# Etale cohomology of rigidification

Let $K/\mathbb Q_p$ be a discretely valued non-archemedean field, let $X$ be a smooth scheme over $\mathcal O_K$. To $X$ one can associate two rigid-analytic spaces over $K$:

1) the analytification $X_K^{\mathrm{an}}$ of the generic fiber,

2) the generic fiber $\mathfrak X^{\mathrm{rig}}$ of the corresponding formal scheme $\mathfrak X$.

One has a natural map $i_X:\mathfrak X^{\mathrm{rig}}\rightarrow X_K^{\mathrm{an}}$ which is an open immersion and an isomorphism in the case of proper $X$. For any $X$ one also has a comparison isomorphism between etale cohomology groups of $X_{\overline K}$ and $X_{K}^{\mathrm{an}}$ for torsion coefficients. So in the proper case one also has an isomorphism $H^i(({X_K})_{et},\mathbb Z/n\mathbb Z)\simeq H^i(\mathfrak X^{\mathrm{rig}}_{et},\mathbb Z/n\mathbb Z)$. Does one have this isomorphism in general or it holds only for $X$ proper? If not what could be a counterexample?

• Have you thought about even the most basic case of the affine line over $O_K$ with $i=1$ and $n=p$ (connected cyclic finite etale covers of $p$-power degree over the analytic closed unit ball and the affine line) for $K \supset \mathbf{Q}_p(\zeta_p)$? – nfdc23 Aug 26 '17 at 0:40
• Ah, I see. For rigidification I can lift the Artin-Schreier covers of the reduction. – user42024 Aug 26 '17 at 7:53
• @nfdc23 Oh. Or I could take something like $\mathbb Q_p(\mu_p)\langle x\rangle[t]/(t^p-px-1)$ which if it were $\mathbb Z_p(\mu_p)\langle x \rangle$ would give an inseperable extension of $\mathbb F_p[x]$ on the reduction modulo the maximal ideal of $\mathbb Z_p(\mu_p)$ – user42024 Aug 26 '17 at 8:29
• Consider the case where the closed fiber is empty (but $X$ isn't). – Laurent Moret-Bailly Aug 28 '17 at 7:02