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Let $\mathfrak{X}$ be a $p$-adic flat formal scheme over $\mathbf{Z}_p$, whose special fiber has an ample line bundle. Then $\mathfrak{X}$ is algebraizable, that is there exists an algebraic $\mathbf{Z}_p$-scheme $X$ such that its $p$-adic formal completion is canonically isomorphic to $\mathfrak{X}$ as a $p$-adic formal scheme, and is unique up to isomorphism.

Grothendieck's Formal Existence Theorem implies the formal completion functor induces an equivalence of categories

$$\text{Coh}(X)\simeq\text{Coh}(\mathfrak{X}).$$

Does it induce an equivalence of étale sites $X_{\rm ét}\simeq\mathfrak{X}_{\rm ét}$, resp. finite étale sites $X_{\rm fét}\simeq\mathfrak{X}_{\rm fét}$? Do their $\ell$-adic cohomologies agree, compatibly with Tate twists?

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  • $\begingroup$ Is $\mathfrak{X}$ proper? $\endgroup$
    – Piotr Achinger
    Mar 13, 2018 at 13:33
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    $\begingroup$ Note that the etale site of $\mathfrak{X}$ is equivalent to the etale site of the special fiber $X_0$, while $X$ has the etale open obtained by inverting $p$. So the etale sites are not equivalent. However, the etale cohomology with torsion coefficients will be the same, by the proper base change theorem. $\endgroup$
    – Piotr Achinger
    Mar 13, 2018 at 13:39
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    $\begingroup$ I would not expect the topoi to be the same, however the cohomologies will agree in the proper case by Piotr's argument, and in the case where $\mathfrak{X}$ is affine and algebraically of finite type over $\operatorname{spec} \mathbb{Z}_p$ by the same argument but using Gabber's affine analog of proper base change. I think the second case would essentially only be finite extensions of $\mathbb{Z}_p$. Your question about the finite etale sites is equivalent to $G$-torsors being representable for finite groups $G$, which I think is true for both etale and adically etale topologies. $\endgroup$
    – Joe Berner
    Mar 13, 2018 at 16:50
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    $\begingroup$ Yes, and the comparison for finite etale (also for the "generic fibers" in some sense) can be seen also using Grothendieck's existence theorem: since if $f \colon Y\to X$ is finite etale, then $Y = {\rm Spec}_X\, \mathcal{A}$ where $\mathcal{A}=f_* \mathcal{O}_Y$ is a coherent $\mathcal{O}_X$-algebra. $\endgroup$
    – Piotr Achinger
    Mar 13, 2018 at 19:01
  • $\begingroup$ (See Prop. IV 2.2 in SGA4.5 Arcata. The entire section IV devoted to the (simplified w.r.t. SGA4) proof of the proper base change theorem should be a good inspiration.) $\endgroup$
    – Piotr Achinger
    Mar 13, 2018 at 19:04

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