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Assume that $X=Spf R$ is p-adic formal scheme over $O_{C_p}$ with generic fiber $X_{\eta}$. I want to know why the nearby cycles map $Ru^\star \mathbb{Z/p}$ is equal to $R\Gamma_{et}(spec R[1/p],\mathbb{Z}/p) $.

this fact is used in the paper "Prisms and Prismatic cohomology" as if it is trivial so I guess I'm missing something.

I know that $X$ has cohomological dimension at most 1 but this does not give the result. also I think $R\Gamma_{et}(spec R[1/p],\mathbb{Z}/p)=R\Gamma_{et}(X_{\eta},\mathbb{Z/p})$. So why the nearby cycles map equals etale cohomology in this case?

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    $\begingroup$ I think you may find the answer to this question helpful: mathoverflow.net/questions/294811/… $\endgroup$
    – Piotr Achinger
    Commented Oct 10, 2020 at 7:10
  • $\begingroup$ Thank you so this is a consequence of something like affine version of proper base change and is true for all torison sheaf $\endgroup$
    – ali
    Commented Oct 10, 2020 at 13:18

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