Let $\pi: X \to S:=\mathrm{Spf}\text{ } A$ be a proper morphism of $\mathbb{Z}_p$-admissible formal schemes and $\mathcal{F}$ be a coherent sheaf on $X$.

Set $S_0=\{x\}$ be a closed point of $S$ and $X_0=\pi^{-1}(x)$. Is there any analogous result to '' the Theorem on formal functions '' in this setting ? i.e., an isomorphism $R^i \pi_{*} (\mathcal{F})^{\wedge } \simeq R^i \widehat{\pi_{*}} (\widehat{\mathcal{F}})$, where

  1. $R^i \pi_{*} (\mathcal{F})^{\wedge }$ is the completion of $R^i \pi_{*} (\mathcal{F})$ at $S_0=\{x\}$.

  2. $\widehat{\pi}: \widehat{X}^{X_0} \to \widehat{S}^{S_0}$ is the completion of $\pi$.

  3. $\widehat{\mathcal{F}}$ is the complétion of $\mathcal{F}$ along $X_0$.

Thank you in advance.



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