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Let $K$ be a valued field of rank one and $K^+$ its valuation ring such that $K^+$ is $\varpi$-adically complete with respect to a pseudo-uniformizer $\varpi\in K^+$. Let $X$ be a smooth finite type $K^+$-scheme and $\hat{X}$ the formal completion. Now pick a closed point $x\in X$, the local ring $\mathscr{O}_{\hat{X},x}$ is coherent but not necessarily $\varpi$-adically complete. For the $\varpi$-adic completion $$ \phi\colon \quad \mathscr{O}_{\hat{X},x}\rightarrow \hat{\mathscr{O}}_{\hat{X},x}, $$ the question: is the target $\hat{\mathscr{O}}_{\hat{X},x}$ a coherent ring?

Indeed, $\phi$ is faithfully flat by [GR, Corollary 11.4.46], but coherence does not ascend along faithfully flat ring extension in general.

[GR]: Foundations for almost ring theory, O. Gabber and L. Ramero, 2018.

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