can hope for analogues of these things, but everything will be a little different... I think the best theory arises when one also employs the v-topology for schemes. $\endgroup$
    – Peter Scholze
    Commented Nov 4, 2021 at 16:00

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