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Let $K$ be a finite extension of $\mathbb{Q} _p$ with a field of integers $\mathcal{O} _K$. Let $X$ be a semistable proper scheme over $\mathcal{O} _K$, and $\mathcal{X}$ the associated p-adic formal scheme.

Question: How are related the cohomology groups $H^i(X _{et}, \mathbb{Z} / p^n \mathbb{Z})$ and $H^i(\mathcal{X} _{et}, \mathbb{Z} / p^n \mathbb{Z})$? Are they isomorphic (if for example, dimension of $X$ is 1)?

Remarks: This kind of thing goes back to Brylinski (see his appendix to the text of Carayol on representations) and then it was generalised by Berkovich "Vanishing cycles for formal schemes" (I and II). But both of them, consider sheaves $\mathbb{Z} / l^n \mathbb{Z}$ with $l \not = p$. Why?

On the other hand, Andreatta and Iovita in their paper "Comparison Isomorphisms for smooth formal schemes", after a theorem 1.5 list a result: $$H^i(X _{et}, \mathcal{L}) \otimes _{\mathbb{Q} _p} B _{cris} \simeq H^i(\mathcal{X} _{et}, \mathcal{L} \otimes _{\mathbb{Z} _p} \mathbb{B} _{cris} ^{\nabla})$$ which has a look of what I would like more or less (actually, I'd like the torsion version of it) but I could not find it explicitly in the text. I will appreciate any help on this.

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The cohomology (with constant coefficients) of $X_{et}$ and $\mathcal{X}_{et}$ coincide, and they both coincide with the cohomology of the special fibre $X_{0,et}$: by the topological invariance of the etale site for $\mathcal{X}$, and the proper base change theorem for $X$ (since $O_K$ is henselian with respect to its maximal ideal). The semistability assumption has no bearing on this discussion. – anon Jan 8 '13 at 5:28

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