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Let $\mathfrak{X}_j$ be an inverse system of qcqs $p$-adic formal scheme, flat over $\mathbf{Z}_p$, with affine transition maps, and assume $\mathcal{O}_{\mathfrak{X}_j}$ is a coherent sheaf of topological rings for all $j$. Call $\mathfrak{X}$ the limit $\varprojlim_j \mathfrak{X}_j$.

Let $\text{Coh}(\mathfrak{X})$, $\text{Coh}(\mathfrak{X}_j)$, be the categories of topologically finitely presented $\mathcal{O}$-modules.

Do we have $2\mbox{-}\varinjlim_j\text{Coh}(\mathfrak{X}_j) = \text{Coh}(\mathfrak{X})$?

In other words, does a topologically finitely presented $\mathcal{O}_{\mathfrak{X}}$-module descend to some finite layer, and do morphisms too?

Of course, for algebraic schemes this is true and standard. A reference?

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    $\begingroup$ No, if "$p$-adic formal scheme" entails $p$-adic completions everywhere (to make the inverse limit categorical). Endomorphisms of the structure sheaf as a module over itself is "global sections", so $\mathfrak{X}_j = {\rm{Spf}}(O_{K_j})$ for the directed system of finite extensions $K_j$ of $\mathbf{Q}_p$ inside an algebraic closure (so $\mathfrak{X}= {\rm{Spf}}(O_{\mathbf{C}_p})$) is a counterexample for morphisms. Via the formal affine lines $\mathfrak{X}_j={\rm{Spf}}(O_{K_j}\{t\})$ it fails on objects: $O_{\mathbf{C}_p}\{t\}/(t-a)$ for $a\in \mathfrak{m}$ not algebraic over $\mathbf{Q}_p$. $\endgroup$
    – nfdc23
    Commented May 15, 2018 at 12:37
  • $\begingroup$ @nfdc23 , would you mind turning your comment into an answer? $\endgroup$
    – Anton Fetisov
    Commented May 16, 2018 at 10:43
  • $\begingroup$ @AntonFetisov: If someone else wants to do so then that is fine by me. The question as posed seemed like asking for way too much (mixing apples and oranges in terms of completeness and direct limits). $\endgroup$
    – nfdc23
    Commented May 19, 2018 at 2:49

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