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Let $H$ be a cohomology theory with respect to some Grothendieck topology (e.g. Zariski, analytic, etale, fppf, Nisnevich, etc.)

Does H satisfy smooth proper base?

If yes, does this mean that "motives" (should?) satisfy smooth proper base change?

I'm mostly interested in the case that the sheaves involved have no torsion.

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    $\begingroup$ The answer is definitely "no" for the first part of your question, and it is rather difficult to formulate a conjecture on motives that would be related to its second part. $\endgroup$
    – Mikhail Bondarko
    Commented Nov 30, 2016 at 21:57
  • $\begingroup$ @MikhailBondarko The cohomology theories I am familiar with (eg etale cohomology) satisfy smooth proper base change if one avoids certain characteristics. Does smooth proper base change fail for Nisnevich or some other Grothendieck topology? $\endgroup$
    – Ciro
    Commented Dec 8, 2016 at 18:34
  • $\begingroup$ Proper base change is only true for etale cohomology with (locally?) CONSTANT coefficients. This is a very specific property! $\endgroup$
    – Mikhail Bondarko
    Commented Dec 8, 2016 at 18:37
  • $\begingroup$ @MikhailBondarko Thank you for your comment. I see now that I was being too vague in my question (and comment). My apologies. Let me ask a slightly more precise question: Does smooth proper base change with finite abelian coefficients fail "for obvious reasons" for other cohomology theories like Nisnevich, h-topology, fppf, etc? $\endgroup$
    – Ciro
    Commented Dec 8, 2016 at 18:44
  • $\begingroup$ For h-topology you will just get the corresponding etale cohomology.:) For other topologies one has to compute... $\endgroup$
    – Mikhail Bondarko
    Commented Dec 8, 2016 at 19:05

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