# nearby cycles map for affine formal schemes

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?

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